Canonical equation layer

WCT Equation Explorer

Every object is rendered from one compiled effective registry. Outcome, verification kind, Lean coverage, empirical state, assumptions, and status history remain separate.

Open full corrected registry ↗SymPy audit Lean coverage Compiled registry

142registered objects

68effective PASS

18conditional

30open

Registry v2.0.0 · generated 2026-08-21T08:57:49+00:00 · WCT SymPy ref main

Showing all 142 objects.

Equation family

Master systems

9 objects

Curvature-locking functional

CONDITIONAL · SYMBOLIC DERIVATION

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.

$$\sigma(s):=\sqrt{\kappa(s)^2+\tau(s)^2}, \qquad \langle f\rangle_w:= \frac{\oint_\Gamma w f\,ds}{\oint_\Gamma w\,ds}, \quad \oint_\Gamma w\,ds>0.$$

$$S_{\rm lock}[\varphi] = \oint_\Gamma w(s)\bigl(\partial_s\varphi-\sigma\bigr)^2\,ds.$$

$$m=\frac{\hbar}{c}\langle\sigma\rangle_w.$$

Checker: check_locking_family · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

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Nonsingular curvature operator and Lyapunov candidate

PASS · LIMIT CHECK

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.

$$R_\varepsilon(\psi) := \frac{\overline{\psi}} {|\psi|^2+\varepsilon^2e^{-2\alpha|\psi|^2}},$$

$$\Theta_\varepsilon[\psi] := -(\Delta\psi)R_\varepsilon(\psi).$$

$$R_\varepsilon(\psi)\longrightarrow \frac1\psi \qquad(\varepsilon\to0).$$

$$\mathcal E_{\rm WCT}[\psi] = \int_\Omega \left( |\nabla\psi|^2+|\Theta_\varepsilon[\psi]|^2 \right)\,dx.$$

Checker: check_regularized_denominator · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Finite-band spectral selector

PASS · SIGN OR EXTREMUM CHECK

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.

$$\partial_tA = \mu A-g|A|^2A-b(\Delta+k_\star^2)^2A, \qquad b>0.$$

$$-b\bigl(|k|^2-k_\star^2\bigr)^2,$$

Checker: check_master_sh_sign · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Dimensional stability threshold

PASS · FORMAL THEOREM

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.

$$H^2(\Omega)\hookrightarrow L^\infty(\Omega) \quad\text{when}\quad 2>\frac n2.$$

$$n\le3.$$

Checker: check_h2_embedding_threshold · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

Assumptions: ASM-STANDARD-SOBOLEV-DOMAIN

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Curvature-bounded computation

CONDITIONAL · UNRESOLVED

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.

$$\psi^{(t+1)}(x) = U\!\left( \psi^{(t)}(x), \{\psi^{(t)}(y):y\in N(x)\} \right).$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

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M6A

Unified linear operator

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$\mathcal L_{\rm WCT} = c_1(\Delta+\sigma^2) -c_2(\Delta+k_\star^2)^2 +i\,c_3m +c_4R^{-(2+n/p)}, \qquad c_2>0.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: DEFINITION · Empirical: NOT APPLICABLE

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M6B

Nonlinear curvature operator

OPEN · COUNTEREXAMPLE TEST

Current effective status: ○ OPEN The operator is well-defined for $\varepsilon>0$; uniqueness of this nonlinear closure remains open.

$$\mathcal N_{\rm curv}[\psi] = -(\Delta\psi) \frac{\overline{\psi}} {|\psi|^2+\varepsilon^2e^{-2\alpha|\psi|^2}}.$$

Checker: check_uniqueness_claim · Scope: UNRESOLVED · Lean: PROVED · Empirical: NOT TESTED

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Full curvature-wavefield equation

PASS · SIGN OR EXTREMUM CHECK

Current effective status: ✅ PASS The explicit negative fourth-order term supplies finite-band ultraviolet damping. Other dynamical claims require separate analysis.

$$\partial_t\psi = \mathcal N_{\rm curv}[\psi] +g|\psi|^2\psi +c_1(\Delta+\sigma^2)\psi -c_2(\Delta+k_\star^2)^2\psi +i\,c_3m\psi +c_4R^{-(2+n/p)}\psi +\eta\psi\circ\xi(t), \qquad c_2>0.$$

Checker: check_master_uwct_sign · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Curvature-acoustic cosmology system

OPEN · UNRESOLVED

Current effective status: ○ OPEN Representative closure relations are

$$\Phi(k,t)=-C_\Phi\frac{\Theta(k,t)}{k^2},$$

$$\delta_g(E) = A_g\cos\!\left( k_\ell\ln\frac{E}{E_0}+\phi \right).$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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Equation family

A. Rest energy, curvature, and loop locking

9 objects

E1A

Curvature-rate density

PASS · DIMENSIONAL CHECK

The canonical registered object for curvature-rate density; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$\sigma_{\rm dens}(s)=\kappa(s)^2+\tau(s)^2, \qquad [\sigma_{\rm dens}]=L^{-2}.$$

Checker: check_e1a · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E1B

Curvature spectral rate

PASS · DIMENSIONAL CHECK

The canonical registered object for curvature spectral rate; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$\sigma(s)=\sqrt{\kappa(s)^2+\tau(s)^2}, \qquad [\sigma]=L^{-1}.$$

Checker: check_e1b · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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Weighted loop average

PASS · ALGEBRAIC IDENTITY

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.

$$\langle f\rangle_w = \frac{\oint_\Gamma w(s)f(s)\,ds} {\oint_\Gamma w(s)\,ds}.$$

Checker: check_weighted_average · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

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Loop-locking action

PASS · VARIATIONAL DERIVATION

A constrained phase-curvature mismatch action. Nonnegative weighting makes its squared mismatch term nonnegative; existence and uniqueness of continuum minimizers require separate analysis.

$$S_{\rm eff}[\varphi] = \oint_\Gamma w(\partial_s\varphi-\sigma)^2\,ds + \Lambda \left( \oint_\Gamma\partial_s\varphi\,ds-2\pi n \right), \qquad n\in\mathbb Z.$$

Checker: check_locking_variation · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Covariant locking solution

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Stationarity gives

$$\partial_s\varphi = \sigma+\frac{\alpha_{\rm lock}}{w},$$

$$\alpha_{\rm lock} = \frac{ 2\pi n-\oint_\Gamma\sigma\,ds }{ \oint_\Gamma ds/w },$$

Checker: check_locking_solution · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Effective-wavenumber chain

PASS · SYMBOLIC DERIVATION

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.

$$L_s:=\oint_\Gamma ds, \qquad k_{\rm wind}:=\frac{2\pi|n|}{L_s}, \qquad k_\sigma:=\frac1{L_s}\oint_\Gamma\sigma\,ds.$$

$$k_{\rm wind}=k_\sigma=\langle\sigma\rangle_w$$

Checker: check_effective_wavenumber_chain_derived · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Assumptions: ASM-COMPATIBLE-ORIENTATION, ASM-EXACT-INTEGRATED-LOCKING, ASM-CONSTANT-POSITIVE-WEIGHT

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_effective_wavenumber_chain_derived.

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Mass-curvature law

PASS · DIMENSIONAL CHECK

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.

$$E_{\rm rest}=\hbar c\,k_{\rm eff}, \qquad m=\frac{\hbar}{c}k_{\rm eff}.$$

$$m=\frac{\hbar}{c}\langle\sigma\rangle_w.$$

Checker: check_mass_curvature_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Solenoidal mass law

PASS · DIMENSIONAL CHECK

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.

$$m = \frac{\hbar}{c} \left\langle \sqrt{\kappa^2+\tau^2} \right\rangle_\Gamma.$$

Checker: check_mass_curvature_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Corrected weighted-lock identity

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS Substituting E4 gives

$$\boxed{ \oint_\Gamma w\,\partial_s\varphi\,ds = \oint_\Gamma w\,\sigma\,ds + \alpha_{\rm lock}L_s }.$$

$$2\pi\oint ds/\oint ds/w$$

Checker: check_e8_identity · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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Equation family

B. Phase-flux and finite-band selection

8 objects

Phase-flux constitutive relation

PASS · ALGEBRAIC IDENTITY

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.

$$\mathbf S(x,t)=u(x,t)\nabla\theta(x,t).$$

$$\partial_tu+\nabla\cdot\mathbf S=0.$$

Checker: check_phase_flux_from_polar_field · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline DEFINITION → effective PASS via derived_overrides.yaml:check_phase_flux_from_polar_field.

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E10

Radial shell quantization

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS Both sides are dimensionless.

$$\int_{r_1}^{r_2}k_r(r)\,dr=2\pi n, \qquad n\in\mathbb Z.$$

Checker: check_shell_quantization_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E11

Phase winding

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS provided $\psi\neq0$ on the loop and the phase is continuous modulo $2\pi$.

$$m(\gamma) = \frac1{2\pi} \oint_\gamma\nabla\theta\cdot d\boldsymbol\ell \in\mathbb Z,$$

Checker: check_winding_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E12

Finite-band dispersion rail

PASS · SIGN OR EXTREMUM CHECK

The quartic finite-band dispersion relation with a stationary maximum at the selected nonzero wavenumber. It verifies the preferred linear shell and damping sign.

$$\lambda_{\rm grow}(k) = r+a|k|^2-b|k|^4, \qquad a,b>0.$$

$$k_\star=\sqrt{\frac{a}{2b}}.$$

$$\lambda_{\rm grow}(k) = \mu-b(|k|^2-k_\star^2)^2, \qquad \mu=r+\frac{a^2}{4b}.$$

Checker: check_dispersion_stationary_point · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E13

Band-pass amplitude evolution

PASS · VARIATIONAL DERIVATION

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.

$$\partial_tA = (r-a\Delta-b\Delta^2)A-\beta|A|^2A,$$

$$\partial_tA = \mu A-b(\Delta+k_\star^2)^2A-\beta|A|^2A.$$

Checker: check_bandpass_gradient_flow · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_bandpass_gradient_flow.

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E14

Band-pass Lyapunov functional

PASS · VARIATIONAL DERIVATION

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.

$$\mathcal E[A] = \int_\Omega \left[ -\mu|A|^2 +b|(\Delta+k_\star^2)A|^2 +\frac{\beta}{2}|A|^4 \right]dx.$$

Checker: check_bandpass_gradient_flow · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_bandpass_gradient_flow.

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E15

Modal growth bound

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$\frac{d}{dt}|\widehat A_k|^2 \le 2\lambda_{\rm grow}(k)|\widehat A_k|^2 -c|\widehat A_k|^4, \qquad c>0.$$

Checker: check_modal_growth_bound_derived · Scope: INTERNAL CONSISTENCY · Lean: OPEN · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_modal_growth_bound_derived.

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E16

Linear spectral concentration

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For the linearized dynamics,

$$P_k(t)=P_k(0)e^{2\lambda_{\rm grow}(k)t}.$$

$$\operatorname*{arg\,max}_kP_k(t)\to k_\star.$$

Checker: check_linear_spectral_growth · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Equation family

C. Curvature feedback and Lyapunov dynamics

7 objects

E17

Nonsingular curvature-feedback operator

PASS · LIMIT CHECK

Current effective status: ✅ PASS For $\varepsilon>0$, the denominator is strictly positive for all complex $\psi$. This replaces

$$R_\varepsilon(\psi) = \frac{\overline\psi} {|\psi|^2+\varepsilon^2e^{-2\alpha|\psi|^2}},$$

$$\boxed{ \Theta_\varepsilon[\psi] = -(\Delta\psi)R_\varepsilon(\psi) }.$$

$$-\Delta\psi/(\psi+\varepsilon e^{-\alpha|\psi|^2})$$

Checker: check_regularized_denominator · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E18

WCT Lyapunov candidate

PASS · VARIATIONAL DERIVATION

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.

$$\mathcal E[\psi] = \int_\Omega \left( c_1|\nabla\psi|^2 +c_2|\Theta_\varepsilon[\psi]|^2 \right)dx.$$

Checker: check_lyapunov_gradient_flow · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_lyapunov_gradient_flow.

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E19

Gap-curvature scaling

CONDITIONAL · CONSISTENCY CHECK

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.

$$\Delta_k^\star\sim\langle\sigma\rangle_w^2, \qquad [\Delta_k^\star]=L^{-2},$$

$$\Delta_\omega^\star:=c^2\Delta_k^\star, \qquad [\Delta_\omega^\star]=T^{-2}.$$

Checker: check_gap_mass_consistency_constraint · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective CONDITIONAL via derived_overrides.yaml:check_gap_mass_consistency_constraint.

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E20

Higher-order cavity quadratic sector

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Let

$$S:=\frac{\Box\psi}{g(\psi)}, \qquad P:=\frac{\Delta\psi}{g(\psi)}.$$

$$Q(S,P)=\kappa S^2+\theta P^2-\gamma SP.$$

$$\kappa\ge0,\qquad \theta\ge0,\qquad \gamma^2\le4\kappa\theta.$$

Checker: check_cavity_quadratic_form · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E21

Second-derivative Euler-Lagrange equation

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For $\mathcal L(\psi,\partial\psi,\partial^2\psi)$,

$$\frac{\delta\mathcal L}{\delta\psi} = \frac{\partial\mathcal L}{\partial\psi} -\partial_\mu \frac{\partial\mathcal L}{\partial(\partial_\mu\psi)} +\partial_\mu\partial_\nu \frac{\partial\mathcal L} {\partial(\partial_\mu\partial_\nu\psi)} =0.$$

Checker: check_second_order_el_template · Scope: INTERNAL CONSISTENCY · Lean: OPEN · Empirical: NOT TESTED

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E22

Effective metric ansatz

CONDITIONAL · CONSISTENCY CHECK

Current effective status: ⚠️ CONDITIONAL The coefficients must carry the units needed to make both corrections dimensionless, and signature/nondegeneracy must be checked.

$$g_{\mu\nu}^{\rm eff} = \eta_{\mu\nu} +\lambda_g \frac{\partial_\mu\overline\psi\,\partial_\nu\psi} {\rho c^2} +\delta_g\,\eta_{\mu\nu}\frac{W_\psi}{W_0}.$$

Checker: check_effective_metric_reality_constraint · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective CONDITIONAL via derived_overrides.yaml:check_effective_metric_reality_constraint.

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E23

Enthalpic curvature relation

CONDITIONAL · UNRESOLVED

Current effective status: ⚠️ CONDITIONAL The constants must reconcile dimensions and the relation requires a constitutive derivation.

$$h(\psi) = C_h\left( W_\psi+\chi|\nabla\psi|^2 \right).$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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Equation family

D. Dimensionality and functional bounds

4 objects

E24

Sobolev embedding threshold

PASS · FORMAL THEOREM

The H²-to-L∞ Sobolev embedding threshold used by the dimensionality argument. It supplies a standard regularity condition, not a complete nonlinear stability theorem.

$$H^2(\Omega)\hookrightarrow L^\infty(\Omega) \quad\Longrightarrow\quad 2>\frac n2.$$

$$n\le3.$$

Checker: check_h2_embedding_threshold · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

Assumptions: ASM-STANDARD-SOBOLEV-DOMAIN

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E25

Critical Sobolev exponent

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION For $n>2$,

$$p_c(n)=\frac{n+2}{n-2}, \qquad p<p_c(n)$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: DEFINITION · Empirical: NOT APPLICABLE

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E26

Corrected curvature $L^2$ bound

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS If $\psi\in H^2(\Omega)$ and

$$|R_\varepsilon(\psi(x))|\le\delta^{-1} \quad\text{a.e.},$$

$$\boxed{ \|\Theta_\varepsilon[\psi]\|_{L^2} \le \delta^{-1}\|\Delta\psi\|_{L^2} }.$$

Checker: check_theta_l2_from_h2 · Scope: INTERNAL CONSISTENCY · Lean: DEFINITION · Empirical: NOT TESTED

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E27

Finite-energy confinement

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$\int_{\mathbb R^n} \left( |\nabla\psi|^2 + |\Theta_\varepsilon[\psi]|^2 \right)dx <\infty.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: DEFINITION · Empirical: NOT APPLICABLE

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Equation family

E. Alpha-drop, entropy reduction, and pruning

7 objects

E28

Corrected alpha-drop exponent

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Let $\rhot(n)\in(0,1]$ be retained fractions. Define

$$\alpha(n) = 1+\frac1n\sum_{t=1}^{m(n)}\log_2\rho_t(n) +\beta(n).$$

$$\beta(n) < -\frac1n\sum_t\log_2\rho_t(n).$$

Checker: check_alpha_drop_corrected · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E29

Entropy-drop pruning

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Iteration gives

$$M_{t+1}\le e^{-\Delta_t}M_t, \qquad \Delta_t\ge0.$$

$$M_T \le M_0\exp\!\left(-\sum_{t=0}^{T-1}\Delta_t\right).$$

Checker: check_state_decay_iteration · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E30

Spectral entropy

PASS · ALGEBRAIC IDENTITY

The normalized Shannon entropy of a finite spectral distribution and its standard bounds. This is a standard information-theoretic identity rather than a WCT-specific physical result.

$$H_k=-\sum_kP_k\ln P_k.$$

$$0\le H_k\le\ln K.$$

Checker: check_entropy_bounds · Scope: STANDARD MATHEMATICS · Lean: OPEN · Empirical: NOT APPLICABLE

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E31

Conditional entropy-production bound

CONDITIONAL · SYMBOLIC DERIVATION

Current effective status: ⚠️ CONDITIONAL Define the entropy drop

$$\Delta H_t:=H_k(t)-H_k(t+1).$$

$$\Delta H_t\ge c_0\mathcal D_t, \qquad \mathcal D_t\ge0.$$

Checker: check_entropy_drop_bound · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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E32

Subexponential exploration condition

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$\limsup_{n\to\infty}\alpha(n)<1.$$

Checker: check_subexponential_exploration_margin_derived · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_subexponential_exploration_margin_derived.

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E33

Corrected support-entropy relation

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For a distribution supported on $Kt$ modes,

$$H_k(t)\le\ln K_t,$$

$$\boxed{ e^{H_k(t)}\le K_t }.$$

Checker: check_support_entropy_bound · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E34

Energy-entropy conversion

OPEN · UNRESOLVED

Current effective status: ○ OPEN For entropy reduction

$$\Delta H_k:=H_{\rm before}-H_{\rm after}\ge0,$$

$$\Delta E_{\rm cost}\ge\lambda\,\Delta H_k, \qquad \lambda>0.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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Equation family

F. WCC, channel capacity, and complexity

9 objects

E35

Curvature-locked fixed point

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION A stationary locked configuration satisfies

$$\partial_t\psi=0, \qquad \nabla\Theta_\varepsilon[\psi]=0, \qquad \frac{d}{dt}S[\psi]=0.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E36

Discrete WCC update

DEFINITION · DEFINITION CHECK

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.

$$\psi^{(t+1)}(x) = U\!\left( \psi^{(t)}(x), \{\psi^{(t)}(y)\}_{y\in N(x)} \right).$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: PROVED · Empirical: NOT APPLICABLE

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E37

Bandlimit from energy

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS with dimensionless $C1$.

$$k_{\max} = C_1\frac{E_{\max}}{\hbar c},$$

Checker: check_bandlimit_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E38

Spatial channel capacity

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS In three spatial dimensions,

$$N_{\rm lanes} \le C_2Vk_{\max}^3,$$

Checker: check_channel_capacity_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E39

Polynomial update bound

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION This defines the assumed computational resource class.

$$T_{\max}(n)\le C_3n^d.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E40

WCC complexity identification

CONDITIONAL · UNRESOLVED

The proposed identification between the WCC resource model and a complexity claim. It remains conditional on encoding, precision, update-cost, and simulation-overhead assumptions.

$$P_{\rm WCC}\cong P, \qquad NP_{\rm WCC}\cong NP.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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E41

Curvature-bounded configuration count

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$|C_{\rm curv}(n)| \le 2^{\alpha(n)n}, \qquad \alpha(n)<1.$$

Checker: check_alpha_drop_counting_bound_derived · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_alpha_drop_counting_bound_derived.

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E42

Theta-information relation

OPEN · UNRESOLVED

Current effective status: ○ OPEN The information functional and coupling $\lambdaI$ require derivation.

$$\frac{d}{dt}I_{\rm coh}[\psi] = -\lambda_I \int_\Omega|\Theta_\varepsilon[\psi]|^2dx.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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E43

Curvature-entropy tradeoff

OPEN · UNRESOLVED

Current effective status: ○ OPEN This remains an analytic/empirical claim.

$$\frac{dH_k}{dt} \le -\mu \int_\Omega|\Theta_\varepsilon[\psi]|^2dx, \qquad \mu>0.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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Equation family

G. Cavity, effective mass, and phase structure

13 objects

E44

Theta eigenmode problem

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Because $\Theta\varepsilon$ is nonlinear, the spectral problem and normalization must be specified carefully.

$$\Theta_\varepsilon[\psi_n] = \lambda_n\psi_n.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E45

Corrected quality factor

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS where

$$\boxed{ Q_{\rm eff} = \omega\frac{U}{P_{\rm loss}} }$$

$$U=\int_\Omega u\,dV$$

Checker: check_q_factor_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E46

Plasma-cavity curvature match

OPEN · UNRESOLVED

Current effective status: ○ OPEN A measurable matching tolerance and transfer mechanism remain open.

$$\langle\sigma\rangle_{w,\rm plasma} \approx \langle\sigma\rangle_{w,\rm cavity}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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E47

Corrected power balance

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS At stationarity,

$$\boxed{ \frac{dW}{dt} = P_{\rm in} + P_{\rm fusion} - P_{\rm loss} - P_{\rm out} }.$$

$$P_{\rm in}+P_{\rm fusion} = P_{\rm loss}+P_{\rm out}.$$

Checker: check_power_balance_form · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E48

Curvature-gap stability criterion

CONDITIONAL · UNRESOLVED

Current effective status: ⚠️ CONDITIONAL The threshold and direction of the inequality must be calibrated to a specified stability observable.

$$\Delta\sigma = \langle\sigma\rangle_{\rm core} - \langle\sigma\rangle_{\rm edge} > \Delta_{\rm crit}.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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E49

Corrected effective-mass gap law

PASS · DIMENSIONAL CHECK

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.

$$\omega_j^2=c^2\lambda_j+\Delta_\omega^\star, \qquad [\Delta_\omega^\star]=T^{-2},$$

$$\omega^2=c^2k^2+\frac{m_{\rm eff}^2c^4}{\hbar^2}$$

$$\boxed{ m_{\rm eff}^2 = \frac{\hbar^2}{c^4}\Delta_\omega^\star }.$$

Checker: check_effective_mass_gap_dimensions · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E50

Phase-coherence functional

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$\mathcal C[\psi] = \int_\Omega \frac{|\psi|^2}{|\nabla\theta|}\,dx.$$

$$|\nabla\theta|\ge\delta>0$$

Checker: check_phase_coherence_lower_gradient_bound_derived · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_phase_coherence_lower_gradient_bound_derived.

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E51

Curvature-gradient commutator

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For a smooth scalar denominator $D\neq0$, define

$$\Theta_D[\psi]:=-\frac{\Delta\psi}{D}.$$

$$[\nabla,\Theta_D]\psi := \nabla(\Theta_D[\psi])-\Theta_D[\nabla\psi] = \frac{\Delta\psi}{D^2}\nabla D.$$

Checker: check_curvature_gradient_commutator · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E52

Curvature gain and gradient loss

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$G_\sigma:=\int_\Omega|\Theta_\varepsilon[\psi]|^2dx, \qquad L_\sigma:=\int_\Omega|\nabla\psi|^2dx.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E53

Curvature pressure density

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS It is the local curvature contribution to E18.

$$p_\Theta(x) := c_2|\Theta_\varepsilon[\psi](x)|^2.$$

Checker: check_pressure_density_embedding · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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E54

Resonance-lock condition

CONDITIONAL · UNRESOLVED

Current effective status: ⚠️ CONDITIONAL Simultaneous satisfaction requires existence and regularity results.

$$\partial_t\psi=0, \qquad \delta\mathcal E[\psi]=0, \qquad \nabla\Theta_\varepsilon[\psi]=0.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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E55

Curvature-induced effective potential

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$V_{\rm eff}(\psi) = V(|\psi|^2) + \kappa|\Theta_\varepsilon[\psi]|^2.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E56

Phase-wall criterion

CONDITIONAL · UNRESOLVED

Current effective status: ⚠️ CONDITIONAL The comparison scale and wall-detection threshold must be defined.

$$|\nabla\theta|_{\rm wall} \sim \sigma_{\rm wall} \gg \langle\sigma\rangle_w.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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Equation family

H. Swift-Hohenberg and spectral projection

8 objects

E57

Swift-Hohenberg shell operator

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Its Fourier symbol is

$$\mathcal{SH}[A] = (\Delta+k_\star^2)^2A.$$

$$(|k|^2-k_\star^2)^2.$$

Checker: check_sh_fourier_symbol · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E58

Band-selective Green kernel

PASS · LIMIT CHECK

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.

$$\mathcal L=r+a(\Delta+k_\star^2)^2,$$

$$G(k) = \frac1{r+a(|k|^2-k_\star^2)^2}.$$

Checker: check_green_kernel_bounded · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Assumptions: ASM-POSITIVE-SPECTRAL-OFFSET

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_green_kernel_bounded.

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E59

Projection onto a dominant annulus

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS With a fixed annulus,

$$\mathcal A^\star := \left\{ k\in\mathbb Z^d: \bigl||k|-k_\star\bigr|\le\Delta k \right\},$$

$$(P_{k_\star}A)(x) = \sum_{k\in\mathcal A^\star} \widehat A_ke^{ik\cdot x}.$$

$$P_{k_\star}^2=P_{k_\star}.$$

Checker: check_projection_idempotence · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E60

Center-manifold amplitude equation

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$\partial_T\mathcal A = \mu\mathcal A-g|\mathcal A|^2\mathcal A.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E61

Pattern-formation threshold

PASS · SIGN OR EXTREMUM CHECK

Current effective status: ✅ PASS In the continuum,

$$r_c = \min_k a(|k|^2-k_\star^2)^2 = 0.$$

Checker: check_pattern_threshold · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E62

Spectral energy concentration

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For nonzero total spectral energy,

$$\eta(t) = \frac{ \sum_{k\in\mathcal A^\star}|\widehat A_k|^2 }{ \sum_k|\widehat A_k|^2 }, \qquad 0\le\eta(t)\le1.$$

Checker: check_spectral_fraction_bounds · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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E63

Entropic mode selection

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$k_\star = \operatorname*{arg\,min}_k \left[ H_k+\lambda_\Theta C_\Theta(k) \right].$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E64

Corrected selected wavelength

PASS · CONSISTENCY CHECK

Current effective status: ✅ PASS From E12,

$$k_\star=\sqrt{\frac{a}{2b}}.$$

$$\boxed{ \lambda_\star = \frac{2\pi}{k_\star} = 2\pi\sqrt{\frac{2b}{a}} }.$$

Checker: check_e12_e64_consistency · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Equation family

I. Sobolev structure and dimensional bounds

6 objects

E65

Critical Sobolev exponent

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For $n>2$,

$$p_c(n)=\frac{n+2}{n-2}.$$

Checker: check_critical_sobolev_exponent · Scope: INTERNAL CONSISTENCY · Lean: DEFINITION · Empirical: NOT TESTED

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E66

Gagliardo-Nirenberg interpolation

CONDITIONAL · UNRESOLVED

Current effective status: ⚠️ CONDITIONAL The allowed $p,\theta,n$, domain, and boundary assumptions must be specified.

$$\|u\|_{L^p} \le C \|\nabla u\|_{L^2}^{\theta} \|u\|_{L^2}^{1-\theta}.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: DEFINITION · Empirical: NOT TESTED

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E67

Failure of $H^2 o L^\infty$ above three dimensions

PASS · FORMAL THEOREM

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.

$$\text{No standalone display equation recorded.}$$

Checker: check_h2_embedding_threshold · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

Assumptions: ASM-STANDARD-SOBOLEV-DOMAIN

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E68

Localized energy estimate

CONDITIONAL · CONSISTENCY CHECK

Current effective status: ⚠️ CONDITIONAL A model-dependent localized estimate is

$$\int_{B_R}|\nabla\psi|^2dx \le CR^{n-2}\|\psi\|_{H^1}^2.$$

Checker: check_localized_energy_concentration_constraint · Scope: STANDARD MATHEMATICS · Lean: DEFINITION · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective CONDITIONAL via derived_overrides.yaml:check_localized_energy_concentration_constraint.

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E69

Corrected high-regularity curvature bound

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS If

$$\psi\in H^s(\Omega), \qquad s>\frac n2+2,$$

$$\Theta_\varepsilon[\psi]\in L^\infty(\Omega).$$

Checker: check_theta_linf_from_high_regularity · Scope: INTERNAL CONSISTENCY · Lean: DEFINITION · Empirical: NOT TESTED

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E70

Dimensional stability criterion

CONDITIONAL · CONSISTENCY CHECK

The conditional WCT dimensional-stability criterion combining the Sobolev threshold with additional confinement hypotheses. It is not a proved biconditional characterization of all stable WCT solutions.

$$n\le3, \qquad H^2\hookrightarrow L^\infty, \qquad p<p_c(n)$$

Checker: check_dimensional_stability_implication · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Assumptions: ASM-STANDARD-SOBOLEV-DOMAIN, ASM-H2-CONFINEMENT-ROUTE

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Equation family

J. Computational resource bounds

6 objects

E71

Physical computation resource bound

CONDITIONAL · DIMENSIONAL CHECK

A proposed physical computation resource bound. Its use in complexity theory requires an explicit machine model, precision accounting, and proof that all relevant resources are included.

$$TVk_{\max}^3 \le C_{\rm phys}.$$

Checker: check_physical_resource_dimensions · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E72

Curvature-pruned search space

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$|S_{\rm eff}(n)| \le 2^{\alpha(n)n}.$$

Checker: check_alpha_drop_counting_bound_derived · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_alpha_drop_counting_bound_derived.

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E73

Polynomial verification

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$V(x,w)\in P, \qquad |w|=\operatorname{poly}(|x|).$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E74

Curvature separation conjecture

OPEN · UNRESOLVED

Current effective status: ○ OPEN The finite-size families $Pn,NPn$ must first be defined.

$$\inf_n \frac{\log|NP_n|}{\log|P_n|} >1.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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E75

Physical-oracle impossibility

OPEN · UNRESOLVED

Current effective status: ○ OPEN This is a complexity claim requiring a formal computational model and reduction.

$$\nexists\, O: O(\psi)=\operatorname*{arg\,min}_\psi\mathcal E[\psi] \quad\text{in polynomial time}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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E76

WCC complexity equivalence

CONDITIONAL · UNRESOLVED

A conditional equivalence claim between curvature-bounded computation and a classical complexity description. The equivalence is not established without explicit simulation and overhead bounds.

$$P_{\rm WCC}=P \quad\Longrightarrow\quad \text{WCC polynomially simulates the declared physical-computation model}.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT TESTED

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Equation family

K. Entropy and information dynamics

6 objects

E77

Mutual-information decay

OPEN · UNRESOLVED

Current effective status: ○ OPEN The probability law, channel, and regularity assumptions remain open.

$$\frac{d}{dt}I(\psi_t;\psi_0) \le -\gamma\mathcal E_\Theta[\psi_t].$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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E78

Fisher-information curvature bound

OPEN · UNRESOLVED

Current effective status: ○ OPEN A common probability density and geometric derivation are required.

$$\mathcal I_F[\psi] \ge c\int_\Omega|\Theta_\varepsilon[\psi]|^2dx.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: STATED_TODO · Empirical: NOT TESTED

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E79

Entropy-production rate

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION The sign convention and units must be fixed when used physically.

$$\dot\Sigma = \frac{dH_k}{dt} + \frac{\mathcal E_\Theta}{T_{\rm eff}}.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E80

Landauer-type bound

CONDITIONAL · DIMENSIONAL CHECK

Current effective status: ⚠️ CONDITIONAL If $\Delta H{\rm bits}$ is measured in bits,

$$\Delta E \ge k_BT_{\rm eff}\ln2\, \Delta H_{\rm bits}.$$

Checker: check_landauer_units · Scope: MODEL CONDITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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E81

Corrected coherence length

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS With normalized spectral weights $pk$,

$$\boxed{ \xi_{\rm coh} = \left( \sum_kp_k|k|^2 \right)^{-1/2} }.$$

$$\boxed{ \xi_{\rm coh} = \sqrt{ \frac{\int_\Omega|\psi|^2dx} {\int_\Omega|\nabla\psi|^2dx} } }.$$

Checker: check_coherence_length_dimensions · Scope: INTERNAL CONSISTENCY · Lean: DEFINITION · Empirical: NOT APPLICABLE

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E82

Information-geometry tensor

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Positive definiteness and coordinate invariance require additional conditions.

$$g_{ij}^{({\rm info})} = \left\langle \partial_i\Theta_\varepsilon\, \partial_j\Theta_\varepsilon \right\rangle.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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Equation family

Curvature-locking equations

10 objects

CLE1

Curvature-locking functional

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Use the inverse-length convention $[\sigma\star]=L^{-1}$:

$$S[\psi] = \int_\mathcal M \left[ |\nabla\psi|^2 + |W_\psi-\sigma_\star^2|^2 \right]\sqrt g\,d^3x, \qquad W_\psi:=-\frac{\Delta\psi}{\psi}.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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CLE2

Corrected curvature-lock Euler-Lagrange equation

PASS · VARIATIONAL DERIVATION

Current effective status: ✅ PASS For the real one-dimensional reduction

$$q:=-\frac{\psi_{xx}}{\psi}-\sigma_\star^2,$$

$$\boxed{ q\frac{\psi_{xx}}{\psi^2} -\psi_{xx} -\frac{d^2}{dx^2}\left(\frac q\psi\right) =0 }.$$

Checker: check_cle2_variation · Scope: INTERNAL CONSISTENCY · Lean: OPEN · Empirical: NOT TESTED

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CLE3

Curvature-locking condition

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Both sides have units $L^{-2}$.

$$W_\psi=\sigma_\star^2.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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CLE4

Locked-field equation

PASS · DIMENSIONAL CHECK

The canonical registered object for locked-field equation; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$-\Delta\psi = \sigma_\star^2\psi.$$

Checker: check_cle_units_chain · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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CLE5

Thin/product-torus Laplacian

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$\Delta\psi \approx \frac1{R^2}\partial_\theta^2\psi + \frac1{r^2}\partial_\phi^2\psi.$$

Checker: check_flat_product_torus_laplacian_derived · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_flat_product_torus_laplacian_derived.

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CLE6

Separation ansatz

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For

$$\psi(\theta,\phi)=f(\theta)g(\phi),$$

$$\frac{f''}{f} + \frac{R^2}{r^2}\frac{g''}{g} = -\sigma_\star^2R^2.$$

Checker: check_separation_substitution · Scope: INTERNAL CONSISTENCY · Lean: OPEN · Empirical: NOT TESTED

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CLE7

Periodic angular mode family

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS The periodic reduced equation

$$f''+m^2f=0$$

$$\boxed{ f(\theta)=A\cos(m\theta)+B\sin(m\theta), \qquad m\in\mathbb Z_{\ge0} }.$$

Checker: check_periodic_ode_family · Scope: INTERNAL CONSISTENCY · Lean: OPEN · Empirical: NOT TESTED

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CLE8

Selected torus eigenmode

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$\psi(\theta,\phi)=Ae^{i\phi}$$

Checker: check_torus_lowest_mode_selection_derived · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_torus_lowest_mode_selection_derived.

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CLE9

Electron radius from curvature

PASS · DIMENSIONAL CHECK

Current effective status: ✅ PASS For $\sigma\star=mec/\hbar$,

$$R=\frac1{\sigma_\star}.$$

$$R=\frac{\hbar}{m_ec}\approx386.16\ {\rm fm}.$$

Checker: check_cle_units_chain · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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CLE10

Curvature scalar chain

PASS · DIMENSIONAL CHECK

The canonical registered object for curvature scalar chain; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$\boxed{ W_\psi = -\frac{\Delta\psi}{\psi} = \sigma_\star^2 }, \qquad R=\sigma_\star^{-1}.$$

Checker: check_cle_units_chain · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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Equation family

Logarithmic and ghost equations

5 objects

Log-periodic ghost modulation

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS For $E>0$ and $E0>0$,

$$\delta_g(E) = A_g\cos\!\left( k_\ell\ln\frac E{E_0}+\phi \right),$$

$$|\delta_g(E)|\le|A_g|.$$

Checker: check_log_modulation · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

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Logarithmic field representation

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS For a positive real field $\psi>0$, let

$$u=\ln\psi, \qquad \psi=e^u.$$

$$\nabla\psi=e^u\nabla u,$$

$$\Delta\psi=e^u(\Delta u+|\nabla u|^2),$$

$$\frac{\Delta\psi}{\psi} = \Delta u+|\nabla u|^2.$$

Checker: check_log_laplacian_identity · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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Log-curvature evolution

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS If

$$\partial_tu = \Delta u+|\nabla u|^2,$$

Checker: check_log_flow_reduction · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT APPLICABLE

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Cole-Hopf reduction

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS With

$$\psi=e^u,$$

$$\partial_t\psi = e^u\partial_tu = e^u(\Delta u+|\nabla u|^2) = \Delta\psi.$$

$$\boxed{\partial_t\psi=\Delta\psi}.$$

Checker: check_cole_hopf · Scope: STANDARD MATHEMATICS · Lean: PROVED · Empirical: NOT APPLICABLE

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Filament-localization condition

CONDITIONAL · UNRESOLVED

Current effective status: ⚠️ CONDITIONAL A norm, tolerance, scale, and dynamical derivation are required.

$$|\nabla u| \sim \kappa_{\rm core}.$$

Checker: classify_conditional · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

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Equation family

Curvature-acoustic cosmology

20 objects

CM1

Fundamental field evolution

OPEN · UNRESOLVED

Current effective status: ○ OPEN Coefficient dimensions and derivation remain open.

$$i\partial_t\psi = -\Theta_\varepsilon[\psi]\,J[\psi],$$

$$J[\psi] = |\psi|^2\Delta\psi\,\varepsilon_{\rm vac}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM2

Curvature-spectral tilt

OPEN · UNRESOLVED

The canonical registered object for curvature-spectral tilt; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$P_{\rm prim}(k)\propto k^{-\alpha_{\rm WCT}},$$

$$n_s-1=-\alpha_{\rm WCT},$$

$$\alpha_{\rm WCT} = -\frac{d\ln|\Theta(k)|}{d\ln k}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM3

Potential from curvature

OPEN · UNRESOLVED

The canonical registered object for potential from curvature; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$\Phi(k,t) = -C_\Phi\frac{\Theta(k,t)}{k^2}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM4

Horizon-entry potential decay

OPEN · UNRESOLVED

Current effective status: ○ OPEN on the domain $\Theta\neq0$.

$$\partial_t\Phi = -\Gamma\Phi,$$

$$\Gamma(k,t) = \left| \frac{\partial_t\Theta(k,t)}{\Theta(k,t)} \right|,$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM5

Curvature-acoustic oscillators

OPEN · UNRESOLVED

The canonical registered object for curvature-acoustic oscillators; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$\ddot\delta_\gamma +c_s^2k^2\delta_\gamma = -k^2\Phi,$$

$$\ddot\delta_b +\mathcal R\,c_s^2k^2\delta_\gamma = -k^2\Phi,$$

$$\mathcal R = \frac{E_{\rm comp}}{E_{\rm rad}}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM6

Sound speed from curvature feedback

OPEN · UNRESOLVED

Current effective status: ○ OPEN Positivity requires the bracketed factor to be nonnegative.

$$c_s^2(t) = \frac1{3(1+\mathcal R(t))} \left[ 1-\beta_{\rm curv} \frac{E_{\rm curv}(t)}{E_{\rm tot}} \right].$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM7

Curvature diffusion

OPEN · UNRESOLVED

Current effective status: ○ OPEN A phenomenological damping replacement is

$$\dot\delta_\gamma = v_\gamma - D_{\rm curv}(t)k^2\delta_\gamma,$$

$$D_{\rm curv}(t) = \frac{\langle|\nabla\psi|^2\rangle} {\langle|\psi|^2\rangle}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM8

Initial conditions

OPEN · UNRESOLVED

Current effective status: ○ OPEN Use CM3 consistently:

$$\delta_\gamma(0)=\delta_b(0)=-2\Phi(k,0),$$

$$\Phi(k,0) = -C_\Phi\frac{\Theta(k,0)}{k^2}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM9

First-order mode system

PASS · ALGEBRAIC IDENTITY

Current effective status: ✅ PASS Baseline status: ○ OPEN

$$\dot\delta_\gamma=v_\gamma, \qquad \dot v_\gamma=-c_s^2k^2\delta_\gamma-k^2\Phi,$$

$$\dot\delta_b=v_b, \qquad \dot v_b=-\mathcal R c_s^2k^2\delta_\gamma-k^2\Phi.$$

Checker: check_cm9_first_order_equivalence · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline OPEN → effective PASS via derived_overrides.yaml:check_cm9_first_order_equivalence.

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CM10

Tight-coupling drag

OPEN · UNRESOLVED

The canonical registered object for tight-coupling drag; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$\delta_b \leftarrow (1-\varepsilon_{\rm drag})\delta_b + \varepsilon_{\rm drag}\delta_\gamma,$$

$$\varepsilon_{\rm drag} = \frac{E_{\rm exch}}{E_{\rm comp}}, \qquad 0\le\varepsilon_{\rm drag}\le1.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM11

Curvature damping envelope

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Baseline status: ○ OPEN

$$D(k) = \exp\!\left(-\frac{k^2}{k_D^2}\right),$$

$$k_D^{-2} = \int_0^{t_\star}D_{\rm curv}(t)\,dt.$$

Checker: check_cm11_gaussian_damping · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline OPEN → effective PASS via derived_overrides.yaml:check_cm11_gaussian_damping.

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CM12

Dimensionless power spectrum

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN

$$\Delta^2(k) = \frac{k^3}{2\pi^2}P(k).$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: DEFINITION · Empirical: NOT APPLICABLE

Status history: baseline OPEN → effective DEFINITION via derived_overrides.yaml:classify_definition.

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CM13

Peak metrics

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN

$$r_{21} = \frac{P(k_2)}{P(k_1)}, \qquad r_{31} = \frac{P(k_3)}{P(k_1)},$$

$$s_{21} = \frac{k_2}{k_1}, \qquad s_{31} = \frac{k_3}{k_1}.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: DEFINITION · Empirical: NOT APPLICABLE

Status history: baseline OPEN → effective DEFINITION via derived_overrides.yaml:classify_definition.

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CM14

Peak-response interpretation

OPEN · UNRESOLVED

Current effective status: ○ OPEN Proposed qualitative relations:

$$\text{faster }\Theta\text{ decay}\Rightarrow s_{ij}\uparrow,$$

$$\text{larger compression}\Rightarrow r_{31}\uparrow,$$

$$\text{larger radiative fraction}\Rightarrow r_{21}\downarrow.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM15

WCT angular scaling

OPEN · UNRESOLVED

The canonical registered object for wct angular scaling; consult the source equation and verification metadata for its assumptions and scientific boundary.

$$k_{\rm phys} = \frac{k}{a_{\rm WCT}(t)},$$

$$a_{\rm WCT}(t) = \left[ \frac{E_{\rm curv}(0)} {E_{\rm curv}(t)} \right]^{1/3}.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM16

Acoustic horizon

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN

$$R_{\rm hor}(t) = \int_0^tc_s(t')\,dt',$$

$$k_{\rm hor} = \frac{2\pi}{R_{\rm hor}}.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: DEFINITION · Empirical: NOT APPLICABLE

Status history: baseline OPEN → effective DEFINITION via derived_overrides.yaml:classify_definition.

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CM17

Curvature-energy closure

OPEN · UNRESOLVED

Current effective status: ○ OPEN for a closed sector with no external source or loss.

$$E_{\rm curv}(t) + E_{\rm grad}(t) = E_{\rm tot},$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM18

Minimal cosmology closure set

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Baseline status: ○ OPEN

$$\mathfrak C_{\rm min} = \{\mathrm{CM1},\mathrm{CM2},\mathrm{CM3}, \mathrm{CM4},\mathrm{CM5},\mathrm{CM7}\}.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

Status history: baseline OPEN → effective DEFINITION via derived_overrides.yaml:classify_definition.

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CM19

Acoustic speed from curvature equation of state

OPEN · UNRESOLVED

Current effective status: ○ OPEN where the derivative must be taken along a specified thermodynamic or dynamical path.

$$c_s^2 = \frac{\partial P_{\rm curv}} {\partial\rho_{\rm curv}},$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CM20

Theta-based expansion ansatz

OPEN · UNRESOLVED

Current effective status: ○ OPEN The constant $K$ must carry the units needed for $H^2$, and the equation requires independent derivation.

$$H(t) = \frac{\dot a_{\rm WCT}}{a_{\rm WCT}} = \sqrt{ \frac{\rho_\Theta(t)}{3|K|} }.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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Equation family

Topology and spectral emergence

9 objects

TOP1

Closed spectral-loop representation

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION This is the chosen configuration representation; emergence of the basis is a separate empirical claim.

$$\gamma(s) = \sum_{k=1}^{K} \left[ a_k\cos(ks)+b_k\sin(ks) \right], \qquad s\in[0,2\pi).$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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TOP2

WCT loop-energy functional

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION

$$\mathcal E_{\rm loop}[\gamma] = \int_\gamma\kappa^2ds + \alpha_{\rm UV} \sum_kk^p(|a_k|^2+|b_k|^2) + V_{\rm SA}[\gamma].$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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TOP3

Irreversible gradient flow

PASS · VARIATIONAL DERIVATION

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$\partial_t\gamma = -\frac{\delta\mathcal E_{\rm loop}}{\delta\gamma}.$$

$$\frac{d\mathcal E_{\rm loop}}{dt} = - \left\| \frac{\delta\mathcal E_{\rm loop}}{\delta\gamma} \right\|^2 \le0.$$

Checker: check_topology_gradient_flow_descent_derived · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_topology_gradient_flow_descent_derived.

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TOP4

Emergent-topology criterion

OPEN · UNRESOLVED

Current effective status: ○ OPEN A proposed physical invariant $I[\gamma]$ satisfies

$$I[\gamma_t]\to I_\infty$$

$$\frac{d\mathcal E}{dt}<0$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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TOP5

WCT dynamical codimension

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION This is not manifold codimension.

$$\operatorname{codim}_{\rm WCT}(\gamma) := \text{minimum number of singular events required to reach the unknot}.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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TOP6

Spectral topology bands

OPEN · UNRESOLVED

Current effective status: ○ OPEN Define

$$\epsilon_\kappa = \frac1L\int_\gamma\kappa^2ds.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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TOP7

Topological mass proxy

CONDITIONAL · CONSISTENCY CHECK

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.

$$m_{\rm WCT} \propto \epsilon_\kappa.$$

Checker: check_topology_curvature_energy_mass_constraint · Scope: MODEL CONDITIONAL · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective CONDITIONAL via derived_overrides.yaml:check_topology_curvature_energy_mass_constraint.

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TOP8

Holonomy non-invariance

OPEN · UNRESOLVED

Current effective status: ○ OPEN For

$$H_\tau[\gamma] = \int_\gamma\tau\,ds,$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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TOP9

Protein-particle structural correspondence

OPEN · UNRESOLVED

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.

$$\text{knotted protein states} \longleftrightarrow \text{stable WCT loop excitations}$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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Equation family

Canonical correction layer

6 objects

CORR1

Full Lyapunov candidate

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION The curvature term alone is only one component.

$$\mathcal E_{\rm WCT}[\psi] = \int \left( |\nabla\psi|^2 + |\Theta_\varepsilon[\psi]|^2 \right)dx.$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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CORR2

Mean-amplitude spectral closure

PASS · SYMBOLIC DERIVATION

Current effective status: ✅ PASS Baseline status: ⚠️ CONDITIONAL

$$D_{\rm eff}^2 := \langle|\psi|^2\rangle+\varepsilon^2,$$

$$C_\Theta(k) \approx \frac{k^4}{D_{\rm eff}^2}.$$

Checker: check_mean_amplitude_closure_error_bound_v2 · Scope: INTERNAL CONSISTENCY · Lean: PROVED · Empirical: NOT TESTED

Status history: baseline CONDITIONAL → effective PASS via derived_overrides.yaml:check_mean_amplitude_closure_error_bound_v2.

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CORR3

Spectral-weight notation

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION Use

$$\lambda_\Theta$$

$$\lambda_{\rm ex}$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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CORR4

Macro-micro control parameter

DEFINITION · DEFINITION CHECK

Current effective status: ◻️ DEFINITION on the domain $H>0$.

$$\Xi = \frac{\int k^4\rho(k)\,dk}{H}, \qquad H=-\sum_kP_k\ln P_k,$$

Checker: classify_definition · Scope: DEFINITIONAL · Lean: OPEN · Empirical: NOT APPLICABLE

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CORR5

Entropy-curvature coupling

OPEN · UNRESOLVED

Current effective status: ○ OPEN This remains the same open claim as E43 unless derived for a specified evolution.

$$\frac{dH}{dt} \le -\mu \int|\Theta_\varepsilon[\psi]|^2dx.$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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CORR6

Isoelectronic-flow alignment

OPEN · UNRESOLVED

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.

$$\text{No standalone display equation recorded.}$$

Checker: classify_open · Scope: UNRESOLVED · Lean: OPEN · Empirical: NOT TESTED

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