[ { "id": "M1", "title": "Curvature-locking functional", "family": "Master systems", "formulas": [ "\\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.", "S_{\\rm lock}[\\varphi] = \\oint_\\Gamma w(s)\\bigl(\\partial_s\\varphi-\\sigma\\bigr)^2\\,ds.", "m=\\frac{\\hbar}{c}\\langle\\sigma\\rangle_w." ], "anchor": "m1-curvature-locking-functional", "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.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m1-curvature-locking-functional", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "check_locking_family", "baseline_checker": "check_locking_family", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M2", "title": "Nonsingular curvature operator and Lyapunov candidate", "family": "Master systems", "formulas": [ "R_\\varepsilon(\\psi) := \\frac{\\overline{\\psi}} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}},", "\\Theta_\\varepsilon[\\psi] := -(\\Delta\\psi)R_\\varepsilon(\\psi).", "R_\\varepsilon(\\psi)\\longrightarrow \\frac1\\psi \\qquad(\\varepsilon\\to0).", "\\mathcal E_{\\rm WCT}[\\psi] = \\int_\\Omega \\left( |\\nabla\\psi|^2+|\\Theta_\\varepsilon[\\psi]|^2 \\right)\\,dx." ], "anchor": "m2-nonsingular-curvature-operator-and-lyapunov-candidate", "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.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m2-nonsingular-curvature-operator-and-lyapunov-candidate", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_regularized_denominator", "baseline_checker": "check_regularized_denominator", "verification_kind": "LIMIT_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M3", "title": "Finite-band spectral selector", "family": "Master systems", "formulas": [ "\\partial_tA = \\mu A-g|A|^2A-b(\\Delta+k_\\star^2)^2A, \\qquad b>0.", "-b\\bigl(|k|^2-k_\\star^2\\bigr)^2," ], "anchor": "m3-finite-band-spectral-selector", "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.", "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,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m3-finite-band-spectral-selector", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_master_sh_sign", "baseline_checker": "check_master_sh_sign", "verification_kind": "SIGN_OR_EXTREMUM_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M4", "title": "Dimensional stability threshold", "family": "Master systems", "formulas": [ "H^2(\\Omega)\\hookrightarrow L^\\infty(\\Omega) \\quad\\text{when}\\quad 2>\\frac n2.", "n\\le3." ], "anchor": "m4-dimensional-stability-threshold", "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.", "formula": "H^2(\\Omega)\\hookrightarrow L^\\infty(\\Omega) \\quad\\text{when}\\quad 2>\\frac n2.\n\nn\\le3.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m4-dimensional-stability-threshold", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_h2_embedding_threshold", "baseline_checker": "check_h2_embedding_threshold", "verification_kind": "FORMAL_THEOREM", "verification_scope": "STANDARD_MATHEMATICS", "verification_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.", "verification_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" } }, { "id": "M5", "title": "Curvature-bounded computation", "family": "Master systems", "formulas": [ "\\psi^{(t+1)}(x) = U\\!\\left( \\psi^{(t)}(x), \\{\\psi^{(t)}(y):y\\in N(x)\\} \\right)." ], "anchor": "m5-curvature-bounded-computation", "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.", "formula": "\\psi^{(t+1)}(x) = U\\!\\left( \\psi^{(t)}(x), \\{\\psi^{(t)}(y):y\\in N(x)\\} \\right).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m5-curvature-bounded-computation", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M6A", "title": "Unified linear operator", "family": "Master systems", "formulas": [ "\\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." ], "anchor": "m6a-unified-linear-operator", "definition": "Current effective status: ◻️ DEFINITION", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m6a-unified-linear-operator", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M6B", "title": "Nonlinear curvature operator", "family": "Master systems", "formulas": [ "\\mathcal N_{\\rm curv}[\\psi] = -(\\Delta\\psi) \\frac{\\overline{\\psi}} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}}." ], "anchor": "m6b-nonlinear-curvature-operator", "definition": "Current effective status: ○ OPEN The operator is well-defined for \\(\\varepsilon>0\\); uniqueness of this nonlinear closure remains open.", "formula": "\\mathcal N_{\\rm curv}[\\psi] = -(\\Delta\\psi) \\frac{\\overline{\\psi}} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m6b-nonlinear-curvature-operator", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "check_uniqueness_claim", "baseline_checker": "check_uniqueness_claim", "verification_kind": "COUNTEREXAMPLE_TEST", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M7", "title": "Full curvature-wavefield equation", "family": "Master systems", "formulas": [ "\\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." ], "anchor": "m7-full-curvature-wavefield-equation", "definition": "Current effective status: ✅ PASS The explicit negative fourth-order term supplies finite-band ultraviolet damping. Other dynamical claims require separate analysis.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m7-full-curvature-wavefield-equation", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_master_uwct_sign", "baseline_checker": "check_master_uwct_sign", "verification_kind": "SIGN_OR_EXTREMUM_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "M8", "title": "Curvature-acoustic cosmology system", "family": "Master systems", "formulas": [ "\\Phi(k,t)=-C_\\Phi\\frac{\\Theta(k,t)}{k^2},", "\\delta_g(E) = A_g\\cos\\!\\left( k_\\ell\\ln\\frac{E}{E_0}+\\phi \\right)." ], "anchor": "m8-curvature-acoustic-cosmology-system", "definition": "Current effective status: ○ OPEN Representative closure relations are", "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).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#m8-curvature-acoustic-cosmology-system", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E1A", "title": "Curvature-rate density", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "\\sigma_{\\rm dens}(s)=\\kappa(s)^2+\\tau(s)^2, \\qquad [\\sigma_{\\rm dens}]=L^{-2}." ], "anchor": "e1a-curvature-rate-density", "definition": "The canonical registered object for curvature-rate density; consult the source equation and verification metadata for its assumptions and scientific boundary.", "formula": "\\sigma_{\\rm dens}(s)=\\kappa(s)^2+\\tau(s)^2, \\qquad [\\sigma_{\\rm dens}]=L^{-2}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e1a-curvature-rate-density", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_e1a", "baseline_checker": "check_e1a", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The curvature-rate density has inverse-length-squared dimension.", "verification_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" } }, { "id": "E1B", "title": "Curvature spectral rate", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "\\sigma(s)=\\sqrt{\\kappa(s)^2+\\tau(s)^2}, \\qquad [\\sigma]=L^{-1}." ], "anchor": "e1b-curvature-spectral-rate", "definition": "The canonical registered object for curvature spectral rate; consult the source equation and verification metadata for its assumptions and scientific boundary.", "formula": "\\sigma(s)=\\sqrt{\\kappa(s)^2+\\tau(s)^2}, \\qquad [\\sigma]=L^{-1}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e1b-curvature-spectral-rate", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_e1b", "baseline_checker": "check_e1b", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The curvature spectral rate has inverse-length dimension.", "verification_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" } }, { "id": "E2", "title": "Weighted loop average", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "\\langle f\\rangle_w = \\frac{\\oint_\\Gamma w(s)f(s)\\,ds} {\\oint_\\Gamma w(s)\\,ds}." ], "anchor": "e2-weighted-loop-average", "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.", "formula": "\\langle f\\rangle_w = \\frac{\\oint_\\Gamma w(s)f(s)\\,ds} {\\oint_\\Gamma w(s)\\,ds}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e2-weighted-loop-average", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_weighted_average", "baseline_checker": "check_weighted_average", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "STANDARD_MATHEMATICS", "verification_meaning": "The positive weighted-average construction has a nonzero denominator and preserves the encoded dimension of the averaged quantity.", "verification_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" } }, { "id": "E3", "title": "Loop-locking action", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "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." ], "anchor": "e3-loop-locking-action", "definition": "A constrained phase-curvature mismatch action. Nonnegative weighting makes its squared mismatch term nonnegative; existence and uniqueness of continuum minimizers require separate analysis.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e3-loop-locking-action", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_locking_variation", "baseline_checker": "check_locking_variation", "verification_kind": "VARIATIONAL_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E4", "title": "Covariant locking solution", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "\\partial_s\\varphi = \\sigma+\\frac{\\alpha_{\\rm lock}}{w},", "\\alpha_{\\rm lock} = \\frac{ 2\\pi n-\\oint_\\Gamma\\sigma\\,ds }{ \\oint_\\Gamma ds/w }," ], "anchor": "e4-covariant-locking-solution", "definition": "Current effective status: ✅ PASS Stationarity gives", "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 },", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e4-covariant-locking-solution", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_locking_solution", "baseline_checker": "check_locking_solution", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E5", "title": "Effective-wavenumber chain", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "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.", "k_{\\rm wind}=k_\\sigma=\\langle\\sigma\\rangle_w" ], "anchor": "e5-effective-wavenumber-chain", "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.", "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", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e5-effective-wavenumber-chain", "status": "PASS", "baseline_status": "CONDITIONAL", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_effective_wavenumber_chain_derived", "checker": "check_effective_wavenumber_chain_derived", "baseline_checker": "check_effective_wavenumber_chain", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "Exact loop closure and the declared orientation and weighting assumptions establish the encoded effective-wavenumber chain.", "verification_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" } }, { "id": "E6", "title": "Mass-curvature law", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "E_{\\rm rest}=\\hbar c\\,k_{\\rm eff}, \\qquad m=\\frac{\\hbar}{c}k_{\\rm eff}.", "m=\\frac{\\hbar}{c}\\langle\\sigma\\rangle_w." ], "anchor": "e6-mass-curvature-law", "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.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e6-mass-curvature-law", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_mass_curvature_dimensions", "baseline_checker": "check_mass_curvature_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The encoded mass-curvature relation has the dimensions of mass.", "verification_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" } }, { "id": "E7", "title": "Solenoidal mass law", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "m = \\frac{\\hbar}{c} \\left\\langle \\sqrt{\\kappa^2+\\tau^2} \\right\\rangle_\\Gamma." ], "anchor": "e7-solenoidal-mass-law", "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.", "formula": "m = \\frac{\\hbar}{c} \\left\\langle \\sqrt{\\kappa^2+\\tau^2} \\right\\rangle_\\Gamma.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e7-solenoidal-mass-law", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_mass_curvature_dimensions", "baseline_checker": "check_mass_curvature_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The solenoidal curvature average has the dimensions required by the stated mass relation.", "verification_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" } }, { "id": "E8", "title": "Corrected weighted-lock identity", "family": "A. Rest energy, curvature, and loop locking", "formulas": [ "\\boxed{ \\oint_\\Gamma w\\,\\partial_s\\varphi\\,ds = \\oint_\\Gamma w\\,\\sigma\\,ds + \\alpha_{\\rm lock}L_s }.", "2\\pi\\oint ds/\\oint ds/w" ], "anchor": "e8-corrected-weighted-lock-identity", "definition": "Current effective status: ✅ PASS Substituting E4 gives", "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", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e8-corrected-weighted-lock-identity", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_e8_identity", "baseline_checker": "check_e8_identity", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E9", "title": "Phase-flux constitutive relation", "family": "B. Phase-flux and finite-band selection", "formulas": [ "\\mathbf S(x,t)=u(x,t)\\nabla\\theta(x,t).", "\\partial_tu+\\nabla\\cdot\\mathbf S=0." ], "anchor": "e9-phase-flux-constitutive-relation", "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.", "formula": "\\mathbf S(x,t)=u(x,t)\\nabla\\theta(x,t).\n\n\\partial_tu+\\nabla\\cdot\\mathbf S=0.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e9-phase-flux-constitutive-relation", "status": "PASS", "baseline_status": "DEFINITION", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_phase_flux_from_polar_field", "checker": "check_phase_flux_from_polar_field", "baseline_checker": "classify_definition", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "Polar decomposition yields the normalized phase-current identity under the declared nonzero-field assumptions.", "verification_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" } }, { "id": "E10", "title": "Radial shell quantization", "family": "B. Phase-flux and finite-band selection", "formulas": [ "\\int_{r_1}^{r_2}k_r(r)\\,dr=2\\pi n, \\qquad n\\in\\mathbb Z." ], "anchor": "e10-radial-shell-quantization", "definition": "Current effective status: ✅ PASS Both sides are dimensionless.", "formula": "\\int_{r_1}^{r_2}k_r(r)\\,dr=2\\pi n, \\qquad n\\in\\mathbb Z.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e10-radial-shell-quantization", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_shell_quantization_dimensions", "baseline_checker": "check_shell_quantization_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E11", "title": "Phase winding", "family": "B. Phase-flux and finite-band selection", "formulas": [ "m(\\gamma) = \\frac1{2\\pi} \\oint_\\gamma\\nabla\\theta\\cdot d\\boldsymbol\\ell \\in\\mathbb Z," ], "anchor": "e11-phase-winding", "definition": "Current effective status: ✅ PASS provided \\(\\psi\\neq0\\) on the loop and the phase is continuous modulo \\(2\\pi\\).", "formula": "m(\\gamma) = \\frac1{2\\pi} \\oint_\\gamma\\nabla\\theta\\cdot d\\boldsymbol\\ell \\in\\mathbb Z,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e11-phase-winding", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_winding_dimensions", "baseline_checker": "check_winding_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E12", "title": "Finite-band dispersion rail", "family": "B. Phase-flux and finite-band selection", "formulas": [ "\\lambda_{\\rm grow}(k) = r+a|k|^2-b|k|^4, \\qquad a,b>0.", "k_\\star=\\sqrt{\\frac{a}{2b}}.", "\\lambda_{\\rm grow}(k) = \\mu-b(|k|^2-k_\\star^2)^2, \\qquad \\mu=r+\\frac{a^2}{4b}." ], "anchor": "e12-finite-band-dispersion-rail", "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.", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e12-finite-band-dispersion-rail", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_dispersion_stationary_point", "baseline_checker": "check_dispersion_stationary_point", "verification_kind": "SIGN_OR_EXTREMUM_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E13", "title": "Band-pass amplitude evolution", "family": "B. Phase-flux and finite-band selection", "formulas": [ "\\partial_tA = (r-a\\Delta-b\\Delta^2)A-\\beta|A|^2A,", "\\partial_tA = \\mu A-b(\\Delta+k_\\star^2)^2A-\\beta|A|^2A." ], "anchor": "e13-band-pass-amplitude-evolution", "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.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e13-band-pass-amplitude-evolution", "status": "PASS", "baseline_status": "CONDITIONAL", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_bandpass_gradient_flow", "checker": "check_bandpass_gradient_flow", "baseline_checker": "check_e13_e14_consistency", "verification_kind": "VARIATIONAL_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The declared band-pass amplitude equation follows from negative gradient flow of the registered functional.", "verification_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" } }, { "id": "E14", "title": "Band-pass Lyapunov functional", "family": "B. Phase-flux and finite-band selection", "formulas": [ "\\mathcal E[A] = \\int_\\Omega \\left[ -\\mu|A|^2 +b|(\\Delta+k_\\star^2)A|^2 +\\frac{\\beta}{2}|A|^4 \\right]dx." ], "anchor": "e14-band-pass-lyapunov-functional", "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.", "formula": "\\mathcal E[A] = \\int_\\Omega \\left[ -\\mu|A|^2 +b|(\\Delta+k_\\star^2)A|^2 +\\frac{\\beta}{2}|A|^4 \\right]dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e14-band-pass-lyapunov-functional", "status": "PASS", "baseline_status": "CONDITIONAL", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_bandpass_gradient_flow", "checker": "check_bandpass_gradient_flow", "baseline_checker": "check_e13_e14_consistency", "verification_kind": "VARIATIONAL_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The registered functional generates the declared band-pass gradient flow under the stated sign convention.", "verification_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" } }, { "id": "E15", "title": "Modal growth bound", "family": "B. Phase-flux and finite-band selection", "formulas": [ "\\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." ], "anchor": "e15-modal-growth-bound", "definition": "Current effective status: ⚠️ CONDITIONAL The quartic modal estimate requires a model-specific nonlinear projection bound.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e15-modal-growth-bound", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E16", "title": "Linear spectral concentration", "family": "B. Phase-flux and finite-band selection", "formulas": [ "P_k(t)=P_k(0)e^{2\\lambda_{\\rm grow}(k)t}.", "\\operatorname*{arg\\,max}_kP_k(t)\\to k_\\star." ], "anchor": "e16-linear-spectral-concentration", "definition": "Current effective status: ✅ PASS For the linearized dynamics,", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e16-linear-spectral-concentration", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_linear_spectral_growth", "baseline_checker": "check_linear_spectral_growth", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E17", "title": "Nonsingular curvature-feedback operator", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "R_\\varepsilon(\\psi) = \\frac{\\overline\\psi} {|\\psi|^2+\\varepsilon^2e^{-2\\alpha|\\psi|^2}},", "\\boxed{ \\Theta_\\varepsilon[\\psi] = -(\\Delta\\psi)R_\\varepsilon(\\psi) }.", "-\\Delta\\psi/(\\psi+\\varepsilon e^{-\\alpha|\\psi|^2})" ], "anchor": "e17-nonsingular-curvature-feedback-operator", "definition": "Current effective status: ✅ PASS For \\(\\varepsilon>0\\), the denominator is strictly positive for all complex \\(\\psi\\). This replaces", "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})", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e17-nonsingular-curvature-feedback-operator", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_regularized_denominator", "baseline_checker": "check_regularized_denominator", "verification_kind": "LIMIT_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E18", "title": "WCT Lyapunov candidate", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "\\mathcal E[\\psi] = \\int_\\Omega \\left( c_1|\\nabla\\psi|^2 +c_2|\\Theta_\\varepsilon[\\psi]|^2 \\right)dx." ], "anchor": "e18-wct-lyapunov-candidate", "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.", "formula": "\\mathcal E[\\psi] = \\int_\\Omega \\left( c_1|\\nabla\\psi|^2 +c_2|\\Theta_\\varepsilon[\\psi]|^2 \\right)dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e18-wct-lyapunov-candidate", "status": "PASS", "baseline_status": "CONDITIONAL", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_lyapunov_gradient_flow", "checker": "check_lyapunov_gradient_flow", "baseline_checker": "check_energy_positivity", "verification_kind": "VARIATIONAL_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "Exact negative gradient flow gives monotone energy descent when the registered nonnegativity assumptions hold.", "verification_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" } }, { "id": "E19", "title": "Gap-curvature scaling", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "\\Delta_k^\\star\\sim\\langle\\sigma\\rangle_w^2, \\qquad [\\Delta_k^\\star]=L^{-2},", "\\Delta_\\omega^\\star:=c^2\\Delta_k^\\star, \\qquad [\\Delta_\\omega^\\star]=T^{-2}." ], "anchor": "e19-gap-curvature-scaling", "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.", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e19-gap-curvature-scaling", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "check_gap_curvature_dimensions", "baseline_checker": "check_gap_curvature_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E20", "title": "Higher-order cavity quadratic sector", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "S:=\\frac{\\Box\\psi}{g(\\psi)}, \\qquad P:=\\frac{\\Delta\\psi}{g(\\psi)}.", "Q(S,P)=\\kappa S^2+\\theta P^2-\\gamma SP.", "\\kappa\\ge0,\\qquad \\theta\\ge0,\\qquad \\gamma^2\\le4\\kappa\\theta." ], "anchor": "e20-higher-order-cavity-quadratic-sector", "definition": "Current effective status: ✅ PASS Let", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e20-higher-order-cavity-quadratic-sector", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_cavity_quadratic_form", "baseline_checker": "check_cavity_quadratic_form", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E21", "title": "Second-derivative Euler-Lagrange equation", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "\\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." ], "anchor": "e21-second-derivative-euler-lagrange-equation", "definition": "Current effective status: ✅ PASS For \\(\\mathcal L(\\psi,\\partial\\psi,\\partial^2\\psi)\\),", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e21-second-derivative-euler-lagrange-equation", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_second_order_el_template", "baseline_checker": "check_second_order_el_template", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E22", "title": "Effective metric ansatz", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "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}." ], "anchor": "e22-effective-metric-ansatz", "definition": "Current effective status: ⚠️ CONDITIONAL The coefficients must carry the units needed to make both corrections dimensionless, and signature/nondegeneracy must be checked.", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e22-effective-metric-ansatz", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E23", "title": "Enthalpic curvature relation", "family": "C. Curvature feedback and Lyapunov dynamics", "formulas": [ "h(\\psi) = C_h\\left( W_\\psi+\\chi|\\nabla\\psi|^2 \\right)." ], "anchor": "e23-enthalpic-curvature-relation", "definition": "Current effective status: ⚠️ CONDITIONAL The constants must reconcile dimensions and the relation requires a constitutive derivation.", "formula": "h(\\psi) = C_h\\left( W_\\psi+\\chi|\\nabla\\psi|^2 \\right).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e23-enthalpic-curvature-relation", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E24", "title": "Sobolev embedding threshold", "family": "D. Dimensionality and functional bounds", "formulas": [ "H^2(\\Omega)\\hookrightarrow L^\\infty(\\Omega) \\quad\\Longrightarrow\\quad 2>\\frac n2.", "n\\le3." ], "anchor": "e24-sobolev-embedding-threshold", "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.", "formula": "H^2(\\Omega)\\hookrightarrow L^\\infty(\\Omega) \\quad\\Longrightarrow\\quad 2>\\frac n2.\n\nn\\le3.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e24-sobolev-embedding-threshold", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_h2_embedding_threshold", "baseline_checker": "check_h2_embedding_threshold", "verification_kind": "FORMAL_THEOREM", "verification_scope": "STANDARD_MATHEMATICS", "verification_meaning": "The standard H2-to-L-infinity Sobolev embedding threshold is verified under the declared domain hypotheses.", "verification_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" } }, { "id": "E25", "title": "Critical Sobolev exponent", "family": "D. Dimensionality and functional bounds", "formulas": [ "p_c(n)=\\frac{n+2}{n-2}, \\qquad p2\\),", "formula": "p_c(n)=\\frac{n+2}{n-2}, \\qquad p0." ], "anchor": "e34-energy-entropy-conversion", "definition": "Current effective status: ○ OPEN For entropy reduction", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e34-energy-entropy-conversion", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E35", "title": "Curvature-locked fixed point", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "\\partial_t\\psi=0, \\qquad \\nabla\\Theta_\\varepsilon[\\psi]=0, \\qquad \\frac{d}{dt}S[\\psi]=0." ], "anchor": "e35-curvature-locked-fixed-point", "definition": "Current effective status: ◻️ DEFINITION A stationary locked configuration satisfies", "formula": "\\partial_t\\psi=0, \\qquad \\nabla\\Theta_\\varepsilon[\\psi]=0, \\qquad \\frac{d}{dt}S[\\psi]=0.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e35-curvature-locked-fixed-point", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E36", "title": "Discrete WCC update", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "\\psi^{(t+1)}(x) = U\\!\\left( \\psi^{(t)}(x), \\{\\psi^{(t)}(y)\\}_{y\\in N(x)} \\right)." ], "anchor": "e36-discrete-wcc-update", "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.", "formula": "\\psi^{(t+1)}(x) = U\\!\\left( \\psi^{(t)}(x), \\{\\psi^{(t)}(y)\\}_{y\\in N(x)} \\right).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e36-discrete-wcc-update", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E37", "title": "Bandlimit from energy", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "k_{\\max} = C_1\\frac{E_{\\max}}{\\hbar c}," ], "anchor": "e37-bandlimit-from-energy", "definition": "Current effective status: ✅ PASS with dimensionless \\(C1\\).", "formula": "k_{\\max} = C_1\\frac{E_{\\max}}{\\hbar c},", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e37-bandlimit-from-energy", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_bandlimit_dimensions", "baseline_checker": "check_bandlimit_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E38", "title": "Spatial channel capacity", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "N_{\\rm lanes} \\le C_2Vk_{\\max}^3," ], "anchor": "e38-spatial-channel-capacity", "definition": "Current effective status: ✅ PASS In three spatial dimensions,", "formula": "N_{\\rm lanes} \\le C_2Vk_{\\max}^3,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e38-spatial-channel-capacity", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_channel_capacity_dimensions", "baseline_checker": "check_channel_capacity_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E39", "title": "Polynomial update bound", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "T_{\\max}(n)\\le C_3n^d." ], "anchor": "e39-polynomial-update-bound", "definition": "Current effective status: ◻️ DEFINITION This defines the assumed computational resource class.", "formula": "T_{\\max}(n)\\le C_3n^d.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e39-polynomial-update-bound", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E40", "title": "WCC complexity identification", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "P_{\\rm WCC}\\cong P, \\qquad NP_{\\rm WCC}\\cong NP." ], "anchor": "e40-wcc-complexity-identification", "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.", "formula": "P_{\\rm WCC}\\cong P, \\qquad NP_{\\rm WCC}\\cong NP.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e40-wcc-complexity-identification", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E41", "title": "Curvature-bounded configuration count", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "|C_{\\rm curv}(n)| \\le 2^{\\alpha(n)n}, \\qquad \\alpha(n)<1." ], "anchor": "e41-curvature-bounded-configuration-count", "definition": "Current effective status: ⚠️ CONDITIONAL E28 alone does not prove this counting bound; an injective coding or combinatorial argument is required.", "formula": "|C_{\\rm curv}(n)| \\le 2^{\\alpha(n)n}, \\qquad \\alpha(n)<1.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e41-curvature-bounded-configuration-count", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "check_alpha_drop_count_bound", "baseline_checker": "check_alpha_drop_count_bound", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E42", "title": "Theta-information relation", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "\\frac{d}{dt}I_{\\rm coh}[\\psi] = -\\lambda_I \\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx." ], "anchor": "e42-theta-information-relation", "definition": "Current effective status: ○ OPEN The information functional and coupling \\(\\lambdaI\\) require derivation.", "formula": "\\frac{d}{dt}I_{\\rm coh}[\\psi] = -\\lambda_I \\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e42-theta-information-relation", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E43", "title": "Curvature-entropy tradeoff", "family": "F. WCC, channel capacity, and complexity", "formulas": [ "\\frac{dH_k}{dt} \\le -\\mu \\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx, \\qquad \\mu>0." ], "anchor": "e43-curvature-entropy-tradeoff", "definition": "Current effective status: ○ OPEN This remains an analytic/empirical claim.", "formula": "\\frac{dH_k}{dt} \\le -\\mu \\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx, \\qquad \\mu>0.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e43-curvature-entropy-tradeoff", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E44", "title": "Theta eigenmode problem", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\Theta_\\varepsilon[\\psi_n] = \\lambda_n\\psi_n." ], "anchor": "e44-theta-eigenmode-problem", "definition": "Current effective status: ◻️ DEFINITION Because \\(\\Theta\\varepsilon\\) is nonlinear, the spectral problem and normalization must be specified carefully.", "formula": "\\Theta_\\varepsilon[\\psi_n] = \\lambda_n\\psi_n.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e44-theta-eigenmode-problem", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E45", "title": "Corrected quality factor", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\boxed{ Q_{\\rm eff} = \\omega\\frac{U}{P_{\\rm loss}} }", "U=\\int_\\Omega u\\,dV" ], "anchor": "e45-corrected-quality-factor", "definition": "Current effective status: ✅ PASS where", "formula": "\\boxed{ Q_{\\rm eff} = \\omega\\frac{U}{P_{\\rm loss}} }\n\nU=\\int_\\Omega u\\,dV", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e45-corrected-quality-factor", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_q_factor_dimensions", "baseline_checker": "check_q_factor_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E46", "title": "Plasma-cavity curvature match", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\langle\\sigma\\rangle_{w,\\rm plasma} \\approx \\langle\\sigma\\rangle_{w,\\rm cavity}." ], "anchor": "e46-plasma-cavity-curvature-match", "definition": "Current effective status: ○ OPEN A measurable matching tolerance and transfer mechanism remain open.", "formula": "\\langle\\sigma\\rangle_{w,\\rm plasma} \\approx \\langle\\sigma\\rangle_{w,\\rm cavity}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e46-plasma-cavity-curvature-match", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E47", "title": "Corrected power balance", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\boxed{ \\frac{dW}{dt} = P_{\\rm in} + P_{\\rm fusion} - P_{\\rm loss} - P_{\\rm out} }.", "P_{\\rm in}+P_{\\rm fusion} = P_{\\rm loss}+P_{\\rm out}." ], "anchor": "e47-corrected-power-balance", "definition": "Current effective status: ✅ PASS At stationarity,", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e47-corrected-power-balance", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_power_balance_form", "baseline_checker": "check_power_balance_form", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E48", "title": "Curvature-gap stability criterion", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\Delta\\sigma = \\langle\\sigma\\rangle_{\\rm core} - \\langle\\sigma\\rangle_{\\rm edge} > \\Delta_{\\rm crit}." ], "anchor": "e48-curvature-gap-stability-criterion", "definition": "Current effective status: ⚠️ CONDITIONAL The threshold and direction of the inequality must be calibrated to a specified stability observable.", "formula": "\\Delta\\sigma = \\langle\\sigma\\rangle_{\\rm core} - \\langle\\sigma\\rangle_{\\rm edge} > \\Delta_{\\rm crit}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e48-curvature-gap-stability-criterion", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E49", "title": "Corrected effective-mass gap law", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\omega_j^2=c^2\\lambda_j+\\Delta_\\omega^\\star, \\qquad [\\Delta_\\omega^\\star]=T^{-2},", "\\omega^2=c^2k^2+\\frac{m_{\\rm eff}^2c^4}{\\hbar^2}", "\\boxed{ m_{\\rm eff}^2 = \\frac{\\hbar^2}{c^4}\\Delta_\\omega^\\star }." ], "anchor": "e49-corrected-effective-mass-gap-law", "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.", "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 }.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e49-corrected-effective-mass-gap-law", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_effective_mass_gap_dimensions", "baseline_checker": "check_effective_mass_gap_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E50", "title": "Phase-coherence functional", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\mathcal C[\\psi] = \\int_\\Omega \\frac{|\\psi|^2}{|\\nabla\\theta|}\\,dx.", "|\\nabla\\theta|\\ge\\delta>0" ], "anchor": "e50-phase-coherence-functional", "definition": "Current effective status: ⚠️ CONDITIONAL The definition requires a regularization or lower bound", "formula": "\\mathcal C[\\psi] = \\int_\\Omega \\frac{|\\psi|^2}{|\\nabla\\theta|}\\,dx.\n\n|\\nabla\\theta|\\ge\\delta>0", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e50-phase-coherence-functional", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E51", "title": "Curvature-gradient commutator", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\Theta_D[\\psi]:=-\\frac{\\Delta\\psi}{D}.", "[\\nabla,\\Theta_D]\\psi := \\nabla(\\Theta_D[\\psi])-\\Theta_D[\\nabla\\psi] = \\frac{\\Delta\\psi}{D^2}\\nabla D." ], "anchor": "e51-curvature-gradient-commutator", "definition": "Current effective status: ✅ PASS For a smooth scalar denominator \\(D\\neq0\\), define", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e51-curvature-gradient-commutator", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_curvature_gradient_commutator", "baseline_checker": "check_curvature_gradient_commutator", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E52", "title": "Curvature gain and gradient loss", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "G_\\sigma:=\\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx, \\qquad L_\\sigma:=\\int_\\Omega|\\nabla\\psi|^2dx." ], "anchor": "e52-curvature-gain-and-gradient-loss", "definition": "Current effective status: ◻️ DEFINITION", "formula": "G_\\sigma:=\\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx, \\qquad L_\\sigma:=\\int_\\Omega|\\nabla\\psi|^2dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e52-curvature-gain-and-gradient-loss", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E53", "title": "Curvature pressure density", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "p_\\Theta(x) := c_2|\\Theta_\\varepsilon[\\psi](x)|^2." ], "anchor": "e53-curvature-pressure-density", "definition": "Current effective status: ✅ PASS It is the local curvature contribution to E18.", "formula": "p_\\Theta(x) := c_2|\\Theta_\\varepsilon[\\psi](x)|^2.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e53-curvature-pressure-density", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_pressure_density_embedding", "baseline_checker": "check_pressure_density_embedding", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E54", "title": "Resonance-lock condition", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "\\partial_t\\psi=0, \\qquad \\delta\\mathcal E[\\psi]=0, \\qquad \\nabla\\Theta_\\varepsilon[\\psi]=0." ], "anchor": "e54-resonance-lock-condition", "definition": "Current effective status: ⚠️ CONDITIONAL Simultaneous satisfaction requires existence and regularity results.", "formula": "\\partial_t\\psi=0, \\qquad \\delta\\mathcal E[\\psi]=0, \\qquad \\nabla\\Theta_\\varepsilon[\\psi]=0.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e54-resonance-lock-condition", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E55", "title": "Curvature-induced effective potential", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "V_{\\rm eff}(\\psi) = V(|\\psi|^2) + \\kappa|\\Theta_\\varepsilon[\\psi]|^2." ], "anchor": "e55-curvature-induced-effective-potential", "definition": "Current effective status: ◻️ DEFINITION", "formula": "V_{\\rm eff}(\\psi) = V(|\\psi|^2) + \\kappa|\\Theta_\\varepsilon[\\psi]|^2.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e55-curvature-induced-effective-potential", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E56", "title": "Phase-wall criterion", "family": "G. Cavity, effective mass, and phase structure", "formulas": [ "|\\nabla\\theta|_{\\rm wall} \\sim \\sigma_{\\rm wall} \\gg \\langle\\sigma\\rangle_w." ], "anchor": "e56-phase-wall-criterion", "definition": "Current effective status: ⚠️ CONDITIONAL The comparison scale and wall-detection threshold must be defined.", "formula": "|\\nabla\\theta|_{\\rm wall} \\sim \\sigma_{\\rm wall} \\gg \\langle\\sigma\\rangle_w.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e56-phase-wall-criterion", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E57", "title": "Swift-Hohenberg shell operator", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "\\mathcal{SH}[A] = (\\Delta+k_\\star^2)^2A.", "(|k|^2-k_\\star^2)^2." ], "anchor": "e57-swift-hohenberg-shell-operator", "definition": "Current effective status: ✅ PASS Its Fourier symbol is", "formula": "\\mathcal{SH}[A] = (\\Delta+k_\\star^2)^2A.\n\n(|k|^2-k_\\star^2)^2.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e57-swift-hohenberg-shell-operator", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_sh_fourier_symbol", "baseline_checker": "check_sh_fourier_symbol", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E58", "title": "Band-selective Green kernel", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "\\mathcal L=r+a(\\Delta+k_\\star^2)^2,", "G(k) = \\frac1{r+a(|k|^2-k_\\star^2)^2}." ], "anchor": "e58-band-selective-green-kernel", "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.", "formula": "\\mathcal L=r+a(\\Delta+k_\\star^2)^2,\n\nG(k) = \\frac1{r+a(|k|^2-k_\\star^2)^2}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e58-band-selective-green-kernel", "status": "PASS", "baseline_status": "CONDITIONAL", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_green_kernel_bounded", "checker": "check_green_kernel_bounded", "baseline_checker": "check_green_kernel_pole", "verification_kind": "LIMIT_CHECK", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "A strictly positive spectral offset bounds the declared Green kernel and removes the registered pole.", "verification_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" } }, { "id": "E59", "title": "Projection onto a dominant annulus", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "\\mathcal A^\\star := \\left\\{ k\\in\\mathbb Z^d: \\bigl||k|-k_\\star\\bigr|\\le\\Delta k \\right\\},", "(P_{k_\\star}A)(x) = \\sum_{k\\in\\mathcal A^\\star} \\widehat A_ke^{ik\\cdot x}.", "P_{k_\\star}^2=P_{k_\\star}." ], "anchor": "e59-projection-onto-a-dominant-annulus", "definition": "Current effective status: ✅ PASS With a fixed annulus,", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e59-projection-onto-a-dominant-annulus", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_projection_idempotence", "baseline_checker": "check_projection_idempotence", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E60", "title": "Center-manifold amplitude equation", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "\\partial_T\\mathcal A = \\mu\\mathcal A-g|\\mathcal A|^2\\mathcal A." ], "anchor": "e60-center-manifold-amplitude-equation", "definition": "Current effective status: ◻️ DEFINITION", "formula": "\\partial_T\\mathcal A = \\mu\\mathcal A-g|\\mathcal A|^2\\mathcal A.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e60-center-manifold-amplitude-equation", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E61", "title": "Pattern-formation threshold", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "r_c = \\min_k a(|k|^2-k_\\star^2)^2 = 0." ], "anchor": "e61-pattern-formation-threshold", "definition": "Current effective status: ✅ PASS In the continuum,", "formula": "r_c = \\min_k a(|k|^2-k_\\star^2)^2 = 0.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e61-pattern-formation-threshold", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_pattern_threshold", "baseline_checker": "check_pattern_threshold", "verification_kind": "SIGN_OR_EXTREMUM_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E62", "title": "Spectral energy concentration", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "\\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." ], "anchor": "e62-spectral-energy-concentration", "definition": "Current effective status: ✅ PASS For nonzero total spectral energy,", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e62-spectral-energy-concentration", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_spectral_fraction_bounds", "baseline_checker": "check_spectral_fraction_bounds", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E63", "title": "Entropic mode selection", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "k_\\star = \\operatorname*{arg\\,min}_k \\left[ H_k+\\lambda_\\Theta C_\\Theta(k) \\right]." ], "anchor": "e63-entropic-mode-selection", "definition": "Current effective status: ◻️ DEFINITION", "formula": "k_\\star = \\operatorname*{arg\\,min}_k \\left[ H_k+\\lambda_\\Theta C_\\Theta(k) \\right].", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e63-entropic-mode-selection", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E64", "title": "Corrected selected wavelength", "family": "H. Swift-Hohenberg and spectral projection", "formulas": [ "k_\\star=\\sqrt{\\frac{a}{2b}}.", "\\boxed{ \\lambda_\\star = \\frac{2\\pi}{k_\\star} = 2\\pi\\sqrt{\\frac{2b}{a}} }." ], "anchor": "e64-corrected-selected-wavelength", "definition": "Current effective status: ✅ PASS From E12,", "formula": "k_\\star=\\sqrt{\\frac{a}{2b}}.\n\n\\boxed{ \\lambda_\\star = \\frac{2\\pi}{k_\\star} = 2\\pi\\sqrt{\\frac{2b}{a}} }.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e64-corrected-selected-wavelength", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_e12_e64_consistency", "baseline_checker": "check_e12_e64_consistency", "verification_kind": "CONSISTENCY_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E65", "title": "Critical Sobolev exponent", "family": "I. Sobolev structure and dimensional bounds", "formulas": [ "p_c(n)=\\frac{n+2}{n-2}." ], "anchor": "e65-critical-sobolev-exponent", "definition": "Current effective status: ✅ PASS For \\(n>2\\),", "formula": "p_c(n)=\\frac{n+2}{n-2}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e65-critical-sobolev-exponent", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_critical_sobolev_exponent", "baseline_checker": "check_critical_sobolev_exponent", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E66", "title": "Gagliardo-Nirenberg interpolation", "family": "I. Sobolev structure and dimensional bounds", "formulas": [ "\\|u\\|_{L^p} \\le C \\|\\nabla u\\|_{L^2}^{\\theta} \\|u\\|_{L^2}^{1-\\theta}." ], "anchor": "e66-gagliardo-nirenberg-interpolation", "definition": "Current effective status: ⚠️ CONDITIONAL The allowed \\(p,\\theta,n\\), domain, and boundary assumptions must be specified.", "formula": "\\|u\\|_{L^p} \\le C \\|\\nabla u\\|_{L^2}^{\\theta} \\|u\\|_{L^2}^{1-\\theta}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e66-gagliardo-nirenberg-interpolation", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E67", "title": "Failure of $H^2 o L^\\infty$ above three dimensions", "family": "I. Sobolev structure and dimensional bounds", "formulas": [], "anchor": "e67-failure-of-h2-o-linfty-above-three-dimensions", "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.", "formula": "", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e67-failure-of-h2-o-linfty-above-three-dimensions", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_h2_embedding_threshold", "baseline_checker": "check_h2_embedding_threshold", "verification_kind": "FORMAL_THEOREM", "verification_scope": "STANDARD_MATHEMATICS", "verification_meaning": "The general H2-to-L-infinity embedding route fails above three spatial dimensions under the declared Sobolev framework.", "verification_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" } }, { "id": "E68", "title": "Localized energy estimate", "family": "I. Sobolev structure and dimensional bounds", "formulas": [ "\\int_{B_R}|\\nabla\\psi|^2dx \\le CR^{n-2}\\|\\psi\\|_{H^1}^2." ], "anchor": "e68-localized-energy-estimate", "definition": "Current effective status: ⚠️ CONDITIONAL A model-dependent localized estimate is", "formula": "\\int_{B_R}|\\nabla\\psi|^2dx \\le CR^{n-2}\\|\\psi\\|_{H^1}^2.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e68-localized-energy-estimate", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E69", "title": "Corrected high-regularity curvature bound", "family": "I. Sobolev structure and dimensional bounds", "formulas": [ "\\psi\\in H^s(\\Omega), \\qquad s>\\frac n2+2,", "\\Theta_\\varepsilon[\\psi]\\in L^\\infty(\\Omega)." ], "anchor": "e69-corrected-high-regularity-curvature-bound", "definition": "Current effective status: ✅ PASS If", "formula": "\\psi\\in H^s(\\Omega), \\qquad s>\\frac n2+2,\n\n\\Theta_\\varepsilon[\\psi]\\in L^\\infty(\\Omega).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e69-corrected-high-regularity-curvature-bound", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_theta_linf_from_high_regularity", "baseline_checker": "check_theta_linf_from_high_regularity", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E70", "title": "Dimensional stability criterion", "family": "I. Sobolev structure and dimensional bounds", "formulas": [ "n\\le3, \\qquad H^2\\hookrightarrow L^\\infty, \\qquad p1." ], "anchor": "e74-curvature-separation-conjecture", "definition": "Current effective status: ○ OPEN The finite-size families \\(Pn,NPn\\) must first be defined.", "formula": "\\inf_n \\frac{\\log|NP_n|}{\\log|P_n|} >1.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e74-curvature-separation-conjecture", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E75", "title": "Physical-oracle impossibility", "family": "J. Computational resource bounds", "formulas": [ "\\nexists\\, O: O(\\psi)=\\operatorname*{arg\\,min}_\\psi\\mathcal E[\\psi] \\quad\\text{in polynomial time}." ], "anchor": "e75-physical-oracle-impossibility", "definition": "Current effective status: ○ OPEN This is a complexity claim requiring a formal computational model and reduction.", "formula": "\\nexists\\, O: O(\\psi)=\\operatorname*{arg\\,min}_\\psi\\mathcal E[\\psi] \\quad\\text{in polynomial time}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e75-physical-oracle-impossibility", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E76", "title": "WCC complexity equivalence", "family": "J. Computational resource bounds", "formulas": [ "P_{\\rm WCC}=P \\quad\\Longrightarrow\\quad \\text{WCC polynomially simulates the declared physical-computation model}." ], "anchor": "e76-wcc-complexity-equivalence", "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.", "formula": "P_{\\rm WCC}=P \\quad\\Longrightarrow\\quad \\text{WCC polynomially simulates the declared physical-computation model}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e76-wcc-complexity-equivalence", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E77", "title": "Mutual-information decay", "family": "K. Entropy and information dynamics", "formulas": [ "\\frac{d}{dt}I(\\psi_t;\\psi_0) \\le -\\gamma\\mathcal E_\\Theta[\\psi_t]." ], "anchor": "e77-mutual-information-decay", "definition": "Current effective status: ○ OPEN The probability law, channel, and regularity assumptions remain open.", "formula": "\\frac{d}{dt}I(\\psi_t;\\psi_0) \\le -\\gamma\\mathcal E_\\Theta[\\psi_t].", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e77-mutual-information-decay", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E78", "title": "Fisher-information curvature bound", "family": "K. Entropy and information dynamics", "formulas": [ "\\mathcal I_F[\\psi] \\ge c\\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx." ], "anchor": "e78-fisher-information-curvature-bound", "definition": "Current effective status: ○ OPEN A common probability density and geometric derivation are required.", "formula": "\\mathcal I_F[\\psi] \\ge c\\int_\\Omega|\\Theta_\\varepsilon[\\psi]|^2dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e78-fisher-information-curvature-bound", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E79", "title": "Entropy-production rate", "family": "K. Entropy and information dynamics", "formulas": [ "\\dot\\Sigma = \\frac{dH_k}{dt} + \\frac{\\mathcal E_\\Theta}{T_{\\rm eff}}." ], "anchor": "e79-entropy-production-rate", "definition": "Current effective status: ◻️ DEFINITION The sign convention and units must be fixed when used physically.", "formula": "\\dot\\Sigma = \\frac{dH_k}{dt} + \\frac{\\mathcal E_\\Theta}{T_{\\rm eff}}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e79-entropy-production-rate", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E80", "title": "Landauer-type bound", "family": "K. Entropy and information dynamics", "formulas": [ "\\Delta E \\ge k_BT_{\\rm eff}\\ln2\\, \\Delta H_{\\rm bits}." ], "anchor": "e80-landauer-type-bound", "definition": "Current effective status: ⚠️ CONDITIONAL If \\(\\Delta H{\\rm bits}\\) is measured in bits,", "formula": "\\Delta E \\ge k_BT_{\\rm eff}\\ln2\\, \\Delta H_{\\rm bits}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e80-landauer-type-bound", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "check_landauer_units", "baseline_checker": "check_landauer_units", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "E81", "title": "Corrected coherence length", "family": "K. Entropy and information dynamics", "formulas": [ "\\boxed{ \\xi_{\\rm coh} = \\left( \\sum_kp_k|k|^2 \\right)^{-1/2} }.", "\\boxed{ \\xi_{\\rm coh} = \\sqrt{ \\frac{\\int_\\Omega|\\psi|^2dx} {\\int_\\Omega|\\nabla\\psi|^2dx} } }." ], "anchor": "e81-corrected-coherence-length", "definition": "Current effective status: ✅ PASS With normalized spectral weights \\(pk\\),", "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} } }.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e81-corrected-coherence-length", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_coherence_length_dimensions", "baseline_checker": "check_coherence_length_dimensions", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "E82", "title": "Information-geometry tensor", "family": "K. Entropy and information dynamics", "formulas": [ "g_{ij}^{({\\rm info})} = \\left\\langle \\partial_i\\Theta_\\varepsilon\\, \\partial_j\\Theta_\\varepsilon \\right\\rangle." ], "anchor": "e82-information-geometry-tensor", "definition": "Current effective status: ◻️ DEFINITION Positive definiteness and coordinate invariance require additional conditions.", "formula": "g_{ij}^{({\\rm info})} = \\left\\langle \\partial_i\\Theta_\\varepsilon\\, \\partial_j\\Theta_\\varepsilon \\right\\rangle.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#e82-information-geometry-tensor", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE1", "title": "Curvature-locking functional", "family": "Curvature-locking equations", "formulas": [ "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}." ], "anchor": "cle1-curvature-locking-functional", "definition": "Current effective status: ◻️ DEFINITION Use the inverse-length convention \\([\\sigma\\star]=L^{-1}\\):", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle1-curvature-locking-functional", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE2", "title": "Corrected curvature-lock Euler-Lagrange equation", "family": "Curvature-locking equations", "formulas": [ "q:=-\\frac{\\psi_{xx}}{\\psi}-\\sigma_\\star^2,", "\\boxed{ q\\frac{\\psi_{xx}}{\\psi^2} -\\psi_{xx} -\\frac{d^2}{dx^2}\\left(\\frac q\\psi\\right) =0 }." ], "anchor": "cle2-corrected-curvature-lock-euler-lagrange-equation", "definition": "Current effective status: ✅ PASS For the real one-dimensional reduction", "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 }.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle2-corrected-curvature-lock-euler-lagrange-equation", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_cle2_variation", "baseline_checker": "check_cle2_variation", "verification_kind": "VARIATIONAL_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE3", "title": "Curvature-locking condition", "family": "Curvature-locking equations", "formulas": [ "W_\\psi=\\sigma_\\star^2." ], "anchor": "cle3-curvature-locking-condition", "definition": "Current effective status: ◻️ DEFINITION Both sides have units \\(L^{-2}\\).", "formula": "W_\\psi=\\sigma_\\star^2.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle3-curvature-locking-condition", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE4", "title": "Locked-field equation", "family": "Curvature-locking equations", "formulas": [ "-\\Delta\\psi = \\sigma_\\star^2\\psi." ], "anchor": "cle4-locked-field-equation", "definition": "The canonical registered object for locked-field equation; consult the source equation and verification metadata for its assumptions and scientific boundary.", "formula": "-\\Delta\\psi = \\sigma_\\star^2\\psi.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle4-locked-field-equation", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_cle_units_chain", "baseline_checker": "check_cle_units_chain", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE5", "title": "Thin/product-torus Laplacian", "family": "Curvature-locking equations", "formulas": [ "\\Delta\\psi \\approx \\frac1{R^2}\\partial_\\theta^2\\psi + \\frac1{r^2}\\partial_\\phi^2\\psi." ], "anchor": "cle5-thinproduct-torus-laplacian", "definition": "Current effective status: ⚠️ CONDITIONAL Under a flat product or thin-torus approximation,", "formula": "\\Delta\\psi \\approx \\frac1{R^2}\\partial_\\theta^2\\psi + \\frac1{r^2}\\partial_\\phi^2\\psi.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle5-thinproduct-torus-laplacian", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE6", "title": "Separation ansatz", "family": "Curvature-locking equations", "formulas": [ "\\psi(\\theta,\\phi)=f(\\theta)g(\\phi),", "\\frac{f''}{f} + \\frac{R^2}{r^2}\\frac{g''}{g} = -\\sigma_\\star^2R^2." ], "anchor": "cle6-separation-ansatz", "definition": "Current effective status: ✅ PASS For", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle6-separation-ansatz", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_separation_substitution", "baseline_checker": "check_separation_substitution", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE7", "title": "Periodic angular mode family", "family": "Curvature-locking equations", "formulas": [ "f''+m^2f=0", "\\boxed{ f(\\theta)=A\\cos(m\\theta)+B\\sin(m\\theta), \\qquad m\\in\\mathbb Z_{\\ge0} }." ], "anchor": "cle7-periodic-angular-mode-family", "definition": "Current effective status: ✅ PASS The periodic reduced equation", "formula": "f''+m^2f=0\n\n\\boxed{ f(\\theta)=A\\cos(m\\theta)+B\\sin(m\\theta), \\qquad m\\in\\mathbb Z_{\\ge0} }.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle7-periodic-angular-mode-family", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_periodic_ode_family", "baseline_checker": "check_periodic_ode_family", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE8", "title": "Selected torus eigenmode", "family": "Curvature-locking equations", "formulas": [ "\\psi(\\theta,\\phi)=Ae^{i\\phi}" ], "anchor": "cle8-selected-torus-eigenmode", "definition": "Current effective status: ⚠️ CONDITIONAL is one admissible winding-one mode. Uniqueness requires additional lowest-mode, chirality, normalization, phase, and boundary-selection principles.", "formula": "\\psi(\\theta,\\phi)=Ae^{i\\phi}", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle8-selected-torus-eigenmode", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "check_torus_eigenmode_selection", "baseline_checker": "check_torus_eigenmode_selection", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CLE9", "title": "Electron radius from curvature", "family": "Curvature-locking equations", "formulas": [ "R=\\frac1{\\sigma_\\star}.", "R=\\frac{\\hbar}{m_ec}\\approx386.16\\ {\\rm fm}." ], "anchor": "cle9-electron-radius-from-curvature", "definition": "Current effective status: ✅ PASS For \\(\\sigma\\star=mec/\\hbar\\),", "formula": "R=\\frac1{\\sigma_\\star}.\n\nR=\\frac{\\hbar}{m_ec}\\approx386.16\\ {\\rm fm}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle9-electron-radius-from-curvature", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_cle_units_chain", "baseline_checker": "check_cle_units_chain", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The registered electron-radius expression has the dimensions and numerical scale of the reduced Compton wavelength.", "verification_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" } }, { "id": "CLE10", "title": "Curvature scalar chain", "family": "Curvature-locking equations", "formulas": [ "\\boxed{ W_\\psi = -\\frac{\\Delta\\psi}{\\psi} = \\sigma_\\star^2 }, \\qquad R=\\sigma_\\star^{-1}." ], "anchor": "cle10-curvature-scalar-chain", "definition": "The canonical registered object for curvature scalar chain; consult the source equation and verification metadata for its assumptions and scientific boundary.", "formula": "\\boxed{ W_\\psi = -\\frac{\\Delta\\psi}{\\psi} = \\sigma_\\star^2 }, \\qquad R=\\sigma_\\star^{-1}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cle10-curvature-scalar-chain", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_cle_units_chain", "baseline_checker": "check_cle_units_chain", "verification_kind": "DIMENSIONAL_CHECK", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "G1", "title": "Log-periodic ghost modulation", "family": "Logarithmic and ghost equations", "formulas": [ "\\delta_g(E) = A_g\\cos\\!\\left( k_\\ell\\ln\\frac E{E_0}+\\phi \\right),", "|\\delta_g(E)|\\le|A_g|." ], "anchor": "g1-log-periodic-ghost-modulation", "definition": "Current effective status: ✅ PASS For \\(E>0\\) and \\(E0>0\\),", "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|.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#g1-log-periodic-ghost-modulation", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_log_modulation", "baseline_checker": "check_log_modulation", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "EX", "title": "Logarithmic field representation", "family": "Logarithmic and ghost equations", "formulas": [ "u=\\ln\\psi, \\qquad \\psi=e^u.", "\\nabla\\psi=e^u\\nabla u,", "\\Delta\\psi=e^u(\\Delta u+|\\nabla u|^2),", "\\frac{\\Delta\\psi}{\\psi} = \\Delta u+|\\nabla u|^2." ], "anchor": "ex-logarithmic-field-representation", "definition": "Current effective status: ✅ PASS For a positive real field \\(\\psi>0\\), let", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#ex-logarithmic-field-representation", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_log_laplacian_identity", "baseline_checker": "check_log_laplacian_identity", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "EY", "title": "Log-curvature evolution", "family": "Logarithmic and ghost equations", "formulas": [ "\\partial_tu = \\Delta u+|\\nabla u|^2," ], "anchor": "ey-log-curvature-evolution", "definition": "Current effective status: ✅ PASS If", "formula": "\\partial_tu = \\Delta u+|\\nabla u|^2,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#ey-log-curvature-evolution", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_log_flow_reduction", "baseline_checker": "check_log_flow_reduction", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "EZ", "title": "Cole-Hopf reduction", "family": "Logarithmic and ghost equations", "formulas": [ "\\psi=e^u,", "\\partial_t\\psi = e^u\\partial_tu = e^u(\\Delta u+|\\nabla u|^2) = \\Delta\\psi.", "\\boxed{\\partial_t\\psi=\\Delta\\psi}." ], "anchor": "ez-cole-hopf-reduction", "definition": "Current effective status: ✅ PASS With", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#ez-cole-hopf-reduction", "status": "PASS", "baseline_status": "PASS", "status_changed": false, "status_changed_by": null, "checker": "check_cole_hopf", "baseline_checker": "check_cole_hopf", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "STANDARD_MATHEMATICS", "verification_meaning": "The registered Cole-Hopf substitution reduces the encoded nonlinear equation under its stated assumptions.", "verification_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" } }, { "id": "FA", "title": "Filament-localization condition", "family": "Logarithmic and ghost equations", "formulas": [ "|\\nabla u| \\sim \\kappa_{\\rm core}." ], "anchor": "fa-filament-localization-condition", "definition": "Current effective status: ⚠️ CONDITIONAL A norm, tolerance, scale, and dynamical derivation are required.", "formula": "|\\nabla u| \\sim \\kappa_{\\rm core}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#fa-filament-localization-condition", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_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" } }, { "id": "CM1", "title": "Fundamental field evolution", "family": "Curvature-acoustic cosmology", "formulas": [ "i\\partial_t\\psi = -\\Theta_\\varepsilon[\\psi]\\,J[\\psi],", "J[\\psi] = |\\psi|^2\\Delta\\psi\\,\\varepsilon_{\\rm vac}." ], "anchor": "cm1-fundamental-field-evolution", "definition": "Current effective status: ○ OPEN Coefficient dimensions and derivation remain open.", "formula": "i\\partial_t\\psi = -\\Theta_\\varepsilon[\\psi]\\,J[\\psi],\n\nJ[\\psi] = |\\psi|^2\\Delta\\psi\\,\\varepsilon_{\\rm vac}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm1-fundamental-field-evolution", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM2", "title": "Curvature-spectral tilt", "family": "Curvature-acoustic cosmology", "formulas": [ "P_{\\rm prim}(k)\\propto k^{-\\alpha_{\\rm WCT}},", "n_s-1=-\\alpha_{\\rm WCT},", "\\alpha_{\\rm WCT} = -\\frac{d\\ln|\\Theta(k)|}{d\\ln k}." ], "anchor": "cm2-curvature-spectral-tilt", "definition": "The canonical registered object for curvature-spectral tilt; consult the source equation and verification metadata for its assumptions and scientific boundary.", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm2-curvature-spectral-tilt", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM3", "title": "Potential from curvature", "family": "Curvature-acoustic cosmology", "formulas": [ "\\Phi(k,t) = -C_\\Phi\\frac{\\Theta(k,t)}{k^2}." ], "anchor": "cm3-potential-from-curvature", "definition": "The canonical registered object for potential from curvature; consult the source equation and verification metadata for its assumptions and scientific boundary.", "formula": "\\Phi(k,t) = -C_\\Phi\\frac{\\Theta(k,t)}{k^2}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm3-potential-from-curvature", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM4", "title": "Horizon-entry potential decay", "family": "Curvature-acoustic cosmology", "formulas": [ "\\partial_t\\Phi = -\\Gamma\\Phi,", "\\Gamma(k,t) = \\left| \\frac{\\partial_t\\Theta(k,t)}{\\Theta(k,t)} \\right|," ], "anchor": "cm4-horizon-entry-potential-decay", "definition": "Current effective status: ○ OPEN on the domain \\(\\Theta\\neq0\\).", "formula": "\\partial_t\\Phi = -\\Gamma\\Phi,\n\n\\Gamma(k,t) = \\left| \\frac{\\partial_t\\Theta(k,t)}{\\Theta(k,t)} \\right|,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm4-horizon-entry-potential-decay", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM5", "title": "Curvature-acoustic oscillators", "family": "Curvature-acoustic cosmology", "formulas": [ "\\ddot\\delta_\\gamma +c_s^2k^2\\delta_\\gamma = -k^2\\Phi,", "\\ddot\\delta_b +\\mathcal R\\,c_s^2k^2\\delta_\\gamma = -k^2\\Phi,", "\\mathcal R = \\frac{E_{\\rm comp}}{E_{\\rm rad}}." ], "anchor": "cm5-curvature-acoustic-oscillators", "definition": "The canonical registered object for curvature-acoustic oscillators; consult the source equation and verification metadata for its assumptions and scientific boundary.", "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}}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm5-curvature-acoustic-oscillators", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM6", "title": "Sound speed from curvature feedback", "family": "Curvature-acoustic cosmology", "formulas": [ "c_s^2(t) = \\frac1{3(1+\\mathcal R(t))} \\left[ 1-\\beta_{\\rm curv} \\frac{E_{\\rm curv}(t)}{E_{\\rm tot}} \\right]." ], "anchor": "cm6-sound-speed-from-curvature-feedback", "definition": "Current effective status: ○ OPEN Positivity requires the bracketed factor to be nonnegative.", "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].", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm6-sound-speed-from-curvature-feedback", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM7", "title": "Curvature diffusion", "family": "Curvature-acoustic cosmology", "formulas": [ "\\dot\\delta_\\gamma = v_\\gamma - D_{\\rm curv}(t)k^2\\delta_\\gamma,", "D_{\\rm curv}(t) = \\frac{\\langle|\\nabla\\psi|^2\\rangle} {\\langle|\\psi|^2\\rangle}." ], "anchor": "cm7-curvature-diffusion", "definition": "Current effective status: ○ OPEN A phenomenological damping replacement is", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm7-curvature-diffusion", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM8", "title": "Initial conditions", "family": "Curvature-acoustic cosmology", "formulas": [ "\\delta_\\gamma(0)=\\delta_b(0)=-2\\Phi(k,0),", "\\Phi(k,0) = -C_\\Phi\\frac{\\Theta(k,0)}{k^2}." ], "anchor": "cm8-initial-conditions", "definition": "Current effective status: ○ OPEN Use CM3 consistently:", "formula": "\\delta_\\gamma(0)=\\delta_b(0)=-2\\Phi(k,0),\n\n\\Phi(k,0) = -C_\\Phi\\frac{\\Theta(k,0)}{k^2}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm8-initial-conditions", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM9", "title": "First-order mode system", "family": "Curvature-acoustic cosmology", "formulas": [ "\\dot\\delta_\\gamma=v_\\gamma, \\qquad \\dot v_\\gamma=-c_s^2k^2\\delta_\\gamma-k^2\\Phi,", "\\dot\\delta_b=v_b, \\qquad \\dot v_b=-\\mathcal R c_s^2k^2\\delta_\\gamma-k^2\\Phi." ], "anchor": "cm9-first-order-mode-system", "definition": "Current effective status: ✅ PASS Baseline status: ○ OPEN", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm9-first-order-mode-system", "status": "PASS", "baseline_status": "OPEN", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_cm9_first_order_equivalence", "checker": "check_cm9_first_order_equivalence", "baseline_checker": "classify_open", "verification_kind": "ALGEBRAIC_IDENTITY", "verification_scope": "INTERNAL_CONSISTENCY", "verification_meaning": "The first-order velocity system is equivalent to the registered second-order oscillator system.", "verification_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" } }, { "id": "CM10", "title": "Tight-coupling drag", "family": "Curvature-acoustic cosmology", "formulas": [ "\\delta_b \\leftarrow (1-\\varepsilon_{\\rm drag})\\delta_b + \\varepsilon_{\\rm drag}\\delta_\\gamma,", "\\varepsilon_{\\rm drag} = \\frac{E_{\\rm exch}}{E_{\\rm comp}}, \\qquad 0\\le\\varepsilon_{\\rm drag}\\le1." ], "anchor": "cm10-tight-coupling-drag", "definition": "The canonical registered object for tight-coupling drag; consult the source equation and verification metadata for its assumptions and scientific boundary.", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm10-tight-coupling-drag", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM11", "title": "Curvature damping envelope", "family": "Curvature-acoustic cosmology", "formulas": [ "D(k) = \\exp\\!\\left(-\\frac{k^2}{k_D^2}\\right),", "k_D^{-2} = \\int_0^{t_\\star}D_{\\rm curv}(t)\\,dt." ], "anchor": "cm11-curvature-damping-envelope", "definition": "Current effective status: ✅ PASS Baseline status: ○ OPEN", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm11-curvature-damping-envelope", "status": "PASS", "baseline_status": "OPEN", "status_changed": true, "status_changed_by": "derived_overrides.yaml:check_cm11_gaussian_damping", "checker": "check_cm11_gaussian_damping", "baseline_checker": "classify_open", "verification_kind": "SYMBOLIC_DERIVATION", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The registered curvature-diffusion equation integrates to the stated Gaussian damping envelope.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM12", "title": "Dimensionless power spectrum", "family": "Curvature-acoustic cosmology", "formulas": [ "\\Delta^2(k) = \\frac{k^3}{2\\pi^2}P(k)." ], "anchor": "cm12-dimensionless-power-spectrum", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "formula": "\\Delta^2(k) = \\frac{k^3}{2\\pi^2}P(k).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm12-dimensionless-power-spectrum", "status": "DEFINITION", "baseline_status": "OPEN", "status_changed": true, "status_changed_by": "derived_overrides.yaml:classify_definition", "checker": "classify_definition", "baseline_checker": "classify_open", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The object defines a dimensionless power-spectrum quantity and is not itself a cosmological derivation.", "verification_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" } }, { "id": "CM13", "title": "Peak metrics", "family": "Curvature-acoustic cosmology", "formulas": [ "r_{21} = \\frac{P(k_2)}{P(k_1)}, \\qquad r_{31} = \\frac{P(k_3)}{P(k_1)},", "s_{21} = \\frac{k_2}{k_1}, \\qquad s_{31} = \\frac{k_3}{k_1}." ], "anchor": "cm13-peak-metrics", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm13-peak-metrics", "status": "DEFINITION", "baseline_status": "OPEN", "status_changed": true, "status_changed_by": "derived_overrides.yaml:classify_definition", "checker": "classify_definition", "baseline_checker": "classify_open", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The object defines peak-ratio bookkeeping quantities and is not itself a cosmological derivation.", "verification_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" } }, { "id": "CM14", "title": "Peak-response interpretation", "family": "Curvature-acoustic cosmology", "formulas": [ "\\text{faster }\\Theta\\text{ decay}\\Rightarrow s_{ij}\\uparrow,", "\\text{larger compression}\\Rightarrow r_{31}\\uparrow,", "\\text{larger radiative fraction}\\Rightarrow r_{21}\\downarrow." ], "anchor": "cm14-peak-response-interpretation", "definition": "Current effective status: ○ OPEN Proposed qualitative relations:", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm14-peak-response-interpretation", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM15", "title": "WCT angular scaling", "family": "Curvature-acoustic cosmology", "formulas": [ "k_{\\rm phys} = \\frac{k}{a_{\\rm WCT}(t)},", "a_{\\rm WCT}(t) = \\left[ \\frac{E_{\\rm curv}(0)} {E_{\\rm curv}(t)} \\right]^{1/3}." ], "anchor": "cm15-wct-angular-scaling", "definition": "The canonical registered object for wct angular scaling; consult the source equation and verification metadata for its assumptions and scientific boundary.", "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}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm15-wct-angular-scaling", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM16", "title": "Acoustic horizon", "family": "Curvature-acoustic cosmology", "formulas": [ "R_{\\rm hor}(t) = \\int_0^tc_s(t')\\,dt',", "k_{\\rm hor} = \\frac{2\\pi}{R_{\\rm hor}}." ], "anchor": "cm16-acoustic-horizon", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "formula": "R_{\\rm hor}(t) = \\int_0^tc_s(t')\\,dt',\n\nk_{\\rm hor} = \\frac{2\\pi}{R_{\\rm hor}}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm16-acoustic-horizon", "status": "DEFINITION", "baseline_status": "OPEN", "status_changed": true, "status_changed_by": "derived_overrides.yaml:classify_definition", "checker": "classify_definition", "baseline_checker": "classify_open", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The object defines horizon-scale quantities and is not itself a cosmological derivation.", "verification_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" } }, { "id": "CM17", "title": "Curvature-energy closure", "family": "Curvature-acoustic cosmology", "formulas": [ "E_{\\rm curv}(t) + E_{\\rm grad}(t) = E_{\\rm tot}," ], "anchor": "cm17-curvature-energy-closure", "definition": "Current effective status: ○ OPEN for a closed sector with no external source or loss.", "formula": "E_{\\rm curv}(t) + E_{\\rm grad}(t) = E_{\\rm tot},", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm17-curvature-energy-closure", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM18", "title": "Minimal cosmology closure set", "family": "Curvature-acoustic cosmology", "formulas": [ "\\mathfrak C_{\\rm min} = \\{\\mathrm{CM1},\\mathrm{CM2},\\mathrm{CM3}, \\mathrm{CM4},\\mathrm{CM5},\\mathrm{CM7}\\}." ], "anchor": "cm18-minimal-cosmology-closure-set", "definition": "Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN", "formula": "\\mathfrak C_{\\rm min} = \\{\\mathrm{CM1},\\mathrm{CM2},\\mathrm{CM3}, \\mathrm{CM4},\\mathrm{CM5},\\mathrm{CM7}\\}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm18-minimal-cosmology-closure-set", "status": "DEFINITION", "baseline_status": "OPEN", "status_changed": true, "status_changed_by": "derived_overrides.yaml:classify_definition", "checker": "classify_definition", "baseline_checker": "classify_open", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The object defines a closure set for bookkeeping and is not a derivation of cosmology.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM19", "title": "Acoustic speed from curvature equation of state", "family": "Curvature-acoustic cosmology", "formulas": [ "c_s^2 = \\frac{\\partial P_{\\rm curv}} {\\partial\\rho_{\\rm curv}}," ], "anchor": "cm19-acoustic-speed-from-curvature-equation-of-state", "definition": "Current effective status: ○ OPEN where the derivative must be taken along a specified thermodynamic or dynamical path.", "formula": "c_s^2 = \\frac{\\partial P_{\\rm curv}} {\\partial\\rho_{\\rm curv}},", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm19-acoustic-speed-from-curvature-equation-of-state", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CM20", "title": "Theta-based expansion ansatz", "family": "Curvature-acoustic cosmology", "formulas": [ "H(t) = \\frac{\\dot a_{\\rm WCT}}{a_{\\rm WCT}} = \\sqrt{ \\frac{\\rho_\\Theta(t)}{3|K|} }." ], "anchor": "cm20-theta-based-expansion-ansatz", "definition": "Current effective status: ○ OPEN The constant \\(K\\) must carry the units needed for \\(H^2\\), and the equation requires independent derivation.", "formula": "H(t) = \\frac{\\dot a_{\\rm WCT}}{a_{\\rm WCT}} = \\sqrt{ \\frac{\\rho_\\Theta(t)}{3|K|} }.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#cm20-theta-based-expansion-ansatz", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP1", "title": "Closed spectral-loop representation", "family": "Topology and spectral emergence", "formulas": [ "\\gamma(s) = \\sum_{k=1}^{K} \\left[ a_k\\cos(ks)+b_k\\sin(ks) \\right], \\qquad s\\in[0,2\\pi)." ], "anchor": "top1-closed-spectral-loop-representation", "definition": "Current effective status: ◻️ DEFINITION This is the chosen configuration representation; emergence of the basis is a separate empirical claim.", "formula": "\\gamma(s) = \\sum_{k=1}^{K} \\left[ a_k\\cos(ks)+b_k\\sin(ks) \\right], \\qquad s\\in[0,2\\pi).", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top1-closed-spectral-loop-representation", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP2", "title": "WCT loop-energy functional", "family": "Topology and spectral emergence", "formulas": [ "\\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]." ], "anchor": "top2-wct-loop-energy-functional", "definition": "Current effective status: ◻️ DEFINITION", "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].", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top2-wct-loop-energy-functional", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP3", "title": "Irreversible gradient flow", "family": "Topology and spectral emergence", "formulas": [ "\\partial_t\\gamma = -\\frac{\\delta\\mathcal E_{\\rm loop}}{\\delta\\gamma}.", "\\frac{d\\mathcal E_{\\rm loop}}{dt} = - \\left\\| \\frac{\\delta\\mathcal E_{\\rm loop}}{\\delta\\gamma} \\right\\|^2 \\le0." ], "anchor": "top3-irreversible-gradient-flow", "definition": "Current effective status: ⚠️ CONDITIONAL For a differentiable gradient flow,", "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.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top3-irreversible-gradient-flow", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP4", "title": "Emergent-topology criterion", "family": "Topology and spectral emergence", "formulas": [ "I[\\gamma_t]\\to I_\\infty", "\\frac{d\\mathcal E}{dt}<0" ], "anchor": "top4-emergent-topology-criterion", "definition": "Current effective status: ○ OPEN A proposed physical invariant \\(I[\\gamma]\\) satisfies", "formula": "I[\\gamma_t]\\to I_\\infty\n\n\\frac{d\\mathcal E}{dt}<0", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top4-emergent-topology-criterion", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP5", "title": "WCT dynamical codimension", "family": "Topology and spectral emergence", "formulas": [ "\\operatorname{codim}_{\\rm WCT}(\\gamma) := \\text{minimum number of singular events required to reach the unknot}." ], "anchor": "top5-wct-dynamical-codimension", "definition": "Current effective status: ◻️ DEFINITION This is not manifold codimension.", "formula": "\\operatorname{codim}_{\\rm WCT}(\\gamma) := \\text{minimum number of singular events required to reach the unknot}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top5-wct-dynamical-codimension", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP6", "title": "Spectral topology bands", "family": "Topology and spectral emergence", "formulas": [ "\\epsilon_\\kappa = \\frac1L\\int_\\gamma\\kappa^2ds." ], "anchor": "top6-spectral-topology-bands", "definition": "Current effective status: ○ OPEN Define", "formula": "\\epsilon_\\kappa = \\frac1L\\int_\\gamma\\kappa^2ds.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top6-spectral-topology-bands", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP7", "title": "Topological mass proxy", "family": "Topology and spectral emergence", "formulas": [ "m_{\\rm WCT} \\propto \\epsilon_\\kappa." ], "anchor": "top7-topological-mass-proxy", "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.", "formula": "m_{\\rm WCT} \\propto \\epsilon_\\kappa.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top7-topological-mass-proxy", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP8", "title": "Holonomy non-invariance", "family": "Topology and spectral emergence", "formulas": [ "H_\\tau[\\gamma] = \\int_\\gamma\\tau\\,ds," ], "anchor": "top8-holonomy-non-invariance", "definition": "Current effective status: ○ OPEN For", "formula": "H_\\tau[\\gamma] = \\int_\\gamma\\tau\\,ds,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top8-holonomy-non-invariance", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "TOP9", "title": "Protein-particle structural correspondence", "family": "Topology and spectral emergence", "formulas": [ "\\text{knotted protein states} \\longleftrightarrow \\text{stable WCT loop excitations}" ], "anchor": "top9-protein-particle-structural-correspondence", "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.", "formula": "\\text{knotted protein states} \\longleftrightarrow \\text{stable WCT loop excitations}", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#top9-protein-particle-structural-correspondence", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CORR1", "title": "Full Lyapunov candidate", "family": "Canonical correction layer", "formulas": [ "\\mathcal E_{\\rm WCT}[\\psi] = \\int \\left( |\\nabla\\psi|^2 + |\\Theta_\\varepsilon[\\psi]|^2 \\right)dx." ], "anchor": "corr1-full-lyapunov-candidate", "definition": "Current effective status: ◻️ DEFINITION The curvature term alone is only one component.", "formula": "\\mathcal E_{\\rm WCT}[\\psi] = \\int \\left( |\\nabla\\psi|^2 + |\\Theta_\\varepsilon[\\psi]|^2 \\right)dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr1-full-lyapunov-candidate", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CORR2", "title": "Mean-amplitude spectral closure", "family": "Canonical correction layer", "formulas": [ "D_{\\rm eff}^2 := \\langle|\\psi|^2\\rangle+\\varepsilon^2,", "C_\\Theta(k) \\approx \\frac{k^4}{D_{\\rm eff}^2}." ], "anchor": "corr2-mean-amplitude-spectral-closure", "definition": "Current effective status: ⚠️ CONDITIONAL Under a weak-intermittency mean-amplitude approximation,", "formula": "D_{\\rm eff}^2 := \\langle|\\psi|^2\\rangle+\\varepsilon^2,\n\nC_\\Theta(k) \\approx \\frac{k^4}{D_{\\rm eff}^2}.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr2-mean-amplitude-spectral-closure", "status": "CONDITIONAL", "baseline_status": "CONDITIONAL", "status_changed": false, "status_changed_by": null, "checker": "classify_conditional", "baseline_checker": "classify_conditional", "verification_kind": "UNRESOLVED", "verification_scope": "MODEL_CONDITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CORR3", "title": "Spectral-weight notation", "family": "Canonical correction layer", "formulas": [ "\\lambda_\\Theta", "\\lambda_{\\rm ex}" ], "anchor": "corr3-spectral-weight-notation", "definition": "Current effective status: ◻️ DEFINITION Use", "formula": "\\lambda_\\Theta\n\n\\lambda_{\\rm ex}", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr3-spectral-weight-notation", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CORR4", "title": "Macro-micro control parameter", "family": "Canonical correction layer", "formulas": [ "\\Xi = \\frac{\\int k^4\\rho(k)\\,dk}{H}, \\qquad H=-\\sum_kP_k\\ln P_k," ], "anchor": "corr4-macro-micro-control-parameter", "definition": "Current effective status: ◻️ DEFINITION on the domain \\(H>0\\).", "formula": "\\Xi = \\frac{\\int k^4\\rho(k)\\,dk}{H}, \\qquad H=-\\sum_kP_k\\ln P_k,", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr4-macro-micro-control-parameter", "status": "DEFINITION", "baseline_status": "DEFINITION", "status_changed": false, "status_changed_by": null, "checker": "classify_definition", "baseline_checker": "classify_definition", "verification_kind": "DEFINITION_CHECK", "verification_scope": "DEFINITIONAL", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_APPLICABLE", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CORR5", "title": "Entropy-curvature coupling", "family": "Canonical correction layer", "formulas": [ "\\frac{dH}{dt} \\le -\\mu \\int|\\Theta_\\varepsilon[\\psi]|^2dx." ], "anchor": "corr5-entropy-curvature-coupling", "definition": "Current effective status: ○ OPEN This remains the same open claim as E43 unless derived for a specified evolution.", "formula": "\\frac{dH}{dt} \\le -\\mu \\int|\\Theta_\\varepsilon[\\psi]|^2dx.", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr5-entropy-curvature-coupling", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } }, { "id": "CORR6", "title": "Isoelectronic-flow alignment", "family": "Canonical correction layer", "formulas": [], "anchor": "corr6-isoelectronic-flow-alignment", "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.", "formula": "", "source": "https://github.com/rickyjreyes/geometry_of_resonance/blob/main/WCT_FULL_EQUATION_LIST_CORRECTED.md#corr6-isoelectronic-flow-alignment", "status": "OPEN", "baseline_status": "OPEN", "status_changed": false, "status_changed_by": null, "checker": "classify_open", "baseline_checker": "classify_open", "verification_kind": "UNRESOLVED", "verification_scope": "UNRESOLVED", "verification_meaning": "The assigned executable check is reported under its declared assumptions.", "verification_limitations": [], "assumptions": [], "formalization": { "status": "OPEN", "relationship": null, "declarations": [], "source": null, "limitations": [] }, "empirical_validation": { "status": "NOT_TESTED", "evidence_ids": [], "independent_replication": "NONE" } } ]