Arxe Theory

1. The three formulas, current status

θ₁₂ and θ₁₃ unchanged from v1. θ₂₃ reverted — see §2.

2. Why θ₂₃ is reverted

The formula 121φ/169 (11²φ/13²) won the numerical fit against 49/75 (Δ0.00091° vs. Δ0.00265° in the original constant search, arxe_mezcla_CKM_PMNS_hipotesis.md §5.2) and was therefore adopted as the standing formula. But it contains 13 — and θ₂₃ connects generation 2 to generation 3, an adjacent jump. The mediator rule (arxe_derivacion_13_largo_alcance.md, corrected in arxe_regla_mediador_13_correccion_v1.md) establishes that 13 shouldn’t appear in adjacent jumps. 49/75, the original pure-arity formula (no constant, arxe_mezcla_CKM_PMNS_hipotesis.md §4.1), doesn’t contain 13 and is consistent with the rule.

Consistency with the mediator rule is prioritized over the marginal improvement in numerical fit (0.007° vs. 0.0009° error — both within a reasonable margin, the difference isn’t large enough to justify breaking an already-derived structural rule). With this reversion, the mediator rule goes from 5/7 to 6/7 — one unresolved exception remains (θ₁₂ CKM = 44/195, documented in arxe_regla_mediador_13_correccion_v1.md §2, with no alternative formula available).

3. Consequence for the “φ in all three PMNS angles, no exception” pattern

This pattern, cited in v1 §3 as the most solid result of the session that produced it, goes from 3/3 to 2/3 with this reversion. This is a real loss of strength for that line of evidence, and it should be stated as such — not minimized. The two lines of evidence (φ-in-PMNS and mediator-13) came into direct conflict for this specific angle, and both can’t be maintained at once with the same formula. The mediator was prioritized for having the deeper structural derivation (closed BC, confinement — see arxe_derivacion_13_largo_alcance.md §§4-6), against a pattern that, while real, doesn’t have the same kind of independent derivation behind it.

4. Specificity test — unchanged for θ₁₂ and θ₁₃, not recalculated for θ₂₃ in this version

The specificity tests for θ₁₂ and θ₁₃ (v1 §2) are unaffected by this change. The specificity test for 49/75 under the canonical protocol (arxe_protocolo_canonico_especificidad_v1.md) is still pending — it wasn’t run in this revision, which was limited to resolving the conflict with the mediator rule.

5. The φ reading of θ₂₃ — not deleted, left for further development

121φ/169 is not discarded as an error — it is documented separately, as a candidate describing a different phenomenon from the one the pure-arity reading describes. Origin of this distinction: the stochastic-spirals line of work (arxe_dos_logicas_estocasticas_pi_phi_v1.md), which found that φ specifically emerges from processes with constitutive memory (the law of large numbers applied to a recursion with memory, not to memoryless sampling). That search did not, in Diego Tentor’s own words, have “the expected consistency” as a general mechanism — but that doesn’t rule out that it may be pointing to something real and distinct: not a pure-arity structure (what arccos(49/75) describes), but a stochastic convergence that overlaps or coexists with that structure, with its own generating mechanism, still undeveloped.

This is explicitly recorded as pending — not as a formula competing for the same status as the pure-arity one, nor as something to resolve now.

6. Pending (unchanged from v1, except as noted)

• δ_PMNS: no formula — see arxe_delta_PMNS_prediccion_registrada_v1.md for the standing pre-registration. With θ₂₃’s reversion, the prediction registered there (that δ_PMNS should carry φ once precision improves) is now backed by a weaker pattern (2/3, not 3/3) — an update note there is warranted too.
• Check whether the same relation with Ω_Λ that θ₁₃ CKM has via 11×13 exists here — not yet explored for PMNS. (Note: that same coincidence was reassessed downward for θ₁₃ CKM in arxe_omega_lambda_resolucion_v1.md — any future exploration should apply the same criterion from the start, not as a later correction.)
• Run the canonical specificity protocol on 49/75 — pending, not done in this version.

ArXe Research — August 2026
Diego Luis Tentor
Numerical verification and reconciliation with the mediator rule assisted by Claude.ai (Anthropic)