The optical axis, tested: a passed null and a single-domain no-go
Paper IV is on Zenodo, and Papers I and II have been re-versioned to v2.0 to record what it found. The framework’s sharpest falsifiable prediction was carried through — one channel passed, the next one excluded the construction that produced it.
Paper IV is on Zenodo — Optical-Axis Anisotropy on the A=1 Discrete Causal Lattice: Birefringence Cancellation and a Single-Domain No-Go. Alongside it, Paper I and Paper II have been re-versioned to v2.0 to record what it found.
The result is negative, and that is the point of publishing it.
The prediction that passed
The series’ sharpest falsifiable claim lived on the optical axis \mathbf{V}_1+\mathbf{V}_2+\mathbf{V}_3 = (1,1,-1). The first channel is vacuum birefringence — two polarizations travelling at different speeds — and 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. It has not been observed. A prediction that forbids an observable outcome, where the outcome is excluded, is a falsifiable prediction that has passed — not an absence of content.
It also discriminates. In the Standard-Model Extension classification, the leading Planck-suppressed birefringent operator has odd mass dimension (dimension five, CPT-odd), and generic discrete-spacetime and Lorentz-violating models populate that sector — which is precisely why the astrophysical null excludes them. This framework sits in the even-dimension, non-birefringent class. The same observation that refutes the birefringent rivals corroborates this one. (The cancellation is a theorem conditional on the geometric constitutive blocks; deriving those as the full dynamical vacuum response remains open.)
The channel that did not
Because birefringence is forbidden, the forward-testable content is direction-dependent dispersion. That is where the construction failed.
The induced photon’s common-mode speed anisotropy — both polarizations sharing a direction-dependent speed — is not small. It is order-unity and dimension-four: a factor-of-about-two anisotropy in the speed of light, independent of energy. Cavity and Michelson–Morley isotropy experiments bound that quantity 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; an earlier reading that treated it as dimension-six-suppressed and therefore safe has been retracted.
Both failures trace to a single feature: the hop set uses three of the four cube body diagonals. The omitted fourth diagonal \mathbf{V}_4 is what leaves the optical axis in the first place — and the four-diagonal set is proportional to the identity, which is to say isotropic. One arbitrary omission produces the axis, the anisotropy, and the exclusion.
What this does and does not touch
Scope matters here, so it is worth being exact.
Not falsified: \mathcal{A}=1 conservation itself. The failure traces to the hop-set realization, not to the conservation principle. The adjugate theorem and the birefringence cancellation stand as algebra. The Bell-test and quantum results are untouched.
Excluded as constructed: the specific single-domain, three-diagonal vacuum — in both the gauge and matter sectors.
Open, and explicitly not withdrawn: whether colour survives a successor geometry (Paper II’s SU(3) arises from a per-site index over \mathbf{V}_1,\mathbf{V}_2,\mathbf{V}_3 — colour space is the three-diagonal set), and the coupling ratio g_3^2/g_2^2 = 3/2, in which the relevant tensor and the universal prefactor cancel.
Correcting the record
Papers I and II were carried out on the substrate this excludes, so both have been re-versioned rather than quietly amended. Each v2.0 carries a revision notice immediately before its abstract and a one-page table separating what is withdrawn, what is now conditional, what is retained, and what is unresolved.
The earlier deposits have not been withdrawn. They remain available at their own DOIs, now carrying superseded-status notices pointing forward. A reader who cited the original can still recover exactly what they read, and can see what changed and why. The publication history is part of the result: the original papers generated a testable consequence, the consequence was followed further, and the implementation did not survive it.
What comes next
The direction is identified rather than hoped for. The exclusion has a specific structural cause — the omitted diagonal — and a correspondingly specific class of successor architectures that do not share it. Those successors are not birefringent, and work on them is under way. We will have more to say about the geometry once the current corrections are fully settled.
A framework that could not fail here would not have been worth much. This one could, in one channel it did, and the record now says so plainly.
Pointers:
- Paper IV (Zenodo, latest): doi.org/10.5281/zenodo.21435951
- Paper I v2.0: doi.org/10.5281/zenodo.21464971 · landing page
- Paper II v2.0: doi.org/10.5281/zenodo.21477781 · landing page
- Claim-by-claim status: claim map
- Series papers: papers page