Two Unconditional Theorems
Both theorems land in the Schrödinger-limit regime of the closure dynamics on the 600-cell substrate. Both are independently reproducible from canonical numerical scripts that run in seconds with no fitted parameters.
For Schrödinger-limit evolution iψ̇ = Hψ with H any real-symmetric nearest-neighbour-supported operator on the 600-cell graph (the Laplacian L = D − A is the canonical example), the explicit edge current
jv→w := −2 Im(ψv∗ Hvw ψw)
satisfies the discrete continuity equation pointwise, is exactly antisymmetric (jv→w + jw→v = 0), and yields exact U(1) total-probability conservation. Promotes the conditional continuity hypothesis from the conditional to unconditional in the Schrödinger limit.
The λ = 0 eigenspace of the 600-cell Laplacian LV₆₀₀ is one-dimensional, lies in the trivial 2I-isotypic sector (where 2I is the binary icosahedral group), is stationary under iψ̇ = Lψ, and L · P0 = 0 as an operator. Non-trivial eigenmodes oscillate at ω = λ exactly. This is the substrate-side zero-mode fact used by the massless m = 0 specialisation of the dispersion ansatz.
Numerical Witnesses — Machine Precision
Both theorems are reproduced by self-contained scripts (numpy + scipy only, no external data, no fitted parameters). Wall-clock seconds on a laptop.
verify_u1_conservation.py — Theorem 1
Builds V600 + Laplacian L, evolves a localised complex initial state ψ under U = exp(−iLΔt) for 200 steps at Δt = 0.05, and at every step verifies (a) the analytic continuity residual at every vertex, (b) total Q conservation, (c) current antisymmetry.
| Quantity | Value |
|---|---|
| n_steps | 200 |
| Q_initial, Q_final | 1.0, 1.0 |
| max Q drift over 200 steps | ~3 × 10−15 |
| max continuity residual (analytic) | ~3 × 10−16 |
| max antisymmetry violation | exactly 0 |
| all_three_pass | true |
verify_photon_dispersion.py — Theorem 2
Builds L, computes the spectrum, verifies λmin = 0 with multiplicity 1 and uniform eigenvector, the operator-level zero L · P0 = 0, the trivial-sector stationarity, and the non-trivial-mode frequency match ω = λ.
| Quantity | Value |
|---|---|
| λmin | ~6 × 10−15 (i.e. ≈ 0) |
| trivial multiplicity | 1 |
| trivial eigenvector uniformity deviation | ~4 × 10−16 |
| ‖L · P0‖F (operator zero) | ~1 × 10−14 |
| trivial-sector evolution drift over 50 ticks | ~5 × 10−15 |
| non-trivial mode ω = λ match | ~5 × 10−15 |
| all_six_pass | true |
Status of Claims at a Glance
The paper applies an explicit claim-boundary discipline. Each statement carries its status:
| Claim type | Content | Status |
|---|---|---|
| Theorem 1 | Schrödinger-limit explicit current, antisymmetry, pointwise discrete continuity, exact U(1) conservation | proved here |
| Theorem 2 | V600 Laplacian zero-mode is 1-dim, trivial 2I sector, stationary under iψ̇=Lψ; non-trivial modes rotate at ω = λ | proved here |
| Hypothesis | Discrete continuity for the full Langevin-with-noise closure dynamics | open |
| Hypothesis | Relativistic-form dispersion ω² = ceff² k² + mρ² with mρ² = βλρ | heuristic identification |
| Hypothesis | Identification of λ = 0 eigenmode with the physical U(1)EM photon sector | conditional on TPH |
| Programme target | ARIA couple_tick verification of Fick / Nernst-Planck / cascade-pressure-balance objectives | not done here |
| Programme target | Microtubule active-substrate biophysics | future biophysics |
Symmetries of the Closure Functional
The closure functional &Fcal;[ψ] = αR[ψ] + βE[ψ] − γC[ψ] has two exact symmetries explicitly catalogued, and two approximate ones explicitly flagged.
U(1) phase
Global U(1) rotation ψ → eiθψ. Yields the conserved total-probability charge Q under Theorem 1.
H4 isometry
The finite Coxeter symmetry of the 600-cell. Yields a conditional block-norm-invariance statement; not a continuous angular-momentum current (H4 has no Lie algebra).
Approximate symmetries explicitly flagged: σ-Galois conjugation in φ and an SO(4) rotation extension. The paper does not claim these as exact.
Five Open Items
The following five items separate the present paper from a closed theorem chain:
- Gauge-field emergence. Gauging the global U(1) of Theorem 1 to a local U(1) introduces an electromagnetic potential Aμ. Whether the closure functional naturally carries a local phase gauge is open; the paper works with the global charge current only.
- Γ-limit coarse-graining. The continuum passage from V600 to a continuum hydrodynamic limit is sketched, not proved. A full proof requires a Γ-convergence argument.
- Learned W as gauge potential. Whether ARIA's learned W can be identified with a gauge potential in the quantised transport law is open; this is the active-side analogue of item 1.
- ARIA repository-level verification. A row-by-row dictionary mapping ARIA's couple_tick to the transport-law objects is not yet carried out.
- Discrete continuity for full Langevin-with-noise. Theorem 1 closes the Schrödinger-limit case; the generator-level derivation for the full Langevin closure dynamics with noise is deferred.
What This Is, And Is Not
Two theorems. Machine-precision numerical witnesses. Five open items, named and bounded.
verify_u1_conservation.py and verify_photon_dispersion.py. Both run in seconds with numpy + scipy only. MIT licence.