{ "schema_version": "2.0.0", "corpus_id": "wct", "name": "Wave Confinement Theory", "description": "Machine-readable, provenance-preserving cross-repository manifest for the Wave Confinement Theory research corpus.", "author": { "name": "Richard J. Reyes", "alternate_name": "Ricky Reyes", "orcid": "0009-0005-5975-8718", "homepage": "https://rickyjreyes.github.io/" }, "canonical_root": "https://rickyjreyes.github.io/research-corpus.json", "compiled_registry": "https://rickyjreyes.github.io/compiled-registry.json", "generated_at": "2026-06-29T23:28:10+00:00", "status_policy": { "PASS": "The assigned executable check succeeds under its declared assumptions.", "CONDITIONAL": "Additional mathematical, domain, regularity, model, or empirical assumptions remain required.", "DEFINITION": "The object is a definition, ansatz, or bookkeeping object rather than a theorem.", "OPEN": "Analysis, formal proof, calibrated simulation, or experiment remains unresolved.", "FAIL": "The encoded statement is contradicted by its assigned checker.", "PROVED": "A named declaration is accepted by the Lean kernel under the hypotheses in its theorem statement." }, "repositories": { "canonical_equations": { "name": "geometry_of_resonance", "url": "https://github.com/rickyjreyes/geometry_of_resonance", "master_equations": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_MASTER_EQUATIONS_UPDATED.md", "full_equation_registry": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md" }, "symbolic_audit": { "name": "wct-sympy", "url": "https://github.com/rickyjreyes/wct-sympy", "registry": "https://github.com/rickyjreyes/wct-sympy/blob/main/equations/full_registry.yaml", "derived_overrides": "https://github.com/rickyjreyes/wct-sympy/blob/main/equations/derived_overrides.yaml", "compiled_registry": "https://github.com/rickyjreyes/wct-sympy/blob/main/compiled-registry.json", "lean_map": "https://github.com/rickyjreyes/wct-sympy/blob/main/interoperability/lean_map.yaml" }, "formalization": { "name": "wct-lean", "url": "https://github.com/rickyjreyes/wct-lean", "theorem_inventory": "https://github.com/rickyjreyes/wct-lean/blob/main/THEOREMS.md", "compiled_root": "https://github.com/rickyjreyes/wct-lean/blob/main/WCTLean/Main.lean" }, "semantic_graph": { "name": "obsidian", "url": "https://github.com/rickyjreyes/obsidian", "entry_point": "https://github.com/rickyjreyes/obsidian/blob/main/Research/00%20Maps/WCT%20Research%20Command%20Center.md" } }, "audit_summary": { "registry_size": 142, "source_of_truth": "compiled-registry.json generated from the baseline registry, effective overrides, verification metadata, assumption registry, canonical equations, and Lean map", "counts": { "PASS": 59, "CONDITIONAL": 27, "DEFINITION": 26, "OPEN": 30, "FAIL": 0 }, "status_conflict_warning": null }, "assumptions": [ { "assumption_id": "ASM-COMPATIBLE-ORIENTATION", "name": "Compatible loop orientation", "kind": "mathematical", "statement": "The winding orientation and the curvature orientation use the same sign convention.", "introduced_by": "E5" }, { "assumption_id": "ASM-EXACT-INTEGRATED-LOCKING", "name": "Exact integrated phase-curvature locking", "kind": "modeling", "statement": "The integrated phase winding equals the integrated curvature contribution without a residual mismatch term.", "introduced_by": "E5" }, { "assumption_id": "ASM-CONSTANT-POSITIVE-WEIGHT", "name": "Constant positive loop weight", "kind": "mathematical", "statement": "The loop weight is positive and constant, so weighted and unweighted loop averages coincide.", "introduced_by": "E5" }, { "assumption_id": "ASM-POSITIVE-SPECTRAL-OFFSET", "name": "Positive spectral offset", "kind": "mathematical", "statement": "The Green-kernel denominator contains a strictly positive offset, preventing a pole on the declared domain.", "introduced_by": "E58" }, { "assumption_id": "ASM-STANDARD-SOBOLEV-DOMAIN", "name": "Standard Sobolev domain hypotheses", "kind": "mathematical", "statement": "The spatial domain satisfies the regularity and extension hypotheses required by the stated Sobolev embedding.", "introduced_by": "M4" }, { "assumption_id": "ASM-H2-CONFINEMENT-ROUTE", "name": "H2 confinement route", "kind": "modeling", "statement": "The declared confinement and stability argument requires control through the H2-to-L-infinity regularity route.", "introduced_by": "E70" } ], "claims": [ { "claim_id": "CLM-CURVATURE-MASS", "title": "Curvature-derived mass", "status": "CONDITIONAL", "equations": [ "E1B", "E2", "E5", "E6", "E7", "E19", "E49" ], "assumptions": [ "ASM-COMPATIBLE-ORIENTATION", "ASM-EXACT-INTEGRATED-LOCKING", "ASM-CONSTANT-POSITIVE-WEIGHT" ], "papers": [ { "publication_id": "geometry-of-resonance", "title": "The Geometry of Resonance: Wave Confinement Theory and the Emergence of Mass, Force, and Spacetime", "local_url": "https://rickyjreyes.github.io/publications/geometry-of-resonance.html", "doi": "10.5281/zenodo.15644222", "relationship": "introduces" }, { "publication_id": "effective-mass-solenoidal-topology", "title": "Emergence of Effective Mass: Solenoidal Topology of Vibrational Energy", "local_url": "https://rickyjreyes.github.io/publications/effective-mass-solenoidal-topology.html", "doi": "10.5281/zenodo.17459463", "relationship": "derives" }, { "publication_id": "rest-energy-density-weighted-loop-curvature", "title": "Rest Energy from Density-Weighted Loop Curvature: A Covariant Locking Principle", "local_url": "https://rickyjreyes.github.io/publications/rest-energy-density-weighted-loop-curvature.html", "doi": "10.5281/zenodo.20533537", "relationship": "qualifies" }, { "publication_id": "logarithmic-curvature-flow-lepton-spectrum", "title": "Logarithmic Curvature Flow, Filament Localization, and the Geometric Origin of the Lepton Mass Spectrum", "local_url": "https://rickyjreyes.github.io/publications/logarithmic-curvature-flow-lepton-spectrum.html", "doi": "10.5281/zenodo.18936949", "relationship": "extends" } ], "repositories": [ "https://github.com/rickyjreyes/geometry_of_resonance", "https://github.com/rickyjreyes/wct-sympy", "https://github.com/rickyjreyes/wct-lean" ], "formal_declarations": [ "effectiveWavenumber_eq_loopAverage", "constantWeightedAverage_eq_loopAverage", "resolved_e5_effectiveWavenumber_chain", "hbar_div_c_mul_k_is_mass" ], "evidence": [ { "evidence_id": "EVID-MASS-SYMPY-E5", "type": "SYMBOLIC_CHECK", "source": "https://rickyjreyes.github.io/sympy/#E5", "status": "CONDITIONAL", "independent": false, "description": "Exact loop closure plus constant positive weight closes the registered effective-wavenumber algebraic chain; it does not establish dynamical realization or physical mass generation." }, { "evidence_id": "EVID-MASS-LEAN-E5", "type": "LEAN_THEOREM", "source": "https://github.com/rickyjreyes/wct-lean/blob/main/THEOREMS.md", "status": "CONDITIONAL", "independent": false, "description": "Lean kernel checks the finite algebraic E5 chain under explicit closure and weighting hypotheses." } ], "empirical_status": "NOT_TESTED", "independent_replication": "NONE", "falsifiers": [ "A calibrated system satisfying the declared locking assumptions does not produce the predicted effective mass relation within stated uncertainty.", "The required curvature scale cannot be selected dynamically without fitting the target mass." ], "unresolved_obligations": [ "Establish that the dynamically selected curvature scale determines physical mass rather than merely supplying a dimensionally valid mapping.", "Close or calibrate the model-dependent proportionality in E19.", "Publish a parameter-frozen empirical test connecting measured curvature or spectral scale to inertial mass." ] }, { "claim_id": "CLM-FINITE-BAND-SELECTION", "title": "Finite-band pattern selection", "status": "CONDITIONAL", "equations": [ "M3", "E12", "E13", "E14", "E15", "E16", "E57", "E58", "E59", "E61", "E62", "E64" ], "assumptions": [ "ASM-POSITIVE-SPECTRAL-OFFSET" ], "papers": [ { "publication_id": "phase-flux-field", "title": "Phase-Flux Field (PFF): Axiomatic Substrate for Wave Confinement Theory", "local_url": "https://rickyjreyes.github.io/publications/phase-flux-field.html", "doi": "10.5281/zenodo.17578766", "relationship": "introduces" }, { "publication_id": "self-emergent-fourier-cymatics", "title": "Self-Emergent Fourier Cymatics: Entropic Eigenmodes out of Chaos", "local_url": "https://rickyjreyes.github.io/publications/self-emergent-fourier-cymatics.html", "doi": "10.5281/zenodo.17732648", "relationship": "tests" }, { "publication_id": "geometry-of-resonance", "title": "The Geometry of Resonance: Wave Confinement Theory and the Emergence of Mass, Force, and Spacetime", "local_url": "https://rickyjreyes.github.io/publications/geometry-of-resonance.html", "doi": "10.5281/zenodo.15644222", "relationship": "extends" } ], "repositories": [ "https://github.com/rickyjreyes/geometry_of_resonance", "https://github.com/rickyjreyes/wct-sympy", "https://github.com/rickyjreyes/wct-lean" ], "formal_declarations": [ "bandpass_oneMode_symbol", "bandGreenDenominator_pos", "bandGreenKernel_le_inverseOffset", "bandGreenKernel_at_shell" ], "evidence": [ { "evidence_id": "EVID-BAND-SYMPY", "type": "SYMBOLIC_CHECK", "source": "https://rickyjreyes.github.io/sympy/#M3", "status": "SUPPORTING", "independent": false, "description": "Executable checks verify the preferred-shell extremum, sign convention, projection identities, selected wavelength, and bounded Green kernel under declared assumptions." }, { "evidence_id": "EVID-BAND-LEAN", "type": "LEAN_THEOREM", "source": "https://github.com/rickyjreyes/wct-lean/blob/main/THEOREMS.md", "status": "CONDITIONAL", "independent": false, "description": "Lean checks the one-mode polynomial symbol and Green-kernel inequalities, not the complete nonlinear pattern-selection theorem." }, { "evidence_id": "EVID-BAND-SIMULATION", "type": "SIMULATION", "source": "https://rickyjreyes.github.io/publications/self-emergent-fourier-cymatics.html", "status": "CONDITIONAL", "independent": false, "description": "Author-generated numerical studies report finite-band spectral concentration for selected models and parameters." } ], "empirical_status": "SIMULATED", "independent_replication": "NONE", "falsifiers": [ "With frozen equations, discretization, parameters, and initial-condition ensemble, the predicted preferred shell fails to appear or is not robust to resolution and domain changes.", "The measured selected wavelength disagrees with the declared dispersion maximum after uncertainty propagation." ], "unresolved_obligations": [ "Distinguish linear spectral selection from nonlinear global pattern selection.", "Connect the selected shell to calibrated experimental observables.", "Publish a frozen cross-resolution and cross-integrator reproduction bundle." ] }, { "claim_id": "CLM-DIMENSIONALITY", "title": "Spatial-dimensionality stability criterion", "status": "CONDITIONAL", "equations": [ "M4", "E24", "E65", "E67", "E69", "E70" ], "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN", "ASM-H2-CONFINEMENT-ROUTE" ], "papers": [ { "publication_id": "hard-upper-bound-spatial-dimensionality", "title": "Hard Upper Bound on Spatial Dimensionality in Wave Confinement Theory", "local_url": "https://rickyjreyes.github.io/publications/hard-upper-bound-spatial-dimensionality.html", "doi": "10.5281/zenodo.17081283", "relationship": "introduces" }, { "publication_id": "geometry-of-resonance", "title": "The Geometry of Resonance: Wave Confinement Theory and the Emergence of Mass, Force, and Spacetime", "local_url": "https://rickyjreyes.github.io/publications/geometry-of-resonance.html", "doi": "10.5281/zenodo.15644222", "relationship": "contextualizes" } ], "repositories": [ "https://github.com/rickyjreyes/geometry_of_resonance", "https://github.com/rickyjreyes/wct-sympy" ], "evidence": [ { "evidence_id": "EVID-DIM-SOBOLEV", "type": "SYMBOLIC_CHECK", "source": "https://rickyjreyes.github.io/equations/#M4", "status": "SUPPORTING", "independent": false, "description": "The registered mathematical result is the standard H2-to-L-infinity threshold for integer dimension n at most 3 under standard domain hypotheses." }, { "evidence_id": "EVID-DIM-E70", "type": "SYMBOLIC_CHECK", "source": "https://rickyjreyes.github.io/equations/#E70", "status": "CONDITIONAL", "independent": false, "description": "E70 records the broader WCT stability interpretation as conditional on the H2 confinement route being necessary." } ], "empirical_status": "NOT_APPLICABLE", "independent_replication": "NONE", "qualification": "The verified Sobolev threshold is not a universal biconditional theorem excluding every possible stable higher-dimensional confinement mechanism.", "falsifiers": [ "A valid stable WCT confinement mechanism in dimension n greater than or equal to 4 that does not require the declared H2-to-L-infinity route falsifies the universal interpretation while leaving the Sobolev theorem intact." ], "unresolved_obligations": [ "Prove that the H2 regularity route is necessary for every admissible stable WCT confinement mechanism, or retain explicitly conditional wording." ] }, { "claim_id": "CLM-KOIDE", "title": "Koide mass relation from curvature-harmonic structure", "status": "OPEN", "expression": "Q = (m_e + m_mu + m_tau) / (sqrt(m_e) + sqrt(m_mu) + sqrt(m_tau))^2", "target_value": "2/3", "equations": [ "TOP7" ], "assumptions": [], "papers": [ { "publication_id": "wct-koide-mass-relation", "title": "Wave Confinement Theory Predicts the Koide Mass Relation: A Curvature-Harmonic Origin of Fermion Mass Triplets", "local_url": "https://rickyjreyes.github.io/publications/wct-koide-mass-relation.html", "doi": "10.5281/zenodo.17887562", "relationship": "introduces" }, { "publication_id": "logarithmic-curvature-flow-lepton-spectrum", "title": "Logarithmic Curvature Flow, Filament Localization, and the Geometric Origin of the Lepton Mass Spectrum", "local_url": "https://rickyjreyes.github.io/publications/logarithmic-curvature-flow-lepton-spectrum.html", "doi": "10.5281/zenodo.18936949", "relationship": "extends" }, { "publication_id": "rest-energy-density-weighted-loop-curvature", "title": "Rest Energy from Density-Weighted Loop Curvature: A Covariant Locking Principle", "local_url": "https://rickyjreyes.github.io/publications/rest-energy-density-weighted-loop-curvature.html", "doi": "10.5281/zenodo.20533537", "relationship": "supplies_mass_mapping" } ], "repositories": [ "https://github.com/rickyjreyes/geometry_of_resonance", "https://github.com/rickyjreyes/wct-sympy", "https://github.com/rickyjreyes/wct-lean" ], "formal_declarations": [ "koide_sqrt_sum_pos", "koide_denominator_positive", "koide_numerator_positive", "koide_well_defined", "koide_positive" ], "evidence": [ { "evidence_id": "EVID-KOIDE-LEAN-DOMAIN", "type": "LEAN_THEOREM", "source": "https://github.com/rickyjreyes/wct-lean/blob/main/THEOREMS.md#koide-wctleankoidelean", "status": "SUPPORTING", "independent": false, "description": "Lean establishes positivity and well-definedness of the Koide expression on the positive-mass domain; it does not derive Q = 2/3." }, { "evidence_id": "EVID-KOIDE-SYMPY", "type": "SYMBOLIC_CHECK", "source": "https://github.com/rickyjreyes/wct-sympy/blob/main/scripts/check_koide.py", "status": "CONDITIONAL", "independent": false, "description": "Author-generated numerical and symbolic checks evaluate the relation but do not resolve whether the target value is derived rather than selected or fitted." } ], "empirical_status": "PHENOMENOLOGICAL", "independent_replication": "NONE", "derivation_classification": "UNRESOLVED", "continuous_free_parameters": "NOT_YET_REGISTERED", "discrete_choices": "NOT_YET_REGISTERED", "normalization_dependence": "NOT_YET_REGISTERED", "state_selection_rule": "NOT_YET_REGISTERED", "unsuccessful_candidate_count": "NOT_YET_REGISTERED", "selected_state_unique": "UNKNOWN", "falsifiers": [ "A preregistered parameter-free derivation fails to return Q = 2/3 for the declared state-selection rule.", "The result changes materially under admissible normalization or unreported discrete model choices.", "The target value is found to be imposed as an input rather than produced by the dynamics or geometry." ], "unresolved_obligations": [ "Determine whether Q = 2/3 is an output or an imposed constraint.", "Record which model choices are fixed before evaluating the ratio.", "Count available continuous and discrete choices.", "Determine whether the selected harmonic state is unique.", "Count unsuccessful candidate triplets.", "Determine dependence on normalization and state selection.", "Classify the result as a scale-free phenomenological derivation, fitted relation, or dynamical prediction." ] }, { "claim_id": "CLM-CURVATURE-COMPUTATION", "title": "Curvature-bounded computation", "status": "CONDITIONAL", "equations": [ "M5", "E36", "E37", "E38", "E39", "E40", "E41", "E71", "E72", "E73", "E76" ], "assumptions": [], "papers": [ { "publication_id": "p-vs-np-curvature-bounded-wave-computation", "title": "P vs NP in Curvature-Bounded Wave Computation: A Model-Relative P_WCC != NP_WCC Separation", "local_url": "https://rickyjreyes.github.io/publications/p-vs-np-curvature-bounded-wave-computation.html", "doi": "10.5281/zenodo.17743607", "relationship": "introduces" }, { "publication_id": "discrete-wave-constrained-computation", "title": "Discrete Wave-Constrained Computation and Classical Complexity: Turing Equivalence for P and NP", "local_url": "https://rickyjreyes.github.io/publications/discrete-wave-constrained-computation.html", "doi": "10.5281/zenodo.17732642", "relationship": "qualifies" }, { "publication_id": "classical-p-vs-np-ill-posed", "title": "The Classical P vs NP Problem Is Mathematically and Physically Ill-Posed", "local_url": "https://rickyjreyes.github.io/publications/classical-p-vs-np-ill-posed.html", "doi": "10.5281/zenodo.17783074", "relationship": "critiques" }, { "publication_id": "wavelock-curvature-locked-one-way-function", "title": "WaveLock: A Curvature-Locked One-Way Function Based on Nonlinear PDE Evolution", "local_url": "https://rickyjreyes.github.io/publications/wavelock-curvature-locked-one-way-function.html", "doi": "10.5281/zenodo.19122146", "relationship": "applies" } ], "repositories": [ "https://github.com/rickyjreyes/geometry_of_resonance", "https://github.com/rickyjreyes/Wavelock", "https://github.com/rickyjreyes/wct-sympy" ], "evidence": [ { "evidence_id": "EVID-COMP-SYMPY", "type": "SYMBOLIC_CHECK", "source": "https://rickyjreyes.github.io/sympy/#M5", "status": "CONDITIONAL", "independent": false, "description": "The registry checks dimensional and counting relations but leaves encoding, precision, update-cost, and equivalence assumptions open." }, { "evidence_id": "EVID-COMP-WAVELOCK", "type": "SIMULATION", "source": "https://github.com/rickyjreyes/Wavelock", "status": "CONDITIONAL", "independent": false, "description": "WaveLock is an author-generated cryptographic research prototype with unresolved security proof obligations; it is not evidence for unrestricted classical P versus NP." } ], "empirical_status": "MODEL_RELATIVE", "independent_replication": "NONE", "qualification": "All separation and equivalence statements are internal to explicitly defined computational models and must not be reported as an unrestricted solution to classical P versus NP.", "falsifiers": [ "A polynomial-overhead simulation invalidates the claimed model-relative separation under the same encoding, precision, and resource assumptions.", "The declared curvature-resource restriction cannot be implemented or measured without exponential hidden overhead." ], "unresolved_obligations": [ "Specify the encoding, precision model, update cost, and finite curvature-resource bound.", "Separate physical-resource claims from classical complexity-class equivalence claims.", "Provide theorem-level simulation relations between the WCC model and a standard computational model." ] } ], "objects": [ { "canonical_id": "M1", "name": "Curvature-locking functional", "family": "Master systems", "formula": "\\sigma(s):=\\sqrt{\\kappa(s)^2+\\tau(s)^2}, \\qquad \\langle f\\rangle_w:= \\frac{\\oint_\\Gamma w f\\,ds}{\\oint_\\Gamma w\\,ds}, \\quad \\oint_\\Gamma w\\,ds>0.\n\nS_{\\rm lock}[\\varphi] = \\oint_\\Gamma w(s)\\bigl(\\partial_s\\varphi-\\sigma\\bigr)^2\\,ds.\n\nm=\\frac{\\hbar}{c}\\langle\\sigma\\rangle_w.", "definition": "A loop-locking variational functional whose stationary configurations relate phase winding to averaged curve curvature. The mass interpretation requires the explicit locking, orientation, and weighting assumptions recorded for E5.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_locking_family" ], "baseline_checker": "check_locking_family", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "finiteLockingAction", "finiteLockingAction_nonnegative", "finiteLockingAction_zero" ], "source": "WCTLean/Models/Locking.lean", "limitations": [ "Finite algebraic model; no curve-integral variational existence theorem is claimed." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m1-curvature-locking-functional" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M1" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M2", "name": "Nonsingular curvature operator and Lyapunov candidate", "family": "Master systems", "formula": "R_\\varepsilon(\\psi) := \\frac{\\overline{\\psi}} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}},\n\n\\Theta_\\varepsilon[\\psi] := -(\\Delta\\psi)R_\\varepsilon(\\psi).\n\nR_\\varepsilon(\\psi)\\longrightarrow \\frac1\\psi \\qquad(\\varepsilon\\to0).\n\n\\mathcal E_{\\rm WCT}[\\psi] = \\int_\\Omega \\left( |\\nabla\\psi|^2+|\\Theta_\\varepsilon[\\psi]|^2 \\right)\\,dx.", "definition": "The complex-safe regularized reciprocal and curvature-feedback operator. Positivity of the modulus-squared denominator removes the historical scalar zero; this does not by itself prove global PDE stability or uniqueness.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "LIMIT_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_regularized_denominator" ], "baseline_checker": "check_regularized_denominator", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "complex_normSq_nonnegative", "complexRegularizerDenominator_positive", "complexRegularizerDenominator_ne_zero", "complexRegularizerDenominator_at_zero", "complexRegularizedReciprocal_at_zero", "thetaComplex_zero_state" ], "source": "WCTLean/Models/ComplexCurvature.lean", "limitations": [ "Local algebraic denominator safety; not a full PDE well-posedness theorem." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m2-nonsingular-curvature-operator-and-lyapunov-candidate" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M3", "name": "Finite-band spectral selector", "family": "Master systems", "formula": "\\partial_tA = \\mu A-g|A|^2A-b(\\Delta+k_\\star^2)^2A, \\qquad b>0.\n\n-b\\bigl(|k|^2-k_\\star^2\\bigr)^2,", "definition": "A Swift–Hohenberg-type finite-band selector whose Fourier symbol damps modes away from the preferred shell. It establishes the linear spectral rail, not global nonlinear pattern selection.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SIGN_OR_EXTREMUM_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_master_sh_sign" ], "baseline_checker": "check_master_sh_sign", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "finiteBandPenalty_nonpositive", "finiteBandPenalty_at_shell", "finiteBandPenalty_strict_off_shell" ], "source": "WCTLean/Models/BandPass.lean", "limitations": [ "Fourier-symbol algebra only; nonlinear pattern selection remains open." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m3-finite-band-spectral-selector" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M4", "name": "Dimensional stability threshold", "family": "Master systems", "formula": "H^2(\\Omega)\\hookrightarrow L^\\infty(\\Omega) \\quad\\text{when}\\quad 2>\\frac n2.\n\nn\\le3.", "definition": "The standard Sobolev threshold H²→L∞ for integer spatial dimension n≤3. This is a regularity threshold and not, by itself, a universal theorem that all stable confinement is impossible above three dimensions.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "FORMAL_THEOREM", "scope": "STANDARD_MATHEMATICS", "checker": [ "check_h2_embedding_threshold" ], "baseline_checker": "check_h2_embedding_threshold", "meaning": "The standard Sobolev embedding threshold H2 to L-infinity holds under the declared domain hypotheses for integer spatial dimension n less than or equal to 3.", "limitations": [ "This standard embedding theorem is not by itself a complete nonlinear WCT stability theorem." ] }, "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN" ], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "h2SubcriticalDimension", "h2Subcritical_iff_le_three", "SobolevCurvatureContract" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Lean proves the integer equivalence n less than 4 iff n at most 3; the Sobolev embedding and curvature bound remain explicit contract fields." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m4-dimensional-stability-threshold" }, "dependencies": { "equations": [], "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN" ], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M4" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M5", "name": "Curvature-bounded computation", "family": "Master systems", "formula": "\\psi^{(t+1)}(x) = U\\!\\left( \\psi^{(t)}(x), \\{\\psi^{(t)}(y):y\\in N(x)\\} \\right).", "definition": "A local discrete update architecture proposed for curvature-bounded computation. Complexity conclusions additionally require a fixed encoding, precision model, update cost, and finite physical-resource bound.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "averagedUpdate", "averagedUpdate_fixed_of_average_fixed", "averagedUpdate_norm_le", "InNormBound", "averagedUpdate_preserves_normBound", "BandLimitedInvariantSet" ], "source": "WCTLean/Models/CompactDynamics.lean", "limitations": [ "Finite-dimensional bounded-dynamics support only; no complexity-class equivalence is claimed." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m5-curvature-bounded-computation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M5" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M6A", "name": "Unified linear operator", "family": "Master systems", "formula": "\\mathcal L_{\\rm WCT} = c_1(\\Delta+\\sigma^2) -c_2(\\Delta+k_\\star^2)^2 +i\\,c_3m +c_4R^{-(2+n/p)}, \\qquad c_2>0.", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declarations": [ "unifiedShellTerm", "unifiedLinearSymbol", "unifiedRealVectorField", "UnifiedCoefficients" ], "source": "WCTLean/Models/UnifiedOperator.lean", "limitations": [ "This is a typed operator skeleton, not a derivation of the full nonlinear UWCT equation." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m6a-unified-linear-operator" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M6A" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M6B", "name": "Nonlinear curvature operator", "family": "Master systems", "formula": "\\mathcal N_{\\rm curv}[\\psi] = -(\\Delta\\psi) \\frac{\\overline{\\psi}} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}}.", "definition": "Current effective status: ○ OPEN The operator is well-defined for \\(\\varepsilon>0\\); uniqueness of this nonlinear closure remains open.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "COUNTEREXAMPLE_TEST", "scope": "UNRESOLVED", "checker": [ "check_uniqueness_claim" ], "baseline_checker": "check_uniqueness_claim", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "thetaComplex", "thetaComplex_canonical_formula", "thetaComplex_denominator_ne_zero" ], "source": "WCTLean/Models/ComplexCurvature.lean", "limitations": [ "The local operator form and positive-epsilon denominator are proved; uniqueness of the nonlinear closure and PDE well-posedness remain open." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m6b-nonlinear-curvature-operator" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M6B" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M7", "name": "Full curvature-wavefield equation", "family": "Master systems", "formula": "\\partial_t\\psi = \\mathcal N_{\\rm curv}[\\psi] +g|\\psi|^2\\psi +c_1(\\Delta+\\sigma^2)\\psi -c_2(\\Delta+k_\\star^2)^2\\psi +i\\,c_3m\\psi +c_4R^{-(2+n/p)}\\psi +\\eta\\psi\\circ\\xi(t), \\qquad c_2>0.", "definition": "Current effective status: ✅ PASS The explicit negative fourth-order term supplies finite-band ultraviolet damping. Other dynamical claims require separate analysis.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SIGN_OR_EXTREMUM_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_master_uwct_sign" ], "baseline_checker": "check_master_uwct_sign", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "unifiedShellTerm_nonpositive", "unifiedShellTerm_zero_on_shell" ], "source": "WCTLean/Models/UnifiedOperator.lean", "limitations": [ "Theorems cover the represented shell term, not the complete master dynamics." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m7-full-curvature-wavefield-equation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M7" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "M8", "name": "Curvature-acoustic cosmology system", "family": "Master systems", "formula": "\\Phi(k,t)=-C_\\Phi\\frac{\\Theta(k,t)}{k^2},\n\n\\delta_g(E) = A_g\\cos\\!\\left( k_\\ell\\ln\\frac{E}{E_0}+\\phi \\right).", "definition": "Current effective status: ○ OPEN Representative closure relations are", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m8-curvature-acoustic-cosmology-system" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "M8" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E1A", "name": "Curvature-rate density", "family": "A. Rest energy, curvature, and loop locking", "formula": "\\sigma_{\\rm dens}(s)=\\kappa(s)^2+\\tau(s)^2, \\qquad [\\sigma_{\\rm dens}]=L^{-2}.", "definition": "The canonical registered object for curvature-rate density; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_e1a" ], "baseline_checker": "check_e1a", "meaning": "The curvature-rate density has inverse-length-squared dimension.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "dimensional_support", "declarations": [ "kappa_sq_dim", "tau_sq_dim" ], "source": "WCTLean/Curvature.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e1a-curvature-rate-density" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E1A" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E1B", "name": "Curvature spectral rate", "family": "A. Rest energy, curvature, and loop locking", "formula": "\\sigma(s)=\\sqrt{\\kappa(s)^2+\\tau(s)^2}, \\qquad [\\sigma]=L^{-1}.", "definition": "The canonical registered object for curvature spectral rate; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_e1b" ], "baseline_checker": "check_e1b", "meaning": "The curvature spectral rate has inverse-length dimension.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "dimensional_support", "declarations": [ "sqrt_inverse_length_squared", "sqrt_kappa_sq_dim" ], "source": "WCTLean/Curvature.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e1b-curvature-spectral-rate" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E1B" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E2", "name": "Weighted loop average", "family": "A. Rest energy, curvature, and loop locking", "formula": "\\langle f\\rangle_w = \\frac{\\oint_\\Gamma w(s)f(s)\\,ds} {\\oint_\\Gamma w(s)\\,ds}.", "definition": "The normalized weighted loop average for a nonnegative weight with nonzero total weight. It preserves the dimension of the averaged quantity but does not select a physical weighting measure.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "STANDARD_MATHEMATICS", "checker": [ "check_weighted_average" ], "baseline_checker": "check_weighted_average", "meaning": "The positive weighted-average construction has a nonzero denominator and preserves the encoded dimension of the averaged quantity.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "densityWeightedAverage_denominator_ne_zero", "weightedSum_dim", "weighted_average_preserves_dimension" ], "source": "WCTLean/ResolvedAudit.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e2-weighted-loop-average" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E3", "name": "Loop-locking action", "family": "A. Rest energy, curvature, and loop locking", "formula": "S_{\\rm eff}[\\varphi] = \\oint_\\Gamma w(\\partial_s\\varphi-\\sigma)^2\\,ds + \\Lambda \\left( \\oint_\\Gamma\\partial_s\\varphi\\,ds-2\\pi n \\right), \\qquad n\\in\\mathbb Z.", "definition": "A constrained phase-curvature mismatch action. Nonnegative weighting makes its squared mismatch term nonnegative; existence and uniqueness of continuum minimizers require separate analysis.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "VARIATIONAL_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_locking_variation" ], "baseline_checker": "check_locking_variation", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "finiteLockingAction_nonnegative", "finiteLockingAction_zero", "lockingMismatch_nonnegative", "lockingMismatch_zero" ], "source": "WCTLean/Models/Locking.lean", "limitations": [ "Finite algebraic support; this is not an integral theorem on general curves." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e3-loop-locking-action" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E4", "name": "Covariant locking solution", "family": "A. Rest energy, curvature, and loop locking", "formula": "\\partial_s\\varphi = \\sigma+\\frac{\\alpha_{\\rm lock}}{w},\n\n\\alpha_{\\rm lock} = \\frac{ 2\\pi n-\\oint_\\Gamma\\sigma\\,ds }{ \\oint_\\Gamma ds/w },", "definition": "Current effective status: ✅ PASS Stationarity gives", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_locking_solution" ], "baseline_checker": "check_locking_solution", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "alphaLock", "alphaLock_closure", "PointwiseLocked", "pointwiseLocked_weighted" ], "source": "WCTLean/Models/Locking.lean", "limitations": [ "The denominator is assumed nonzero and no stationary curve solution is constructed." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e4-covariant-locking-solution" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E4" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E5", "name": "Effective-wavenumber chain", "family": "A. Rest energy, curvature, and loop locking", "formula": "L_s:=\\oint_\\Gamma ds, \\qquad k_{\\rm wind}:=\\frac{2\\pi|n|}{L_s}, \\qquad k_\\sigma:=\\frac1{L_s}\\oint_\\Gamma\\sigma\\,ds.\n\nk_{\\rm wind}=k_\\sigma=\\langle\\sigma\\rangle_w", "definition": "The effective-wavenumber identification connecting phase winding, integrated curvature, and the weighted curvature average. The chain is derived only under compatible orientation, exact integrated locking, and a positive constant-weight condition.", "status": { "baseline": "CONDITIONAL", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_effective_wavenumber_chain_derived" }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_effective_wavenumber_chain_derived" ], "baseline_checker": "check_effective_wavenumber_chain", "meaning": "Exact loop closure and the declared orientation and weighting assumptions establish the encoded effective-wavenumber chain.", "limitations": [ "This does not independently establish that physical particle masses are generated by the model." ] }, "assumptions": [ "ASM-COMPATIBLE-ORIENTATION", "ASM-EXACT-INTEGRATED-LOCKING", "ASM-CONSTANT-POSITIVE-WEIGHT" ], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "effectiveWavenumber_eq_loopAverage", "constantWeightedAverage_eq_loopAverage", "resolved_e5_effectiveWavenumber_chain" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "Does not prove dynamic attainment of exact lock." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e5-effective-wavenumber-chain" }, "dependencies": { "equations": [], "assumptions": [ "ASM-COMPATIBLE-ORIENTATION", "ASM-EXACT-INTEGRATED-LOCKING", "ASM-CONSTANT-POSITIVE-WEIGHT" ], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E5" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E6", "name": "Mass-curvature law", "family": "A. Rest energy, curvature, and loop locking", "formula": "E_{\\rm rest}=\\hbar c\\,k_{\\rm eff}, \\qquad m=\\frac{\\hbar}{c}k_{\\rm eff}.\n\nm=\\frac{\\hbar}{c}\\langle\\sigma\\rangle_w.", "definition": "A dimensionally consistent mapping from an effective inverse-length scale to rest energy and mass. The PASS establishes dimensional closure, not that WCT dynamics generates observed particle masses.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_mass_curvature_dimensions" ], "baseline_checker": "check_mass_curvature_dimensions", "meaning": "The encoded mass-curvature relation has the dimensions of mass.", "limitations": [ "Dimensional closure does not establish the physical identification of mass with curvature." ] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "dimensional_support", "declarations": [ "hbar_div_c_mul_k_is_mass", "weighted_average_preserves_dimension" ], "source": "WCTLean/Dimension.lean", "limitations": [ "Dimensions only; not the physical identification." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e6-mass-curvature-law" }, "dependencies": [ "E1B", "E2", "E5" ], "source_papers": [ { "title": "Rest Energy from Density-Weighted Loop Curvature: A Covariant Locking Principle", "doi": "10.5281/zenodo.20533537", "url": "https://zenodo.org/records/20533537" } ], "evidence": [ { "type": "publication", "status": "reported", "source": "https://zenodo.org/records/20533537" } ], "cross_layer_consistency": { "status": "partial", "notes": [ "E5 remains conditional unless exact locking and compatible weighting assumptions are supplied." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E6", "M1", "E7" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E7", "name": "Solenoidal mass law", "family": "A. Rest energy, curvature, and loop locking", "formula": "m = \\frac{\\hbar}{c} \\left\\langle \\sqrt{\\kappa^2+\\tau^2} \\right\\rangle_\\Gamma.", "definition": "The solenoidal form of the mass-curvature mapping using an averaged curve-curvature magnitude. Its physical prediction requires a specified averaging measure and a dynamically selected geometry.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_mass_curvature_dimensions" ], "baseline_checker": "check_mass_curvature_dimensions", "meaning": "The solenoidal curvature average has the dimensions required by the stated mass relation.", "limitations": [ "The averaging measure and physical identification require separate justification." ] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "dimensional_support", "declarations": [ "hbar_div_c_mul_k_is_mass", "weighted_average_preserves_dimension" ], "source": "WCTLean/Dimension.lean", "limitations": [ "Dimensions only." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e7-solenoidal-mass-law" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E7" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E8", "name": "Corrected weighted-lock identity", "family": "A. Rest energy, curvature, and loop locking", "formula": "\\boxed{ \\oint_\\Gamma w\\,\\partial_s\\varphi\\,ds = \\oint_\\Gamma w\\,\\sigma\\,ds + \\alpha_{\\rm lock}L_s }.\n\n2\\pi\\oint ds/\\oint ds/w", "definition": "Current effective status: ✅ PASS Substituting E4 gives", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_e8_identity" ], "baseline_checker": "check_e8_identity", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "pointwiseLocked_weighted", "correctedWeightedLockIdentity" ], "source": "WCTLean/Models/Locking.lean", "limitations": [ "Finite counting-measure analogue; general curve integration is not formalized." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e8-corrected-weighted-lock-identity" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E8" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E9", "name": "Phase-flux constitutive relation", "family": "B. Phase-flux and finite-band selection", "formula": "\\mathbf S(x,t)=u(x,t)\\nabla\\theta(x,t).\n\n\\partial_tu+\\nabla\\cdot\\mathbf S=0.", "definition": "The phase-current identity obtained from a supplied polar representation of the complex field. The finite algebraic identity does not replace a function-space proof of full polar differentiation and conservation dynamics.", "status": { "baseline": "DEFINITION", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_phase_flux_from_polar_field" }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_phase_flux_from_polar_field" ], "baseline_checker": "classify_definition", "meaning": "Polar decomposition yields the normalized phase-current identity under the declared nonzero-field assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "polarCurrentProduct_im", "phaseCurrent_of_polar_factorization", "conservationResidual_zero_iff" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "Full function-space polar differentiation remains open." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e9-phase-flux-constitutive-relation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E9" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E10", "name": "Radial shell quantization", "family": "B. Phase-flux and finite-band selection", "formula": "\\int_{r_1}^{r_2}k_r(r)\\,dr=2\\pi n, \\qquad n\\in\\mathbb Z.", "definition": "Current effective status: ✅ PASS Both sides are dimensionless.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_shell_quantization_dimensions" ], "baseline_checker": "check_shell_quantization_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "RadialShellQuantized", "radialShellQuantized_add", "radialShellQuantized_iff_windingQuantized" ], "source": "WCTLean/Models/PhaseFlux.lean", "limitations": [ "Exact integer-law closure only; no dynamical shell-selection theorem is claimed." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e10-radial-shell-quantization" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E10" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E11", "name": "Phase winding", "family": "B. Phase-flux and finite-band selection", "formula": "m(\\gamma) = \\frac1{2\\pi} \\oint_\\gamma\\nabla\\theta\\cdot d\\boldsymbol\\ell \\in\\mathbb Z,", "definition": "Current effective status: ✅ PASS provided \\(\\psi\\neq0\\) on the loop and the phase is continuous modulo \\(2\\pi\\).", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_winding_dimensions" ], "baseline_checker": "check_winding_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "WindingQuantized", "windingQuantized_add", "radialShellQuantized_iff_windingQuantized" ], "source": "WCTLean/Models/PhaseFlux.lean", "limitations": [ "Exact integer-law closure only; no field-defect existence theorem is claimed." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e11-phase-winding" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E11" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E12", "name": "Finite-band dispersion rail", "family": "B. Phase-flux and finite-band selection", "formula": "\\lambda_{\\rm grow}(k) = r+a|k|^2-b|k|^4, \\qquad a,b>0.\n\nk_\\star=\\sqrt{\\frac{a}{2b}}.\n\n\\lambda_{\\rm grow}(k) = \\mu-b(|k|^2-k_\\star^2)^2, \\qquad \\mu=r+\\frac{a^2}{4b}.", "definition": "The quartic finite-band dispersion relation with a stationary maximum at the selected nonzero wavenumber. It verifies the preferred linear shell and damping sign.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SIGN_OR_EXTREMUM_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_dispersion_stationary_point" ], "baseline_checker": "check_dispersion_stationary_point", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "swiftHohenbergLinearSymbol_le_mu", "swiftHohenbergLinearSymbol_at_shell", "quarticGrowthSymbol_completedSquare" ], "source": "WCTLean/Models/BandPass.lean", "limitations": [ "Symbol-level result; no nonlinear PDE selection theorem." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e12-finite-band-dispersion-rail" }, "dependencies": [], "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "consistent", "notes": [ "The canonical registry uses lambda_grow rather than sigma(k) to avoid collision with curvature sigma." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E12", "M3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E13", "name": "Band-pass amplitude evolution", "family": "B. Phase-flux and finite-band selection", "formula": "\\partial_tA = (r-a\\Delta-b\\Delta^2)A-\\beta|A|^2A,\n\n\\partial_tA = \\mu A-b(\\Delta+k_\\star^2)^2A-\\beta|A|^2A.", "definition": "The registered finite-band amplitude evolution equation. Its gradient-flow form follows under the stated functional and sign convention; nonlinear existence and long-time selection remain separate obligations.", "status": { "baseline": "CONDITIONAL", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_bandpass_gradient_flow" }, "verification": { "outcome": "PASS", "kind": "VARIATIONAL_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_bandpass_gradient_flow" ], "baseline_checker": "check_e13_e14_consistency", "meaning": "The declared band-pass amplitude equation follows from negative gradient flow of the registered functional.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "bandpass_oneMode_symbol" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "Not the full nonlinear pattern-selection theorem." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e13-band-pass-amplitude-evolution" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E13" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E14", "name": "Band-pass Lyapunov functional", "family": "B. Phase-flux and finite-band selection", "formula": "\\mathcal E[A] = \\int_\\Omega \\left[ -\\mu|A|^2 +b|(\\Delta+k_\\star^2)A|^2 +\\frac{\\beta}{2}|A|^4 \\right]dx.", "definition": "The energy functional associated with the finite-band amplitude equation. The variational relation is supported under exact negative gradient flow, while full functional-analytic Lyapunov theory remains conditional.", "status": { "baseline": "CONDITIONAL", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_bandpass_gradient_flow" }, "verification": { "outcome": "PASS", "kind": "VARIATIONAL_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_bandpass_gradient_flow" ], "baseline_checker": "check_e13_e14_consistency", "meaning": "The registered functional generates the declared band-pass gradient flow under the stated sign convention.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "bandpass_oneMode_symbol", "square_nonnegative", "positive_quadratic_energy_nonnegative" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "Not full Lyapunov descent." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e14-band-pass-lyapunov-functional" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E14" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E15", "name": "Modal growth bound", "family": "B. Phase-flux and finite-band selection", "formula": "\\frac{d}{dt}|\\widehat A_k|^2 \\le 2\\lambda_{\\rm grow}(k)|\\widehat A_k|^2 -c|\\widehat A_k|^4, \\qquad c>0.", "definition": "Current effective status: ⚠️ CONDITIONAL The quartic modal estimate requires a model-specific nonlinear projection bound.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e15-modal-growth-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E15" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E16", "name": "Linear spectral concentration", "family": "B. Phase-flux and finite-band selection", "formula": "P_k(t)=P_k(0)e^{2\\lambda_{\\rm grow}(k)t}.\n\n\\operatorname*{arg\\,max}_kP_k(t)\\to k_\\star.", "definition": "Current effective status: ✅ PASS For the linearized dynamics,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_linear_spectral_growth" ], "baseline_checker": "check_linear_spectral_growth", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e16-linear-spectral-concentration" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E16" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E17", "name": "Nonsingular curvature-feedback operator", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "R_\\varepsilon(\\psi) = \\frac{\\overline\\psi} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}},\n\n\\boxed{ \\Theta_\\varepsilon[\\psi] = -(\\Delta\\psi)R_\\varepsilon(\\psi) }.\n\n-\\Delta\\psi/(\\psi+\\varepsilon e^{-\\alpha|\\psi|^2})", "definition": "Current effective status: ✅ PASS For \\(\\varepsilon>0\\), the denominator is strictly positive for all complex \\(\\psi\\). This replaces", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "LIMIT_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_regularized_denominator" ], "baseline_checker": "check_regularized_denominator", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "complexRegularizerDenominator_positive", "complexRegularizerDenominator_ne_zero", "complexRegularizerDenominator_at_zero", "complexRegularizedReciprocal_at_zero", "thetaComplex_zero_state", "resolved_e17_counterexample" ], "source": "WCTLean/Models/ComplexCurvature.lean", "limitations": [ "The corrected local complex denominator is proved safe for positive epsilon; full operator regularity and epsilon-to-zero control remain open." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e17-nonsingular-curvature-feedback-operator" }, "dependencies": [], "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "consistent", "notes": [ "The corrected modulus-squared operator supersedes the historical scalar denominator." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E17", "M2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E18", "name": "WCT Lyapunov candidate", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "\\mathcal E[\\psi] = \\int_\\Omega \\left( c_1|\\nabla\\psi|^2 +c_2|\\Theta_\\varepsilon[\\psi]|^2 \\right)dx.", "definition": "A WCT energy candidate whose nonnegative terms and exact negative-gradient-flow hypothesis imply monotone descent. The full chain rule and PDE well-posedness are not established by the algebraic PASS.", "status": { "baseline": "CONDITIONAL", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_lyapunov_gradient_flow" }, "verification": { "outcome": "PASS", "kind": "VARIATIONAL_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_lyapunov_gradient_flow" ], "baseline_checker": "check_energy_positivity", "meaning": "Exact negative gradient flow gives monotone energy descent when the registered nonnegativity assumptions hold.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "resolved_e18_energy_nonnegative", "resolved_e18_gradientFlow_descent" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "The complete functional chain rule remains an explicit hypothesis." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e18-wct-lyapunov-candidate" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E18" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E19", "name": "Gap-curvature scaling", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "\\Delta_k^\\star\\sim\\langle\\sigma\\rangle_w^2, \\qquad [\\Delta_k^\\star]=L^{-2},\n\n\\Delta_\\omega^\\star:=c^2\\Delta_k^\\star, \\qquad [\\Delta_\\omega^\\star]=T^{-2}.", "definition": "A proposed scaling between a spectral gap and curvature scale. Its dimensional structure is consistent, but the proportionality, spectral derivation, and physical calibration remain model dependent.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "DIMENSIONAL_CHECK", "scope": "MODEL_CONDITIONAL", "checker": [ "check_gap_curvature_dimensions" ], "baseline_checker": "check_gap_curvature_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e19-gap-curvature-scaling" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E19" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E20", "name": "Higher-order cavity quadratic sector", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "S:=\\frac{\\Box\\psi}{g(\\psi)}, \\qquad P:=\\frac{\\Delta\\psi}{g(\\psi)}.\n\nQ(S,P)=\\kappa S^2+\\theta P^2-\\gamma SP.\n\n\\kappa\\ge0,\\qquad \\theta\\ge0,\\qquad \\gamma^2\\le4\\kappa\\theta.", "definition": "Current effective status: ✅ PASS Let", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_cavity_quadratic_form" ], "baseline_checker": "check_cavity_quadratic_form", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "cavityQuadratic", "cavityQuadratic_nonnegative" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Scalar quadratic-sector positivity; not the full higher-order Euler-Lagrange equation." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e20-higher-order-cavity-quadratic-sector" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E20" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E21", "name": "Second-derivative Euler-Lagrange equation", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "\\frac{\\delta\\mathcal L}{\\delta\\psi} = \\frac{\\partial\\mathcal L}{\\partial\\psi} -\\partial_\\mu \\frac{\\partial\\mathcal L}{\\partial(\\partial_\\mu\\psi)} +\\partial_\\mu\\partial_\\nu \\frac{\\partial\\mathcal L} {\\partial(\\partial_\\mu\\partial_\\nu\\psi)} =0.", "definition": "Current effective status: ✅ PASS For \\(\\mathcal L(\\psi,\\partial\\psi,\\partial^2\\psi)\\),", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_second_order_el_template" ], "baseline_checker": "check_second_order_el_template", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e21-second-derivative-euler-lagrange-equation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E21" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E22", "name": "Effective metric ansatz", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "g_{\\mu\\nu}^{\\rm eff} = \\eta_{\\mu\\nu} +\\lambda_g \\frac{\\partial_\\mu\\overline\\psi\\,\\partial_\\nu\\psi} {\\rho c^2} +\\delta_g\\,\\eta_{\\mu\\nu}\\frac{W_\\psi}{W_0}.", "definition": "Current effective status: ⚠️ CONDITIONAL The coefficients must carry the units needed to make both corrections dimensionless, and signature/nondegeneracy must be checked.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e22-effective-metric-ansatz" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E22" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E23", "name": "Enthalpic curvature relation", "family": "C. Curvature feedback and Lyapunov dynamics", "formula": "h(\\psi) = C_h\\left( W_\\psi+\\chi|\\nabla\\psi|^2 \\right).", "definition": "Current effective status: ⚠️ CONDITIONAL The constants must reconcile dimensions and the relation requires a constitutive derivation.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e23-enthalpic-curvature-relation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E23" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E24", "name": "Sobolev embedding threshold", "family": "D. Dimensionality and functional bounds", "formula": "H^2(\\Omega)\\hookrightarrow L^\\infty(\\Omega) \\quad\\Longrightarrow\\quad 2>\\frac n2.\n\nn\\le3.", "definition": "The H²-to-L∞ Sobolev embedding threshold used by the dimensionality argument. It supplies a standard regularity condition, not a complete nonlinear stability theorem.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "FORMAL_THEOREM", "scope": "STANDARD_MATHEMATICS", "checker": [ "check_h2_embedding_threshold" ], "baseline_checker": "check_h2_embedding_threshold", "meaning": "The standard H2-to-L-infinity Sobolev embedding threshold is verified under the declared domain hypotheses.", "limitations": [ "The theorem supplies a regularity route and does not characterize every possible confinement mechanism." ] }, "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN" ], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "h2SubcriticalDimension", "h2Subcritical_iff_le_three", "SobolevCurvatureContract" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Lean proves n less than 4 iff n at most 3; the Sobolev embedding and curvature bound are contract fields, not proved analysis." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e24-sobolev-embedding-threshold" }, "dependencies": { "equations": [], "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN" ], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E24" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E25", "name": "Critical Sobolev exponent", "family": "D. Dimensionality and functional bounds", "formula": "p_c(n)=\\frac{n+2}{n-2}, \\qquad p2\\),", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declarations": [ "criticalSobolevExponent" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Typed definition only." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e25-critical-sobolev-exponent" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E25" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E26", "name": "Corrected curvature $L^2$ bound", "family": "D. Dimensionality and functional bounds", "formula": "|R_\\varepsilon(\\psi(x))|\\le\\delta^{-1} \\quad\\text{a.e.},\n\n\\boxed{ \\|\\Theta_\\varepsilon[\\psi]\\|_{L^2} \\le \\delta^{-1}\\|\\Delta\\psi\\|_{L^2} }.", "definition": "Current effective status: ✅ PASS If \\(\\psi\\in H^2(\\Omega)\\) and", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_theta_l2_from_h2" ], "baseline_checker": "check_theta_l2_from_h2", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "analytic_contract_only", "declarations": [ "SobolevCurvatureContract" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Contract exposes the required theorem and does not prove it." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e26-corrected-curvature-l2-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E26" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E27", "name": "Finite-energy confinement", "family": "D. Dimensionality and functional bounds", "formula": "\\int_{\\mathbb R^n} \\left( |\\nabla\\psi|^2 + |\\Theta_\\varepsilon[\\psi]|^2 \\right)dx <\\infty.", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declarations": [ "FiniteConfinementEnergy" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Predicate definition only; existence of a confined state is not proved." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e27-finite-energy-confinement" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E27" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E28", "name": "Corrected alpha-drop exponent", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "\\alpha(n) = 1+\\frac1n\\sum_{t=1}^{m(n)}\\log_2\\rho_t(n) +\\beta(n).\n\n\\beta(n) < -\\frac1n\\sum_t\\log_2\\rho_t(n).", "definition": "Current effective status: ✅ PASS Let \\(\\rhot(n)\\in(0,1]\\) be retained fractions. Define", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_alpha_drop_corrected" ], "baseline_checker": "check_alpha_drop_corrected", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "alphaDropValue", "alphaDrop_lt_one" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Scalar implication under the encoded beta hypothesis." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e28-corrected-alpha-drop-exponent" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E28" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E29", "name": "Entropy-drop pruning", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "M_{t+1}\\le e^{-\\Delta_t}M_t, \\qquad \\Delta_t\\ge0.\n\nM_T \\le M_0\\exp\\!\\left(-\\sum_{t=0}^{T-1}\\Delta_t\\right).", "definition": "Current effective status: ✅ PASS Iteration gives", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_state_decay_iteration" ], "baseline_checker": "check_state_decay_iteration", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "entropyPrunedUpperBound", "entropyPrunedUpperBound_nonnegative", "entropyPrunedUpperBound_le_modeCount", "entropyPrunedUpperBound_antitone" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Scalar bound under nonnegative mode-count assumptions; the entropy drop is not derived from the full PDE." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e29-entropy-drop-pruning" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E29" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E30", "name": "Spectral entropy", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "H_k=-\\sum_kP_k\\ln P_k.\n\n0\\le H_k\\le\\ln K.", "definition": "The normalized Shannon entropy of a finite spectral distribution and its standard bounds. This is a standard information-theoretic identity rather than a WCT-specific physical result.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "STANDARD_MATHEMATICS", "checker": [ "check_entropy_bounds" ], "baseline_checker": "check_entropy_bounds", "meaning": "The normalized spectral Shannon entropy satisfies its standard finite-support bounds.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e30-spectral-entropy" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E30" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E31", "name": "Conditional entropy-production bound", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "\\Delta H_t:=H_k(t)-H_k(t+1).\n\n\\Delta H_t\\ge c_0\\mathcal D_t, \\qquad \\mathcal D_t\\ge0.", "definition": "Current effective status: ⚠️ CONDITIONAL Define the entropy drop", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_entropy_drop_bound" ], "baseline_checker": "check_entropy_drop_bound", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e31-conditional-entropy-production-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E31" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E32", "name": "Subexponential exploration condition", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "\\limsup_{n\\to\\infty}\\alpha(n)<1.", "definition": "Current effective status: ⚠️ CONDITIONAL This follows only if the retained-fraction and \\(\\beta(n)\\) bounds in E28 hold uniformly with sufficient margin.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_alpha_drop_count_bound" ], "baseline_checker": "check_alpha_drop_count_bound", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e32-subexponential-exploration-condition" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E32" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E33", "name": "Corrected support-entropy relation", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "H_k(t)\\le\\ln K_t,\n\n\\boxed{ e^{H_k(t)}\\le K_t }.", "definition": "Current effective status: ✅ PASS For a distribution supported on \\(Kt\\) modes,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_support_entropy_bound" ], "baseline_checker": "check_support_entropy_bound", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "exp_entropy_le_support" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Order-theoretic consequence of the supplied entropy bound." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e33-corrected-support-entropy-relation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E33" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E34", "name": "Energy-entropy conversion", "family": "E. Alpha-drop, entropy reduction, and pruning", "formula": "\\Delta H_k:=H_{\\rm before}-H_{\\rm after}\\ge0,\n\n\\Delta E_{\\rm cost}\\ge\\lambda\\,\\Delta H_k, \\qquad \\lambda>0.", "definition": "Current effective status: ○ OPEN For entropy reduction", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e34-energy-entropy-conversion" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E34" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E35", "name": "Curvature-locked fixed point", "family": "F. WCC, channel capacity, and complexity", "formula": "\\partial_t\\psi=0, \\qquad \\nabla\\Theta_\\varepsilon[\\psi]=0, \\qquad \\frac{d}{dt}S[\\psi]=0.", "definition": "Current effective status: ◻️ DEFINITION A stationary locked configuration satisfies", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e35-curvature-locked-fixed-point" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E35" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E36", "name": "Discrete WCC update", "family": "F. WCC, channel capacity, and complexity", "formula": "\\psi^{(t+1)}(x) = U\\!\\left( \\psi^{(t)}(x), \\{\\psi^{(t)}(y)\\}_{y\\in N(x)} \\right).", "definition": "The local discrete WCC state-update rule on a prescribed neighborhood. It defines the computational dynamics but does not establish a classical complexity-class equivalence.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "averagedUpdate", "averagedUpdate_fixed_of_average_fixed", "averagedUpdate_norm_le" ], "source": "WCTLean/Models/CompactDynamics.lean", "limitations": [ "Concrete finite-dimensional special case; narrower than the full neighborhood-dependent WCC update." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e36-discrete-wcc-update" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E36" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E37", "name": "Bandlimit from energy", "family": "F. WCC, channel capacity, and complexity", "formula": "k_{\\max} = C_1\\frac{E_{\\max}}{\\hbar c},", "definition": "Current effective status: ✅ PASS with dimensionless \\(C1\\).", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_bandlimit_dimensions" ], "baseline_checker": "check_bandlimit_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e37-bandlimit-from-energy" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E37" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E38", "name": "Spatial channel capacity", "family": "F. WCC, channel capacity, and complexity", "formula": "N_{\\rm lanes} \\le C_2Vk_{\\max}^3,", "definition": "Current effective status: ✅ PASS In three spatial dimensions,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_channel_capacity_dimensions" ], "baseline_checker": "check_channel_capacity_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e38-spatial-channel-capacity" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E38" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E39", "name": "Polynomial update bound", "family": "F. WCC, channel capacity, and complexity", "formula": "T_{\\max}(n)\\le C_3n^d.", "definition": "Current effective status: ◻️ DEFINITION This defines the assumed computational resource class.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e39-polynomial-update-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E39" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E40", "name": "WCC complexity identification", "family": "F. WCC, channel capacity, and complexity", "formula": "P_{\\rm WCC}\\cong P, \\qquad NP_{\\rm WCC}\\cong NP.", "definition": "The proposed identification between the WCC resource model and a complexity claim. It remains conditional on encoding, precision, update-cost, and simulation-overhead assumptions.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e40-wcc-complexity-identification" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E40" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E41", "name": "Curvature-bounded configuration count", "family": "F. WCC, channel capacity, and complexity", "formula": "|C_{\\rm curv}(n)| \\le 2^{\\alpha(n)n}, \\qquad \\alpha(n)<1.", "definition": "Current effective status: ⚠️ CONDITIONAL E28 alone does not prove this counting bound; an injective coding or combinatorial argument is required.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_alpha_drop_count_bound" ], "baseline_checker": "check_alpha_drop_count_bound", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e41-curvature-bounded-configuration-count" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E41" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E42", "name": "Theta-information relation", "family": "F. WCC, channel capacity, and complexity", "formula": "\\frac{d}{dt}I_{\\rm coh}[\\psi] = -\\lambda_I \\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx.", "definition": "Current effective status: ○ OPEN The information functional and coupling \\(\\lambdaI\\) require derivation.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e42-theta-information-relation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E42" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E43", "name": "Curvature-entropy tradeoff", "family": "F. WCC, channel capacity, and complexity", "formula": "\\frac{dH_k}{dt} \\le -\\mu \\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx, \\qquad \\mu>0.", "definition": "Current effective status: ○ OPEN This remains an analytic/empirical claim.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e43-curvature-entropy-tradeoff" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E43" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E44", "name": "Theta eigenmode problem", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\Theta_\\varepsilon[\\psi_n] = \\lambda_n\\psi_n.", "definition": "Current effective status: ◻️ DEFINITION Because \\(\\Theta\\varepsilon\\) is nonlinear, the spectral problem and normalization must be specified carefully.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e44-theta-eigenmode-problem" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E44" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E45", "name": "Corrected quality factor", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\boxed{ Q_{\\rm eff} = \\omega\\frac{U}{P_{\\rm loss}} }\n\nU=\\int_\\Omega u\\,dV", "definition": "Current effective status: ✅ PASS where", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_q_factor_dimensions" ], "baseline_checker": "check_q_factor_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "effectiveQualityFactor", "effectiveQualityFactor_positive" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e45-corrected-quality-factor" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E45" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E46", "name": "Plasma-cavity curvature match", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\langle\\sigma\\rangle_{w,\\rm plasma} \\approx \\langle\\sigma\\rangle_{w,\\rm cavity}.", "definition": "Current effective status: ○ OPEN A measurable matching tolerance and transfer mechanism remain open.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e46-plasma-cavity-curvature-match" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E46" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E47", "name": "Corrected power balance", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\boxed{ \\frac{dW}{dt} = P_{\\rm in} + P_{\\rm fusion} - P_{\\rm loss} - P_{\\rm out} }.\n\nP_{\\rm in}+P_{\\rm fusion} = P_{\\rm loss}+P_{\\rm out}.", "definition": "Current effective status: ✅ PASS At stationarity,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_power_balance_form" ], "baseline_checker": "check_power_balance_form", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "powerBalanceResidual", "stationaryPowerBalance_iff" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e47-corrected-power-balance" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E47" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E48", "name": "Curvature-gap stability criterion", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\Delta\\sigma = \\langle\\sigma\\rangle_{\\rm core} - \\langle\\sigma\\rangle_{\\rm edge} > \\Delta_{\\rm crit}.", "definition": "Current effective status: ⚠️ CONDITIONAL The threshold and direction of the inequality must be calibrated to a specified stability observable.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e48-curvature-gap-stability-criterion" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E48" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E49", "name": "Corrected effective-mass gap law", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\omega_j^2=c^2\\lambda_j+\\Delta_\\omega^\\star, \\qquad [\\Delta_\\omega^\\star]=T^{-2},\n\n\\omega^2=c^2k^2+\\frac{m_{\\rm eff}^2c^4}{\\hbar^2}\n\n\\boxed{ m_{\\rm eff}^2 = \\frac{\\hbar^2}{c^4}\\Delta_\\omega^\\star }.", "definition": "The dimensionally corrected relation between an effective spectral gap and squared effective mass. Dimensional consistency does not determine the gap dynamically or calibrate an observed mass spectrum.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_effective_mass_gap_dimensions" ], "baseline_checker": "check_effective_mass_gap_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "effectiveMassSqFromGap", "effectiveMassSqFromGap_eq" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Definitional identity; does not derive a physical mass gap." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e49-corrected-effective-mass-gap-law" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E49" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E50", "name": "Phase-coherence functional", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\mathcal C[\\psi] = \\int_\\Omega \\frac{|\\psi|^2}{|\\nabla\\theta|}\\,dx.\n\n|\\nabla\\theta|\\ge\\delta>0", "definition": "Current effective status: ⚠️ CONDITIONAL The definition requires a regularization or lower bound", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e50-phase-coherence-functional" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E50" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E51", "name": "Curvature-gradient commutator", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\Theta_D[\\psi]:=-\\frac{\\Delta\\psi}{D}.\n\n[\\nabla,\\Theta_D]\\psi := \\nabla(\\Theta_D[\\psi])-\\Theta_D[\\nabla\\psi] = \\frac{\\Delta\\psi}{D^2}\\nabla D.", "definition": "Current effective status: ✅ PASS For a smooth scalar denominator \\(D\\neq0\\), define", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_curvature_gradient_commutator" ], "baseline_checker": "check_curvature_gradient_commutator", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "curvatureGradientCommutator", "curvatureGradientCommutator_eq" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Scalar component under a nonzero-denominator hypothesis." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e51-curvature-gradient-commutator" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E51" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E52", "name": "Curvature gain and gradient loss", "family": "G. Cavity, effective mass, and phase structure", "formula": "G_\\sigma:=\\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx, \\qquad L_\\sigma:=\\int_\\Omega|\\nabla\\psi|^2dx.", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e52-curvature-gain-and-gradient-loss" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E52" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E53", "name": "Curvature pressure density", "family": "G. Cavity, effective mass, and phase structure", "formula": "p_\\Theta(x) := c_2|\\Theta_\\varepsilon[\\psi](x)|^2.", "definition": "Current effective status: ✅ PASS It is the local curvature contribution to E18.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_pressure_density_embedding" ], "baseline_checker": "check_pressure_density_embedding", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "curvaturePressureDensity", "curvaturePressureDensity_nonnegative" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e53-curvature-pressure-density" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E53" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E54", "name": "Resonance-lock condition", "family": "G. Cavity, effective mass, and phase structure", "formula": "\\partial_t\\psi=0, \\qquad \\delta\\mathcal E[\\psi]=0, \\qquad \\nabla\\Theta_\\varepsilon[\\psi]=0.", "definition": "Current effective status: ⚠️ CONDITIONAL Simultaneous satisfaction requires existence and regularity results.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e54-resonance-lock-condition" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E54" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E55", "name": "Curvature-induced effective potential", "family": "G. Cavity, effective mass, and phase structure", "formula": "V_{\\rm eff}(\\psi) = V(|\\psi|^2) + \\kappa|\\Theta_\\varepsilon[\\psi]|^2.", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e55-curvature-induced-effective-potential" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E55" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E56", "name": "Phase-wall criterion", "family": "G. Cavity, effective mass, and phase structure", "formula": "|\\nabla\\theta|_{\\rm wall} \\sim \\sigma_{\\rm wall} \\gg \\langle\\sigma\\rangle_w.", "definition": "Current effective status: ⚠️ CONDITIONAL The comparison scale and wall-detection threshold must be defined.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e56-phase-wall-criterion" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E56" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E57", "name": "Swift-Hohenberg shell operator", "family": "H. Swift-Hohenberg and spectral projection", "formula": "\\mathcal{SH}[A] = (\\Delta+k_\\star^2)^2A.\n\n(|k|^2-k_\\star^2)^2.", "definition": "Current effective status: ✅ PASS Its Fourier symbol is", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_sh_fourier_symbol" ], "baseline_checker": "check_sh_fourier_symbol", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "finiteBandPenalty_at_shell" ], "source": "WCTLean/Models/BandPass.lean", "limitations": [ "Scalar Fourier-shell identity only." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e57-swift-hohenberg-shell-operator" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E57" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E58", "name": "Band-selective Green kernel", "family": "H. Swift-Hohenberg and spectral projection", "formula": "\\mathcal L=r+a(\\Delta+k_\\star^2)^2,\n\nG(k) = \\frac1{r+a(|k|^2-k_\\star^2)^2}.", "definition": "The band-selective Green kernel. A positive spectral offset yields positivity and the bound G(k)≤1/r; the offset and its physical interpretation remain model assumptions.", "status": { "baseline": "CONDITIONAL", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_green_kernel_bounded" }, "verification": { "outcome": "PASS", "kind": "LIMIT_CHECK", "scope": "MODEL_CONDITIONAL", "checker": [ "check_green_kernel_bounded" ], "baseline_checker": "check_green_kernel_pole", "meaning": "A strictly positive spectral offset bounds the declared Green kernel and removes the registered pole.", "limitations": [] }, "assumptions": [ "ASM-POSITIVE-SPECTRAL-OFFSET" ], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "bandGreenDenominator_pos", "bandGreenDenominator_ge_offset", "bandGreenKernel_pos", "bandGreenKernel_le_inverseOffset", "bandGreenKernel_at_shell" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e58-band-selective-green-kernel" }, "dependencies": { "equations": [], "assumptions": [ "ASM-POSITIVE-SPECTRAL-OFFSET" ], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E58" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E59", "name": "Projection onto a dominant annulus", "family": "H. Swift-Hohenberg and spectral projection", "formula": "\\mathcal A^\\star := \\left\\{ k\\in\\mathbb Z^d: \\bigl||k|-k_\\star\\bigr|\\le\\Delta k \\right\\},\n\n(P_{k_\\star}A)(x) = \\sum_{k\\in\\mathcal A^\\star} \\widehat A_ke^{ik\\cdot x}.\n\nP_{k_\\star}^2=P_{k_\\star}.", "definition": "Current effective status: ✅ PASS With a fixed annulus,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_projection_idempotence" ], "baseline_checker": "check_projection_idempotence", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "maskProjection", "maskProjection_idempotent" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Coordinate-mask model; not a complete Hilbert-space Fourier projection theorem." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e59-projection-onto-a-dominant-annulus" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E59" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E60", "name": "Center-manifold amplitude equation", "family": "H. Swift-Hohenberg and spectral projection", "formula": "\\partial_T\\mathcal A = \\mu\\mathcal A-g|\\mathcal A|^2\\mathcal A.", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e60-center-manifold-amplitude-equation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E60" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E61", "name": "Pattern-formation threshold", "family": "H. Swift-Hohenberg and spectral projection", "formula": "r_c = \\min_k a(|k|^2-k_\\star^2)^2 = 0.", "definition": "Current effective status: ✅ PASS In the continuum,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SIGN_OR_EXTREMUM_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_pattern_threshold" ], "baseline_checker": "check_pattern_threshold", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "swiftHohenbergLinearSymbol_at_shell", "selectedShell_positive_at_onset" ], "source": "WCTLean/Models/BandPass.lean", "limitations": [ "Linear symbol statement only." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e61-pattern-formation-threshold" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E61" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E62", "name": "Spectral energy concentration", "family": "H. Swift-Hohenberg and spectral projection", "formula": "\\eta(t) = \\frac{ \\sum_{k\\in\\mathcal A^\\star}|\\widehat A_k|^2 }{ \\sum_k|\\widehat A_k|^2 }, \\qquad 0\\le\\eta(t)\\le1.", "definition": "Current effective status: ✅ PASS For nonzero total spectral energy,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_spectral_fraction_bounds" ], "baseline_checker": "check_spectral_fraction_bounds", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "spectralEnergyFraction", "spectralEnergyFraction_mem_unitInterval" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e62-spectral-energy-concentration" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E62" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E63", "name": "Entropic mode selection", "family": "H. Swift-Hohenberg and spectral projection", "formula": "k_\\star = \\operatorname*{arg\\,min}_k \\left[ H_k+\\lambda_\\Theta C_\\Theta(k) \\right].", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e63-entropic-mode-selection" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E63" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E64", "name": "Corrected selected wavelength", "family": "H. Swift-Hohenberg and spectral projection", "formula": "k_\\star=\\sqrt{\\frac{a}{2b}}.\n\n\\boxed{ \\lambda_\\star = \\frac{2\\pi}{k_\\star} = 2\\pi\\sqrt{\\frac{2b}{a}} }.", "definition": "Current effective status: ✅ PASS From E12,", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "CONSISTENCY_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_e12_e64_consistency" ], "baseline_checker": "check_e12_e64_consistency", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "selectedWavenumberSq", "selectedWavelength", "selectedWavelength_positive" ], "source": "WCTLean/Models/BandPass.lean", "limitations": [ "Positivity and formula typing only; no measured wavelength prediction." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e64-corrected-selected-wavelength" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E64" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E65", "name": "Critical Sobolev exponent", "family": "I. Sobolev structure and dimensional bounds", "formula": "p_c(n)=\\frac{n+2}{n-2}.", "definition": "Current effective status: ✅ PASS For \\(n>2\\),", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_critical_sobolev_exponent" ], "baseline_checker": "check_critical_sobolev_exponent", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declarations": [ "criticalSobolevExponent" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Typed definition only." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e65-critical-sobolev-exponent" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E65" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E66", "name": "Gagliardo-Nirenberg interpolation", "family": "I. Sobolev structure and dimensional bounds", "formula": "\\|u\\|_{L^p} \\le C \\|\\nabla u\\|_{L^2}^{\\theta} \\|u\\|_{L^2}^{1-\\theta}.", "definition": "Current effective status: ⚠️ CONDITIONAL The allowed \\(p,\\theta,n\\), domain, and boundary assumptions must be specified.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "analytic_contract_only", "declarations": [ "GagliardoNirenbergBound" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Proposition encoding the required inequality; no proof of the analytic estimate." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e66-gagliardo-nirenberg-interpolation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E66" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E67", "name": "Failure of $H^2 o L^\\infty$ above three dimensions", "family": "I. Sobolev structure and dimensional bounds", "formula": "", "definition": "The failure, in general, of the H²-to-L∞ embedding above three spatial dimensions. This blocks that regularity route but does not exclude every possible higher-dimensional confinement mechanism.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "FORMAL_THEOREM", "scope": "STANDARD_MATHEMATICS", "checker": [ "check_h2_embedding_threshold" ], "baseline_checker": "check_h2_embedding_threshold", "meaning": "The general H2-to-L-infinity embedding route fails above three spatial dimensions under the declared Sobolev framework.", "limitations": [ "Failure of this regularity route does not exclude every conceivable higher-dimensional confinement mechanism." ] }, "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN" ], "formalization": { "status": "PROVED", "relationship": "supporting_lemma_only", "declarations": [ "h2Subcritical_iff_le_three", "SobolevCurvatureContract" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Arithmetic equivalence is proved; embedding is an explicit contract assumption." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e67-failure-of-h2-o-linfty-above-three-dimensions" }, "dependencies": { "equations": [], "assumptions": [ "ASM-STANDARD-SOBOLEV-DOMAIN" ], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E67" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E68", "name": "Localized energy estimate", "family": "I. Sobolev structure and dimensional bounds", "formula": "\\int_{B_R}|\\nabla\\psi|^2dx \\le CR^{n-2}\\|\\psi\\|_{H^1}^2.", "definition": "Current effective status: ⚠️ CONDITIONAL A model-dependent localized estimate is", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "analytic_contract_only", "declarations": [ "LocalEnergyBound" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Proposition definition only; the localized estimate is not proved." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e68-localized-energy-estimate" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E68" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E69", "name": "Corrected high-regularity curvature bound", "family": "I. Sobolev structure and dimensional bounds", "formula": "\\psi\\in H^s(\\Omega), \\qquad s>\\frac n2+2,\n\n\\Theta_\\varepsilon[\\psi]\\in L^\\infty(\\Omega).", "definition": "Current effective status: ✅ PASS If", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_theta_linf_from_high_regularity" ], "baseline_checker": "check_theta_linf_from_high_regularity", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "analytic_contract_only", "declarations": [ "SobolevCurvatureContract" ], "source": "WCTLean/Contracts/Analytic.lean", "limitations": [ "Contract only; no analytic boundedness proof." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e69-corrected-high-regularity-curvature-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E69" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E70", "name": "Dimensional stability criterion", "family": "I. Sobolev structure and dimensional bounds", "formula": "n\\le3, \\qquad H^2\\hookrightarrow L^\\infty, \\qquad p1.", "definition": "Current effective status: ○ OPEN The finite-size families \\(Pn,NPn\\) must first be defined.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e74-curvature-separation-conjecture" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E74" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E75", "name": "Physical-oracle impossibility", "family": "J. Computational resource bounds", "formula": "\\nexists\\, O: O(\\psi)=\\operatorname*{arg\\,min}_\\psi\\mathcal E[\\psi] \\quad\\text{in polynomial time}.", "definition": "Current effective status: ○ OPEN This is a complexity claim requiring a formal computational model and reduction.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e75-physical-oracle-impossibility" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E75" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E76", "name": "WCC complexity equivalence", "family": "J. Computational resource bounds", "formula": "P_{\\rm WCC}=P \\quad\\Longrightarrow\\quad \\text{WCC polynomially simulates the declared physical-computation model}.", "definition": "A conditional equivalence claim between curvature-bounded computation and a classical complexity description. The equivalence is not established without explicit simulation and overhead bounds.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e76-wcc-complexity-equivalence" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E76" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E77", "name": "Mutual-information decay", "family": "K. Entropy and information dynamics", "formula": "\\frac{d}{dt}I(\\psi_t;\\psi_0) \\le -\\gamma\\mathcal E_\\Theta[\\psi_t].", "definition": "Current effective status: ○ OPEN The probability law, channel, and regularity assumptions remain open.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e77-mutual-information-decay" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E77" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E78", "name": "Fisher-information curvature bound", "family": "K. Entropy and information dynamics", "formula": "\\mathcal I_F[\\psi] \\ge c\\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx.", "definition": "Current effective status: ○ OPEN A common probability density and geometric derivation are required.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "STATED_TODO", "relationship": "adjacent_todo_only", "declarations": [ "cos_orthogonality_TODO" ], "source": "WCTLean/Fourier.lean", "limitations": [ "The declaration records a proposition and is not a proof." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e78-fisher-information-curvature-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E78" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E79", "name": "Entropy-production rate", "family": "K. Entropy and information dynamics", "formula": "\\dot\\Sigma = \\frac{dH_k}{dt} + \\frac{\\mathcal E_\\Theta}{T_{\\rm eff}}.", "definition": "Current effective status: ◻️ DEFINITION The sign convention and units must be fixed when used physically.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e79-entropy-production-rate" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E79" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E80", "name": "Landauer-type bound", "family": "K. Entropy and information dynamics", "formula": "\\Delta E \\ge k_BT_{\\rm eff}\\ln2\\, \\Delta H_{\\rm bits}.", "definition": "Current effective status: ⚠️ CONDITIONAL If \\(\\Delta H{\\rm bits}\\) is measured in bits,", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "DIMENSIONAL_CHECK", "scope": "MODEL_CONDITIONAL", "checker": [ "check_landauer_units" ], "baseline_checker": "check_landauer_units", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e80-landauer-type-bound" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E80" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E81", "name": "Corrected coherence length", "family": "K. Entropy and information dynamics", "formula": "\\boxed{ \\xi_{\\rm coh} = \\left( \\sum_kp_k|k|^2 \\right)^{-1/2} }.\n\n\\boxed{ \\xi_{\\rm coh} = \\sqrt{ \\frac{\\int_\\Omega|\\psi|^2dx} {\\int_\\Omega|\\nabla\\psi|^2dx} } }.", "definition": "Current effective status: ✅ PASS With normalized spectral weights \\(pk\\),", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_coherence_length_dimensions" ], "baseline_checker": "check_coherence_length_dimensions", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declarations": [ "spectralCoherenceLength", "gradientCoherenceLength" ], "source": "WCTLean/Models/AlgebraicChecks.lean", "limitations": [ "Definitions only; equivalence between the two coherence measures is not proved." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e81-corrected-coherence-length" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E81" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "E82", "name": "Information-geometry tensor", "family": "K. Entropy and information dynamics", "formula": "g_{ij}^{({\\rm info})} = \\left\\langle \\partial_i\\Theta_\\varepsilon\\, \\partial_j\\Theta_\\varepsilon \\right\\rangle.", "definition": "Current effective status: ◻️ DEFINITION Positive definiteness and coordinate invariance require additional conditions.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e82-information-geometry-tensor" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "E82" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE1", "name": "Curvature-locking functional", "family": "Curvature-locking equations", "formula": "S[\\psi] = \\int_\\mathcal M \\left[ |\\nabla\\psi|^2 + |W_\\psi-\\sigma_\\star^2|^2 \\right]\\sqrt g\\,d^3x, \\qquad W_\\psi:=-\\frac{\\Delta\\psi}{\\psi}.", "definition": "Current effective status: ◻️ DEFINITION Use the inverse-length convention \\([\\sigma\\star]=L^{-1}\\):", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle1-curvature-locking-functional" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE1" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE2", "name": "Corrected curvature-lock Euler-Lagrange equation", "family": "Curvature-locking equations", "formula": "q:=-\\frac{\\psi_{xx}}{\\psi}-\\sigma_\\star^2,\n\n\\boxed{ q\\frac{\\psi_{xx}}{\\psi^2} -\\psi_{xx} -\\frac{d^2}{dx^2}\\left(\\frac q\\psi\\right) =0 }.", "definition": "Current effective status: ✅ PASS For the real one-dimensional reduction", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "VARIATIONAL_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_cle2_variation" ], "baseline_checker": "check_cle2_variation", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle2-corrected-curvature-lock-euler-lagrange-equation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE3", "name": "Curvature-locking condition", "family": "Curvature-locking equations", "formula": "W_\\psi=\\sigma_\\star^2.", "definition": "Current effective status: ◻️ DEFINITION Both sides have units \\(L^{-2}\\).", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle3-curvature-locking-condition" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE4", "name": "Locked-field equation", "family": "Curvature-locking equations", "formula": "-\\Delta\\psi = \\sigma_\\star^2\\psi.", "definition": "The canonical registered object for locked-field equation; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_cle_units_chain" ], "baseline_checker": "check_cle_units_chain", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle4-locked-field-equation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE4" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE5", "name": "Thin/product-torus Laplacian", "family": "Curvature-locking equations", "formula": "\\Delta\\psi \\approx \\frac1{R^2}\\partial_\\theta^2\\psi + \\frac1{r^2}\\partial_\\phi^2\\psi.", "definition": "Current effective status: ⚠️ CONDITIONAL Under a flat product or thin-torus approximation,", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle5-thinproduct-torus-laplacian" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE5" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE6", "name": "Separation ansatz", "family": "Curvature-locking equations", "formula": "\\psi(\\theta,\\phi)=f(\\theta)g(\\phi),\n\n\\frac{f''}{f} + \\frac{R^2}{r^2}\\frac{g''}{g} = -\\sigma_\\star^2R^2.", "definition": "Current effective status: ✅ PASS For", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_separation_substitution" ], "baseline_checker": "check_separation_substitution", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle6-separation-ansatz" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE6" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE7", "name": "Periodic angular mode family", "family": "Curvature-locking equations", "formula": "f''+m^2f=0\n\n\\boxed{ f(\\theta)=A\\cos(m\\theta)+B\\sin(m\\theta), \\qquad m\\in\\mathbb Z_{\\ge0} }.", "definition": "Current effective status: ✅ PASS The periodic reduced equation", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_periodic_ode_family" ], "baseline_checker": "check_periodic_ode_family", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle7-periodic-angular-mode-family" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE7" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE8", "name": "Selected torus eigenmode", "family": "Curvature-locking equations", "formula": "\\psi(\\theta,\\phi)=Ae^{i\\phi}", "definition": "Current effective status: ⚠️ CONDITIONAL is one admissible winding-one mode. Uniqueness requires additional lowest-mode, chirality, normalization, phase, and boundary-selection principles.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_torus_eigenmode_selection" ], "baseline_checker": "check_torus_eigenmode_selection", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle8-selected-torus-eigenmode" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE8" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE9", "name": "Electron radius from curvature", "family": "Curvature-locking equations", "formula": "R=\\frac1{\\sigma_\\star}.\n\nR=\\frac{\\hbar}{m_ec}\\approx386.16\\ {\\rm fm}.", "definition": "Current effective status: ✅ PASS For \\(\\sigma\\star=mec/\\hbar\\),", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_cle_units_chain" ], "baseline_checker": "check_cle_units_chain", "meaning": "The registered electron-radius expression has the dimensions and numerical scale of the reduced Compton wavelength.", "limitations": [ "Reproducing a known length scale does not establish an independent WCT derivation of electron structure." ] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle9-electron-radius-from-curvature" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE9" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CLE10", "name": "Curvature scalar chain", "family": "Curvature-locking equations", "formula": "\\boxed{ W_\\psi = -\\frac{\\Delta\\psi}{\\psi} = \\sigma_\\star^2 }, \\qquad R=\\sigma_\\star^{-1}.", "definition": "The canonical registered object for curvature scalar chain; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "DIMENSIONAL_CHECK", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_cle_units_chain" ], "baseline_checker": "check_cle_units_chain", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle10-curvature-scalar-chain" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CLE10" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "G1", "name": "Log-periodic ghost modulation", "family": "Logarithmic and ghost equations", "formula": "\\delta_g(E) = A_g\\cos\\!\\left( k_\\ell\\ln\\frac E{E_0}+\\phi \\right),\n\n|\\delta_g(E)|\\le|A_g|.", "definition": "Current effective status: ✅ PASS For \\(E>0\\) and \\(E0>0\\),", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_log_modulation" ], "baseline_checker": "check_log_modulation", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "ghostMode", "abs_ghostMode_le" ], "source": "WCTLean/Models/GhostModes.lean", "limitations": [ "The global amplitude bound is proved; no empirical fit or physical-origin theorem is claimed." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#g1-log-periodic-ghost-modulation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "G1" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "EX", "name": "Logarithmic field representation", "family": "Logarithmic and ghost equations", "formula": "u=\\ln\\psi, \\qquad \\psi=e^u.\n\n\\nabla\\psi=e^u\\nabla u,\n\n\\Delta\\psi=e^u(\\Delta u+|\\nabla u|^2),\n\n\\frac{\\Delta\\psi}{\\psi} = \\Delta u+|\\nabla u|^2.", "definition": "Current effective status: ✅ PASS For a positive real field \\(\\psi>0\\), let", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_log_laplacian_identity" ], "baseline_checker": "check_log_laplacian_identity", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "logField", "exp_logField_eq" ], "source": "WCTLean/Models/LogFlow.lean", "limitations": [ "Restricted to positive real field values." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#ex-logarithmic-field-representation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "EX" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "EY", "name": "Log-curvature evolution", "family": "Logarithmic and ghost equations", "formula": "\\partial_tu = \\Delta u+|\\nabla u|^2,", "definition": "Current effective status: ✅ PASS If", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_log_flow_reduction" ], "baseline_checker": "check_log_flow_reduction", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "logCurvatureResidual", "coleHopf_residual_factorization", "diffusionResidual_zero_iff_logCurvatureResidual_zero" ], "source": "WCTLean/Models/LogFlow.lean", "limitations": [ "Temporal and Laplacian chain rules are explicit hypotheses; function-space regularity remains open." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#ey-log-curvature-evolution" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "EY" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "EZ", "name": "Cole-Hopf reduction", "family": "Logarithmic and ghost equations", "formula": "\\psi=e^u,\n\n\\partial_t\\psi = e^u\\partial_tu = e^u(\\Delta u+|\\nabla u|^2) = \\Delta\\psi.\n\n\\boxed{\\partial_t\\psi=\\Delta\\psi}.", "definition": "Current effective status: ✅ PASS With", "status": { "baseline": "PASS", "effective": "PASS", "changed": false, "changed_by": null }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "STANDARD_MATHEMATICS", "checker": [ "check_cole_hopf" ], "baseline_checker": "check_cole_hopf", "meaning": "The registered Cole-Hopf substitution reduces the encoded nonlinear equation under its stated assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "diffusionResidual", "coleHopf_residual_factorization", "diffusionResidual_zero_iff_logCurvatureResidual_zero" ], "source": "WCTLean/Models/LogFlow.lean", "limitations": [ "The algebraic bridge is proved under displayed hypotheses; PDE existence and regularity are not." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#ez-cole-hopf-reduction" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "EZ" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "FA", "name": "Filament-localization condition", "family": "Logarithmic and ghost equations", "formula": "|\\nabla u| \\sim \\kappa_{\\rm core}.", "definition": "Current effective status: ⚠️ CONDITIONAL A norm, tolerance, scale, and dynamical derivation are required.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "FilamentLocalizationCondition", "filamentLocalizationMismatch", "filamentLocalizationCondition_iff_mismatch_zero" ], "source": "WCTLean/Models/LogFlow.lean", "limitations": [ "Scalar equivalence only; no filament-existence or localization theorem is claimed." ] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#fa-filament-localization-condition" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "FA" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM1", "name": "Fundamental field evolution", "family": "Curvature-acoustic cosmology", "formula": "i\\partial_t\\psi = -\\Theta_\\varepsilon[\\psi]\\,J[\\psi],\n\nJ[\\psi] = |\\psi|^2\\Delta\\psi\\,\\varepsilon_{\\rm vac}.", "definition": "Current effective status: ○ OPEN Coefficient dimensions and derivation remain open.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm1-fundamental-field-evolution" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM1" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM2", "name": "Curvature-spectral tilt", "family": "Curvature-acoustic cosmology", "formula": "P_{\\rm prim}(k)\\propto k^{-\\alpha_{\\rm WCT}},\n\nn_s-1=-\\alpha_{\\rm WCT},\n\n\\alpha_{\\rm WCT} = -\\frac{d\\ln|\\Theta(k)|}{d\\ln k}.", "definition": "The canonical registered object for curvature-spectral tilt; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm2-curvature-spectral-tilt" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM3", "name": "Potential from curvature", "family": "Curvature-acoustic cosmology", "formula": "\\Phi(k,t) = -C_\\Phi\\frac{\\Theta(k,t)}{k^2}.", "definition": "The canonical registered object for potential from curvature; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm3-potential-from-curvature" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM4", "name": "Horizon-entry potential decay", "family": "Curvature-acoustic cosmology", "formula": "\\partial_t\\Phi = -\\Gamma\\Phi,\n\n\\Gamma(k,t) = \\left| \\frac{\\partial_t\\Theta(k,t)}{\\Theta(k,t)} \\right|,", "definition": "Current effective status: ○ OPEN on the domain \\(\\Theta\\neq0\\).", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm4-horizon-entry-potential-decay" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM4" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM5", "name": "Curvature-acoustic oscillators", "family": "Curvature-acoustic cosmology", "formula": "\\ddot\\delta_\\gamma +c_s^2k^2\\delta_\\gamma = -k^2\\Phi,\n\n\\ddot\\delta_b +\\mathcal R\\,c_s^2k^2\\delta_\\gamma = -k^2\\Phi,\n\n\\mathcal R = \\frac{E_{\\rm comp}}{E_{\\rm rad}}.", "definition": "The canonical registered object for curvature-acoustic oscillators; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm5-curvature-acoustic-oscillators" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM5" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM6", "name": "Sound speed from curvature feedback", "family": "Curvature-acoustic cosmology", "formula": "c_s^2(t) = \\frac1{3(1+\\mathcal R(t))} \\left[ 1-\\beta_{\\rm curv} \\frac{E_{\\rm curv}(t)}{E_{\\rm tot}} \\right].", "definition": "Current effective status: ○ OPEN Positivity requires the bracketed factor to be nonnegative.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm6-sound-speed-from-curvature-feedback" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM6" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM7", "name": "Curvature diffusion", "family": "Curvature-acoustic cosmology", "formula": "\\dot\\delta_\\gamma = v_\\gamma - D_{\\rm curv}(t)k^2\\delta_\\gamma,\n\nD_{\\rm curv}(t) = \\frac{\\langle|\\nabla\\psi|^2\\rangle} {\\langle|\\psi|^2\\rangle}.", "definition": "Current effective status: ○ OPEN A phenomenological damping replacement is", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm7-curvature-diffusion" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM7" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM8", "name": "Initial conditions", "family": "Curvature-acoustic cosmology", "formula": "\\delta_\\gamma(0)=\\delta_b(0)=-2\\Phi(k,0),\n\n\\Phi(k,0) = -C_\\Phi\\frac{\\Theta(k,0)}{k^2}.", "definition": "Current effective status: ○ OPEN Use CM3 consistently:", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm8-initial-conditions" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM8" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM9", "name": "First-order mode system", "family": "Curvature-acoustic cosmology", "formula": "\\dot\\delta_\\gamma=v_\\gamma, \\qquad \\dot v_\\gamma=-c_s^2k^2\\delta_\\gamma-k^2\\Phi,\n\n\\dot\\delta_b=v_b, \\qquad \\dot v_b=-\\mathcal R c_s^2k^2\\delta_\\gamma-k^2\\Phi.", "definition": "Current effective status: ✅ PASS Baseline status: ○ OPEN", "status": { "baseline": "OPEN", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_cm9_first_order_equivalence" }, "verification": { "outcome": "PASS", "kind": "ALGEBRAIC_IDENTITY", "scope": "INTERNAL_CONSISTENCY", "checker": [ "check_cm9_first_order_equivalence" ], "baseline_checker": "classify_open", "meaning": "The first-order velocity system is equivalent to the registered second-order oscillator system.", "limitations": [] }, "assumptions": [], "formalization": { "status": "PROVED", "relationship": "formalizes_same_claim", "declarations": [ "photonSecondOrder_iff_firstOrder", "baryonSecondOrder_iff_firstOrder" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm9-first-order-mode-system" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM9" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM10", "name": "Tight-coupling drag", "family": "Curvature-acoustic cosmology", "formula": "\\delta_b \\leftarrow (1-\\varepsilon_{\\rm drag})\\delta_b + \\varepsilon_{\\rm drag}\\delta_\\gamma,\n\n\\varepsilon_{\\rm drag} = \\frac{E_{\\rm exch}}{E_{\\rm comp}}, \\qquad 0\\le\\varepsilon_{\\rm drag}\\le1.", "definition": "The canonical registered object for tight-coupling drag; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm10-tight-coupling-drag" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM10" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM11", "name": "Curvature damping envelope", "family": "Curvature-acoustic cosmology", "formula": "D(k) = \\exp\\!\\left(-\\frac{k^2}{k_D^2}\\right),\n\nk_D^{-2} = \\int_0^{t_\\star}D_{\\rm curv}(t)\\,dt.", "definition": "Current effective status: ✅ PASS Baseline status: ○ OPEN", "status": { "baseline": "OPEN", "effective": "PASS", "changed": true, "changed_by": "derived_overrides.yaml:check_cm11_gaussian_damping" }, "verification": { "outcome": "PASS", "kind": "SYMBOLIC_DERIVATION", "scope": "MODEL_CONDITIONAL", "checker": [ "check_cm11_gaussian_damping" ], "baseline_checker": "classify_open", "meaning": "The registered curvature-diffusion equation integrates to the stated Gaussian damping envelope.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm11-curvature-damping-envelope" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM11" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM12", "name": "Dimensionless power spectrum", "family": "Curvature-acoustic cosmology", "formula": "\\Delta^2(k) = \\frac{k^3}{2\\pi^2}P(k).", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "status": { "baseline": "OPEN", "effective": "DEFINITION", "changed": true, "changed_by": "derived_overrides.yaml:classify_definition" }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_open", "meaning": "The object defines a dimensionless power-spectrum quantity and is not itself a cosmological derivation.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declaration_type": "definition", "declarations": [ "dimensionlessPowerSpectrum" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "Kernel acceptance of a definition is not a cosmological derivation." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm12-dimensionless-power-spectrum" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM12" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM13", "name": "Peak metrics", "family": "Curvature-acoustic cosmology", "formula": "r_{21} = \\frac{P(k_2)}{P(k_1)}, \\qquad r_{31} = \\frac{P(k_3)}{P(k_1)},\n\ns_{21} = \\frac{k_2}{k_1}, \\qquad s_{31} = \\frac{k_3}{k_1}.", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "status": { "baseline": "OPEN", "effective": "DEFINITION", "changed": true, "changed_by": "derived_overrides.yaml:classify_definition" }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_open", "meaning": "The object defines peak-ratio bookkeeping quantities and is not itself a cosmological derivation.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declaration_type": "definition", "declarations": [ "peakPowerRatio21", "peakPowerRatio31", "peakScaleRatio21", "peakScaleRatio31" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "Kernel acceptance of these definitions is not a fit to cosmological data." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm13-peak-metrics" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM13" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM14", "name": "Peak-response interpretation", "family": "Curvature-acoustic cosmology", "formula": "\\text{faster }\\Theta\\text{ decay}\\Rightarrow s_{ij}\\uparrow,\n\n\\text{larger compression}\\Rightarrow r_{31}\\uparrow,\n\n\\text{larger radiative fraction}\\Rightarrow r_{21}\\downarrow.", "definition": "Current effective status: ○ OPEN Proposed qualitative relations:", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm14-peak-response-interpretation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM14" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM15", "name": "WCT angular scaling", "family": "Curvature-acoustic cosmology", "formula": "k_{\\rm phys} = \\frac{k}{a_{\\rm WCT}(t)},\n\na_{\\rm WCT}(t) = \\left[ \\frac{E_{\\rm curv}(0)} {E_{\\rm curv}(t)} \\right]^{1/3}.", "definition": "The canonical registered object for wct angular scaling; consult the source equation and verification metadata for its assumptions and scientific boundary.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm15-wct-angular-scaling" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM15" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM16", "name": "Acoustic horizon", "family": "Curvature-acoustic cosmology", "formula": "R_{\\rm hor}(t) = \\int_0^tc_s(t')\\,dt',\n\nk_{\\rm hor} = \\frac{2\\pi}{R_{\\rm hor}}.", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "status": { "baseline": "OPEN", "effective": "DEFINITION", "changed": true, "changed_by": "derived_overrides.yaml:classify_definition" }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_open", "meaning": "The object defines horizon-scale quantities and is not itself a cosmological derivation.", "limitations": [] }, "assumptions": [], "formalization": { "status": "DEFINITION", "relationship": "definition_only", "declaration_type": "definition", "declarations": [ "horizonWavenumber" ], "source": "WCTLean/DerivedAudit.lean", "limitations": [ "The horizon integral and cosmological dynamics are not formalized here." ] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm16-acoustic-horizon" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM16" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM17", "name": "Curvature-energy closure", "family": "Curvature-acoustic cosmology", "formula": "E_{\\rm curv}(t) + E_{\\rm grad}(t) = E_{\\rm tot},", "definition": "Current effective status: ○ OPEN for a closed sector with no external source or loss.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm17-curvature-energy-closure" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM17" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM18", "name": "Minimal cosmology closure set", "family": "Curvature-acoustic cosmology", "formula": "\\mathfrak C_{\\rm min} = \\{\\mathrm{CM1},\\mathrm{CM2},\\mathrm{CM3}, \\mathrm{CM4},\\mathrm{CM5},\\mathrm{CM7}\\}.", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "status": { "baseline": "OPEN", "effective": "DEFINITION", "changed": true, "changed_by": "derived_overrides.yaml:classify_definition" }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_open", "meaning": "The object defines a closure set for bookkeeping and is not a derivation of cosmology.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm18-minimal-cosmology-closure-set" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM18" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM19", "name": "Acoustic speed from curvature equation of state", "family": "Curvature-acoustic cosmology", "formula": "c_s^2 = \\frac{\\partial P_{\\rm curv}} {\\partial\\rho_{\\rm curv}},", "definition": "Current effective status: ○ OPEN where the derivative must be taken along a specified thermodynamic or dynamical path.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm19-acoustic-speed-from-curvature-equation-of-state" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM19" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CM20", "name": "Theta-based expansion ansatz", "family": "Curvature-acoustic cosmology", "formula": "H(t) = \\frac{\\dot a_{\\rm WCT}}{a_{\\rm WCT}} = \\sqrt{ \\frac{\\rho_\\Theta(t)}{3|K|} }.", "definition": "Current effective status: ○ OPEN The constant \\(K\\) must carry the units needed for \\(H^2\\), and the equation requires independent derivation.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm20-theta-based-expansion-ansatz" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CM20" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP1", "name": "Closed spectral-loop representation", "family": "Topology and spectral emergence", "formula": "\\gamma(s) = \\sum_{k=1}^{K} \\left[ a_k\\cos(ks)+b_k\\sin(ks) \\right], \\qquad s\\in[0,2\\pi).", "definition": "Current effective status: ◻️ DEFINITION This is the chosen configuration representation; emergence of the basis is a separate empirical claim.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top1-closed-spectral-loop-representation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP1" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP2", "name": "WCT loop-energy functional", "family": "Topology and spectral emergence", "formula": "\\mathcal E_{\\rm loop}[\\gamma] = \\int_\\gamma\\kappa^2ds + \\alpha_{\\rm UV} \\sum_kk^p(|a_k|^2+|b_k|^2) + V_{\\rm SA}[\\gamma].", "definition": "Current effective status: ◻️ DEFINITION", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top2-wct-loop-energy-functional" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP3", "name": "Irreversible gradient flow", "family": "Topology and spectral emergence", "formula": "\\partial_t\\gamma = -\\frac{\\delta\\mathcal E_{\\rm loop}}{\\delta\\gamma}.\n\n\\frac{d\\mathcal E_{\\rm loop}}{dt} = - \\left\\| \\frac{\\delta\\mathcal E_{\\rm loop}}{\\delta\\gamma} \\right\\|^2 \\le0.", "definition": "Current effective status: ⚠️ CONDITIONAL For a differentiable gradient flow,", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top3-irreversible-gradient-flow" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP4", "name": "Emergent-topology criterion", "family": "Topology and spectral emergence", "formula": "I[\\gamma_t]\\to I_\\infty\n\n\\frac{d\\mathcal E}{dt}<0", "definition": "Current effective status: ○ OPEN A proposed physical invariant \\(I[\\gamma]\\) satisfies", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top4-emergent-topology-criterion" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP4" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP5", "name": "WCT dynamical codimension", "family": "Topology and spectral emergence", "formula": "\\operatorname{codim}_{\\rm WCT}(\\gamma) := \\text{minimum number of singular events required to reach the unknot}.", "definition": "Current effective status: ◻️ DEFINITION This is not manifold codimension.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top5-wct-dynamical-codimension" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP5" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP6", "name": "Spectral topology bands", "family": "Topology and spectral emergence", "formula": "\\epsilon_\\kappa = \\frac1L\\int_\\gamma\\kappa^2ds.", "definition": "Current effective status: ○ OPEN Define", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top6-spectral-topology-bands" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP6" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP7", "name": "Topological mass proxy", "family": "Topology and spectral emergence", "formula": "m_{\\rm WCT} \\propto \\epsilon_\\kappa.", "definition": "A conditional proportionality between a topological curvature-energy proxy and WCT mass within a fixed normalization and topology class. The absolute scale and broader state-selection rule require calibration and derivation.", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top7-topological-mass-proxy" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP7" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP8", "name": "Holonomy non-invariance", "family": "Topology and spectral emergence", "formula": "H_\\tau[\\gamma] = \\int_\\gamma\\tau\\,ds,", "definition": "Current effective status: ○ OPEN For", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top8-holonomy-non-invariance" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP8" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "TOP9", "name": "Protein-particle structural correspondence", "family": "Topology and spectral emergence", "formula": "\\text{knotted protein states} \\longleftrightarrow \\text{stable WCT loop excitations}", "definition": "Current effective status: ○ OPEN is a proposed analogy restricted to irreversible curvature flow with self-avoidance and spectral suppression. It is not an established physical equivalence.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top9-protein-particle-structural-correspondence" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "TOP9" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CORR1", "name": "Full Lyapunov candidate", "family": "Canonical correction layer", "formula": "\\mathcal E_{\\rm WCT}[\\psi] = \\int \\left( |\\nabla\\psi|^2 + |\\Theta_\\varepsilon[\\psi]|^2 \\right)dx.", "definition": "Current effective status: ◻️ DEFINITION The curvature term alone is only one component.", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr1-full-lyapunov-candidate" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CORR1" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CORR2", "name": "Mean-amplitude spectral closure", "family": "Canonical correction layer", "formula": "D_{\\rm eff}^2 := \\langle|\\psi|^2\\rangle+\\varepsilon^2,\n\nC_\\Theta(k) \\approx \\frac{k^4}{D_{\\rm eff}^2}.", "definition": "Current effective status: ⚠️ CONDITIONAL Under a weak-intermittency mean-amplitude approximation,", "status": { "baseline": "CONDITIONAL", "effective": "CONDITIONAL", "changed": false, "changed_by": null }, "verification": { "outcome": "CONDITIONAL", "kind": "UNRESOLVED", "scope": "MODEL_CONDITIONAL", "checker": [ "classify_conditional" ], "baseline_checker": "classify_conditional", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr2-mean-amplitude-spectral-closure" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CORR2" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CORR3", "name": "Spectral-weight notation", "family": "Canonical correction layer", "formula": "\\lambda_\\Theta\n\n\\lambda_{\\rm ex}", "definition": "Current effective status: ◻️ DEFINITION Use", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr3-spectral-weight-notation" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CORR3" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CORR4", "name": "Macro-micro control parameter", "family": "Canonical correction layer", "formula": "\\Xi = \\frac{\\int k^4\\rho(k)\\,dk}{H}, \\qquad H=-\\sum_kP_k\\ln P_k,", "definition": "Current effective status: ◻️ DEFINITION on the domain \\(H>0\\).", "status": { "baseline": "DEFINITION", "effective": "DEFINITION", "changed": false, "changed_by": null }, "verification": { "outcome": "DEFINITION", "kind": "DEFINITION_CHECK", "scope": "DEFINITIONAL", "checker": [ "classify_definition" ], "baseline_checker": "classify_definition", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr4-macro-micro-control-parameter" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CORR4" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CORR5", "name": "Entropy-curvature coupling", "family": "Canonical correction layer", "formula": "\\frac{dH}{dt} \\le -\\mu \\int|\\Theta_\\varepsilon[\\psi]|^2dx.", "definition": "Current effective status: ○ OPEN This remains the same open claim as E43 unless derived for a specified evolution.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr5-entropy-curvature-coupling" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CORR5" ], "status": "lookup_by_stable_id" } }, { "canonical_id": "CORR6", "name": "Isoelectronic-flow alignment", "family": "Canonical correction layer", "formula": "", "definition": "Current effective status: ○ OPEN The imaginary-time isoelectronic flow is proposed as a reduced sector of M7 with ultraviolet smoothing and norm enforcement. A derivation of the reduction and its error bounds remains open.", "status": { "baseline": "OPEN", "effective": "OPEN", "changed": false, "changed_by": null }, "verification": { "outcome": "OPEN", "kind": "UNRESOLVED", "scope": "UNRESOLVED", "checker": [ "classify_open" ], "baseline_checker": "classify_open", "meaning": "The assigned executable check is reported under its declared assumptions.", "limitations": [] }, "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" }, "sources": { "canonical_equation": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr6-isoelectronic-flow-alignment" }, "dependencies": { "equations": [], "assumptions": [], "claims": [], "papers": [] }, "source_papers": [], "evidence": [], "cross_layer_consistency": { "status": "compiled", "notes": [ "Symbolic status, verification kind, Lean coverage, and empirical state are generated from the compiled registry." ] }, "semantic_graph": { "repository": "https://github.com/rickyjreyes/obsidian", "lookup_keys": [ "CORR6" ], "status": "lookup_by_stable_id" } } ], "machine_reading_guidance": { "canonical_id_policy": "Use canonical_id as the primary key. Aliases must not be silently collapsed.", "status_precedence": [ "Per-object derived override", "Executable SymPy baseline registry", "Canonical equation narrative" ], "verification_rule": "Read verification.outcome together with verification.kind and verification.scope. PASS alone does not state what kind of check succeeded.", "formalization_rule": "Never report a SymPy PASS as a Lean proof. Only named kernel-accepted declarations may be labeled PROVED.", "empirical_rule": "Empirical validation is independent of symbolic and formal status.", "missing_data_rule": "Use null, empty arrays, UNKNOWN, or OPEN rather than inventing identifiers, proofs, dates, dependencies, or evidence." }, "content_type": "application/json" }