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
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
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
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
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
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
CM7
Curvature diffusion
OPEN · UNRESOLVEDCurrent 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
CM8
Initial conditions
OPEN · UNRESOLVEDCurrent 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
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.
CM10
Tight-coupling drag
OPEN · UNRESOLVEDThe 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
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: OPEN · Empirical: NOT TESTED
Status history: baseline OPEN → effective PASS via derived_overrides.yaml:check_cm11_gaussian_damping.
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.
CM13
Peak metrics
DEFINITION · DEFINITION CHECKCurrent 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.
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
CM15
WCT angular scaling
OPEN · UNRESOLVEDThe 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
CM16
Acoustic horizon
DEFINITION · DEFINITION CHECKCurrent 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.
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
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.
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
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