Theory Gates

The equations, normalization choices, and claim boundaries behind the dashboard

Telegraph correspondence

The scalar telegraph equation is

\[ \partial_t^2 u + 2a\,\partial_t u = v^2 \Delta u. \]

The kinetic part of the manuscript’s Wheeler–DeWitt equation matches it under

\[ t \leftrightarrow \alpha,\qquad (x_1,x_2) \leftrightarrow (\beta_+,\beta_-),\qquad 2a=-B. \]

An ordinary Poisson collision rate requires \(a\geq0\), hence \(B\leq0\). Positive \(B\) is anti-damping and has no ordinary positive-rate interpretation.

Why the one- and two-dimensional results differ

The one-dimensional Goldstein–Kac process has two velocities, \(+v\) and \(-v\), and Poisson velocity flips. Eliminating the two directional populations gives the scalar telegraph equation exactly.

In two dimensions, resetting a continuously distributed direction at Poisson times produces a causal velocity-jump process, but eliminating its angular distribution does not close to the same scalar second-order PDE. NUM-2 tests that distinction directly. The causal front passes while the scalar-density identification fails.

The T15b planar persistent random flight retains its heading variable. Its position-heading density satisfies

\[ \partial_t f+c\,\mathbf e(\theta)\cdot\nabla f =-\lambda f+\frac{\lambda}{2\pi}\rho, \qquad \rho=\int_0^{2\pi}f\,d\theta. \]

Its exact Fourier–Laplace density transform is

\[ \widetilde\rho(\mathbf k,s) =\frac{1}{\sqrt{(s+\lambda)^2+c^2|\mathbf k|^2}-\lambda}. \]

The Masoliver–Lindenberg fluid approximation is

\[ \rho_{tt}+\frac{\lambda}{2}\rho_t =\frac{c^2}{4}\Delta\rho, \qquad D=\frac{c^2}{2\lambda}. \]

The stored transform comparisons show decreasing relative error in the tested small-\(s\), small-\(k\) regime. The v2 seeded mode comparison reaches \(\lambda t=128\); the largest final-mode difference is 2.84 Monte Carlo standard errors. These results support an asymptotic fluid-limit comparison, not an exact finite-scale scalar closure.

Bianchi IX normalization

Define the dimensionless Misner shape potential used by the numerical solver:

\[ V_M(\beta_+,\beta_-) =\frac13e^{-8\beta_+} -\frac43e^{-2\beta_+}\cosh(2\sqrt3\beta_-) +\frac23e^{4\beta_+}\left[\cosh(4\sqrt3\beta_-)-1\right]. \]

It satisfies \(V_M(0,0)=-1\). Write the spatial scalar curvature as

\[ R(\alpha,\beta_\pm)=-R_0e^{-2\alpha}V_M(\beta_\pm), \]

where \(R_0\) is the scalar curvature of the chosen isotropic invariant metric at \(\alpha=0\). Then

\[ -24\pi^2e^{6\alpha}R =24\pi^2R_0e^{4\alpha}V_M. \]

For a unit round three-sphere convention, \(R_0=6\) and the coefficient is \(144\pi^2\). Other normalizations of the invariant one-forms rescale \(R_0\). The manuscript uses the unit-round-\(S^3\) convention and therefore states the \(144\pi^2\) coefficient explicitly.

Signed Bianchi IX evolution

The potential coefficient changes sign over anisotropy space, and the evolved scalar field becomes negative under refinement. A fixed positive population cannot represent it; positive branching and killing alone also cannot create a negative field.

NUM-4 therefore evolves the PDE as a signed amplitude. In the fixed-box outer-wall sequence, the largest adjacent signed-mass change is 4.85%, while the finest common-interior signed-field \(L^1\) difference is still 19.5%. This is a useful PDE calculation, but it remains provisional rather than a converged microscopic signed-trajectory construction.

Signed-particle amplitude prototype

NUM-6 starts from the central-well finite-difference update already used by NUM-4. It represents the recurrence on the pair \((\Psi_n,\Psi_{n-1})\) by sparse signed walkers: a walker samples a matrix entry, carries its real signed weight, and participates in state-wise cancellation and resampling after every \(\alpha\) step. This is an amplitude-level representation, not a positive stochastic process.

At fixed grid and step size, the \(L^1\) error against the deterministic update falls from 35.8 at 20,000 particles to 2.00 at 80,000 and 0.848 at 320,000. The declared acceptance gate is \(L^1<0.10\). Thus the prototype demonstrates a concrete signed microscopic estimator, but not an efficient or validated trajectory model: the residual sign problem remains the limiting result.

Intertwiner-sector diffusion boundary

NUM-5 fixes four external spins at \(j_1=j_2=j_3=j_4=3\) and builds the state space from the intermediate spins allowed simultaneously by the pair couplings \((j_1,j_2)\to k\) and \((j_3,j_4)\to k\). The resulting channels \(k=0,1,\ldots,6\) are all valid four-valent SU(2) intertwiners. A continuous-time nearest-neighbor walk on this path is compared with its exact graph heat kernel.

This is a basic constraint-preserving single-node spin-network diffusion model. It does not establish shared-edge consistency on a many-node graph, Hamiltonian-constraint dynamics, or convergence from quantum geometry to minisuperspace.

Reduced Bianchi I geometry-labelled benchmark (NUM-7)

NUM-7 tests the one-anisotropy LRS Bianchi I equation derived in T15a. It compares an event-driven walk in \(\beta\) with the exact Fourier solution of the matched density/current equations for both \(B=0,a=0\) and the manuscript’s derivative-only \(B=-3/2,a=3/4\) prescription. Three compact smooth profiles include a nonzero-current initial condition \(J_0=0.4u_0\).

All six profile/ordering cases pass the frozen relative density gate of \(0.05\): at 320,000 walkers, the mean final-time \(L^1\) errors range from 0.0281 to 0.0355, and the eight-seed 95% confidence upper endpoints range from 0.0294 to 0.0366. At 640,000 walkers, errors range from 0.0187 to 0.0243. This validates the chosen positive stochastic realization for these initial data and prescriptions. It does not show that the walk is a microscopic law of quantum geometry, nor that arbitrary complex WDW solutions are probability densities.

Claim matrix

Claim Gate Current status
1D velocity flips realize scalar telegraph dynamics Density convergence and causal front Passed
Isotropic 2D direction resets realize scalar 2D telegraph dynamics Density convergence under grid/ensemble refinement Failed
Factor ordering is an ordinary collision rate for all \(B\) Nonnegative rate \(a=-B/2\) Failed for \(B>0\)
Bianchi IX is a positive density ensemble Nonnegative field under refinement Failed
Signed Bianchi IX PDE evolution is numerically available Stable signed-field refinement diagnostic Provisional: 4.85% mass change, 19.5% field difference
Signed particles efficiently reproduce fixed-grid Bianchi IX evolution \(L^1<0.10\) under particle refinement Provisional: 0.848 at 320,000 particles
Single-node intertwiner walk matches constrained graph diffusion Monte Carlo versus exact heat kernel Passed
LRS Bianchi I persistent walk matches its reduced WDW/telegraph equation in the selected positive sector Relative density \(L^1\leq0.05\) with 95% upper confidence bound below gate for all profiles/orderings Passed (NUM-7; qualification above)
T15b Euclidean persistent random flight matches its exact MSD and causal front checks Five seeded snapshot ensembles; all MSD estimates within \(3\) standard errors and no front violations Passed
T15b planar density is exactly the scalar 2D telegraph equation Exact position-heading transport law and finite-scale density comparison Not claimed; the telegraph equation is the ML fluid-limit approximation
T15b ML telegraph approximation approaches the planar walk in the tested fluid limit Transform error decreases along tested rays; seeded Fourier modes extend to \(\lambda t=128\) Provisional: highest final mode differs by \(2.84\) Monte Carlo standard errors
Many-node spin-network diffusion realizes the WDW limit Shared-edge constraints, microscopic dynamics, and continuum convergence Open

References