Paper IV — Optical-Axis Anisotropy: Birefringence Cancellation and a Single-Domain No-Go

Paper IV tests the A=1 lattice’s sharpest falsifiable prediction on the (1,1,-1) optical axis. Vacuum birefringence is forbidden and the data agree — a passed null that also excludes the dimension-5 birefringent rivals. The dispersion channel then excludes the single-domain three-diagonal construction itself, in both the gauge and matter sectors. A=1 conservation is untouched.
ImportantAn honest no-go (v1.0, July 2026)

This paper carries a negative result, published as a result. The three-diagonal, single-domain vacuum construction is excluded as constructed in both the gauge and matter sectors, while the \mathcal{A}=1 conservation principle itself is untouched. The root of the exclusion is a single feature — the omission of the fourth cube body diagonal \mathbf{V}_4, the same omission that defines the optical axis — and the escape is open, structural future work. Full claim-by-claim status in the claim map.

Abstract

Paper IV of the A=1 Discrete Causal Lattice series takes the series’ sharpest falsifiable claim — an anisotropy on the optical axis \mathbf{V}_1+\mathbf{V}_2+\mathbf{V}_3=(1,1,-1) — and carries it through two channels. The first channel is vacuum birefringence: two polarizations travelling at different speeds. The framework does not merely fail to predict it; it forbids it. The induced photon action’s adjugate closure Q_B=\operatorname{adj}(P) produces a doubled transverse root, so the polarization split cancels identically. Vacuum birefringence is among the most tightly bounded quantities in physics and has not been observed — so a prediction that forbids it, where the outcome is excluded, is a falsifiable prediction that has passed, and one that additionally excludes the dimension-5, CPT-odd birefringent operators that generic discrete-spacetime and Lorentz-violating models populate. This cancellation is a theorem conditional on the geometric constitutive blocks; deriving those as the full dynamical vacuum response remains open. The second, forward-testable channel is direction-dependent dispersion, and it is where the construction fails: the induced photon’s common-mode speed anisotropy — both polarizations sharing a direction-dependent speed — is order-unity and dimension-four, a factor-of-about-two anisotropy in the speed of light independent of energy, which cavity and Michelson–Morley isotropy experiments bound to around one part in 10^{18}. For a single fixed trigonal domain the induced electromagnetic sector is therefore excluded as constructed; the matter sector is excluded as constructed too, and an earlier reading that treated it as dimension-six-suppressed and therefore safe is retracted. Both failures trace to one feature: the hop set uses three of the four cube body diagonals, and the omitted fourth diagonal \mathbf{V}_4 is what leaves the optical axis in the first place — the four-diagonal set being proportional to the identity, i.e. isotropic.

What this paper establishes

  • A passed null, not an absence of content. Optical-axis birefringence is forbidden by the adjugate closure Q_B=\operatorname{adj}(P) (the doubled transverse root cancels the polarization split), and the astrophysical null observation corroborates the framework while excluding the dimension-5 birefringent rivals — one observation that refutes them and supports this. The cancellation is conditional on the geometric constitutive blocks (their derivation as the full dynamical response is a PART row).

  • A single-domain no-go. Because birefringence is forbidden, the forward-testable content is dispersion, and there the induced-photon common-mode speed anisotropy is order-unity and dimension-four — excluded by cavity / Michelson–Morley bounds. The single-domain, three-diagonal vacuum is therefore excluded as constructed in the gauge sector; the matter sector is likewise excluded (the “dimension-six-safe” reading is retracted).

  • One structural root. Both exclusions trace to the omission of the fourth cube body diagonal \mathbf{V}_4 — the very omission that defines the optical axis. The four-diagonal set is proportional to the identity (isotropic); one arbitrary omission produces the axis, the anisotropy, and the exclusion.

  • What is not touched. \mathcal{A}=1 conservation itself is not falsified — the failure is in the hop-set realization, not the principle. The adjugate theorem and the birefringence cancellation stand as algebra, and the Bell-test and quantum results are untouched.

The escape, open and not withdrawn

The exclusion is specific, so the class of successor architectures that do not share it is specific too. The parked escape is an O_h-restoring four-orientation vacuum (a vacuum that restores the full cubic symmetry while matter keeps its axis) together with a token\tomatter emergence map. Whether colour \mathbb{C}^3\to SU(3) survives a \mathbf{V}_4-inclusive geometry — colour space is the three-diagonal set — and the coupling ratio g_3^2/g_2^2=3/2 are left open, not withdrawn. This is unresolved foundational research; Paper IV makes no claim which way it resolves.

Methodological role

Paper IV is the series’ falsifiability instrument: it takes a sharp, no-free-parameter prediction and follows it until one channel passes and the next excludes the very construction that produced it. It is published because it is negative — a framework that could not fail here would not have been worth much. Papers I and II were carried out on the substrate this excludes and have been re-versioned to v2.0 to record it; the earlier deposits remain available and unwithdrawn, carrying superseded-status notices that point forward.

Audit-row impact

See the Claim map for the one-page, reviewer-facing table. In brief:

  • PASS / blocks PART — adjugate closure Q_B=\operatorname{adj}(P) and null optical-axis birefringence, conditional on the geometric blocks.
  • FAIL — single-domain gauge viability (order-unity, dimension-four common-mode speed anisotropy, excluded by cavity / Michelson–Morley bounds).
  • FAIL — matter sector “dimension-six-suppressed, therefore safe” (retracted; also excluded as constructed).
  • — — \mathcal{A}=1 conservation: unresolved, not falsified by this result.

Citation and version

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