Arxe Theory

1. Verification, redone with the actual formulas

5 of 7, not 7 of 7. Two real exceptions, not apparent ones — verified by direct factorization, not by reading.

2. The two exceptions have different causes

θ₁₂ CKM = 44/195 — a verification error, not a superseded formula

44/195 isn’t a formula that replaced another one — it’s in the original table (§3.1 of arxe_mezcla_CKM_PMNS_hipotesis.md), the same source that later generated the mediator rule. In other words: whoever wrote the “7/7” verification table had this formula available in the same corpus and didn’t cross-check it carefully. This isn’t a propagation problem (like Ω_Λ) or a silent substitution — it’s a simple counting error in the table that was cited as verification.

θ₂₃ PMNS = 121φ/169 — a superseded formula, never reconciled

This case does have a substitution history, with precedent in the corpus (arxe_mezcla_CKM_PMNS_hipotesis.md §4.1 vs. §5.2):

• Original version (§4.1): arccos(49/75) = arccos(7²/(3×5²)) — no constant, no 13, consistent with the mediator rule.
• Later version (§5.2): a “best constant per angle” search among {π,φ,e,γ,√2,√3} found that φ fits better (Δ0.00091° vs. Δ0.00265° for the no-constant version) — and that search produced 121φ/169, which does contain 13.

The φ version won out because it beat the numerical fit and because it fit the separate, also-verified pattern that φ wins in all three PMNS angles without exception (arxe_PMNS_epistemic_status_v1.md §3). But that substitution was never checked against the mediator rule — and doing so now, they come into tension: the “φ in all three PMNS, no exception” pattern and the “13 not in adjacent angles” rule cannot both be satisfied at once by the current θ₂₃ PMNS formula.

3. What this means for the mediator rule — and for δ

The rule goes from “verified, 7/7, zero exceptions” to “verified at 5/7, with two exceptions — one a transcription error, the other a real, unreconciled conflict between two distinct corpus rules.” That isn’t the same as “the rule is wrong” — δ, θ₁₃ CKM, and θ₁₃ PMNS remain consistent with it, and its own structural logic (BC_closed≥2, T⁻³ confinement) doesn’t depend on case-counting, it depends on the argument in arxe_derivacion_13_largo_alcance.md §§4-6, which is untouched here. But the empirical backing attributed to it — “7/7, zero exceptions” — is not sustainable as written, and needs correcting before it’s cited again as strong evidence, including in arxe_delta_CKM_grammar_filter_resolution_v1.md and in arxe_CKM_epistemic_status_v8.md, where it was used precisely to exclude competitors for δ.

This doesn’t invalidate the exclusion of those competitors (98/45 still lacked 13, and that absence remains relevant regardless of whether the pattern is 5/7 or 7/7) — but if the axiomatic-relativity principle (arxe_criterios_lectura_gramatical_v1.md §3) is applied with the same rigor demanded of everything else today, it must be said that the rule supporting that exclusion is weaker than presented, and that its own internal consistency (5/7, not 7/7) is also, now, in question.

4. What to do with the two exceptions — decision made

θ₂₃ PMNS — resolved. Reverted to 49/75 (no constant, consistent with the mediator), prioritizing the structural rule (BC_closed/confinement derivation) over the φ version’s marginal improvement in numerical fit (Δ0.007° vs. Δ0.0009°, a difference insufficient to justify breaking the rule). The φ version is not discarded as an error — it remains documented separately (arxe_PMNS_epistemic_status_v2.md §5) as a candidate for a different phenomenon: stochastic convergence with constitutive memory (line of work from arxe_dos_logicas_estocasticas_pi_phi_v1.md), not pure arity — pending further development, not competing with the structural reading.

Consequence accepted, not hidden: the pattern “φ wins in all three PMNS angles, no exception” goes from 3/3 to 2/3. This is a real loss of strength for that line of evidence — documented as such in arxe_PMNS_epistemic_status_v2.md §3, not minimized.

4b. θ₁₂ CKM — unresolved exception, no alternative formula forced

Unlike θ₂₃ PMNS, 44/195 has no precedent of an alternative formula in the corpus — it’s the only reading available since the origin (§3.1 of arxe_mezcla_CKM_PMNS_hipotesis.md), with a strong numerical margin (53.4× under the canonical protocol). It would not be appropriate to invent an alternative reading just so the mediator rule closes at 7/7 — that would be exactly the kind of ad hoc fitting being avoided throughout this whole review. It is recorded as a genuine, unexplained exception, and the mediator rule is cited from now on as “6/7, one open exception (θ₁₂ CKM),” not as “no exceptions.”

5. Immediate action

Correct arxe_derivacion_13_largo_alcance.md §7: replace “7 of 7, zero exceptions” with “6 of 7 — one open exception: θ₁₂ CKM contains 13 despite being an adjacent jump, with no available explanation.” Do not retroactively edit the θ₂₃ PMNS row of that table as if there had never been a conflict — annotate the reversion with a date, the same way it was done elsewhere in the corpus today.

ArXe Research — August 2026
Diego Luis Tentor
Systematic verification assisted by Claude.ai (Anthropic)