Fermionic Mixing: Overview
What this program is about
The Standard Model gives seven mixing angles between quark generations (CKM matrix) and neutrino generations (PMNS matrix). Their values are measured to high precision, but nothing in the Standard Model explains why they take those specific values — they are free parameters, fit to data, not derived from anything more basic.
This program asks whether those seven angles can be read as combinations of a small, fixed set of integers (2, 3, 5, 7, 11, 13), each carrying a structural meaning within the ArXe framework, rather than as arbitrary numbers. The test applied throughout is not “does some combination reproduce the measured value” — with enough free numbers, that’s nearly guaranteed and proves nothing. The test is: does a restricted, small lexicon, checked against numerical precision, grammatical consistency, and specificity (how many competing combinations exist within the same restricted set), single out one answer — or not.
Current state, in one table
| Angle | Formula | Status |
|---|---|---|
| δ_CKM | arctan(169/77) | Official (v10). Two other candidates (273/125, 286/147) are not wrong — each is native to a different experimental target/fit; 169/77 was adopted as the one matching the current PDG global fit. |
| θ₁₃ CKM | arcsin(15/(e·11²·13)) | Strongest formula in the corpus |
| θ₂₃ CKM | 3²γ/5³ | Verified |
| θ₁₂ CKM | 44/195 | Verified — see the open exception below |
| θ₁₂ PMNS | arctan(20φ/49) | Verified |
| θ₁₃ PMNS | arcsin(78φ/847) | Verified |
| θ₂₃ PMNS | arccos(49/75) | Reverted from an earlier φ-based formula (121φ/169) after that version was found to contradict the mediator rule below. The φ-based version isn’t deleted — it’s kept on record as a candidate for a different phenomenon, undeveloped. |
The one structural rule that ties this together
The mediator rule: the integer 13 appears in a formula whenever — and, with one exception, only when — that formula mixes non-adjacent generations (1↔3), not adjacent ones (1↔2, 2↔3). Verified across the seven angles: 6 out of 7. The rule was originally reported as 7/7 in earlier working notes; that was wrong, corrected once checked carefully.
The open exception: θ₁₂ CKM (44/195) contains 13 despite mixing adjacent generations. No alternative formula for θ₁₂ CKM exists in the corpus that would remove the exception. It’s documented as a genuine, unresolved anomaly — not explained away.
What’s derived, not just fitted
Two of the constants used in these formulas — e and γ (Euler’s number and the Euler-Mascheroni constant) — aren’t inserted by hand. They’re derived from the boundary-condition structure of the underlying framework as the natural attractors of two different processes: e for independent, memoryless open boundary conditions; γ for boundary conditions with decreasing memory. That derivation is the formal backbone under θ₁₃ CKM and θ₂₃ CKM specifically, not a post-hoc justification for numbers that were already chosen to fit.
Method, briefly
- Numerical filter: does the candidate reproduce the measured value within experimental precision?
- Grammar filter: does the combination of integers make sense under the framework’s own reading rules (which integer means what, how they combine), checked independently of whether the candidate happens to fit numerically?
- Specificity check: how many other combinations, within the same restricted set of allowed integers, would also fit? A formula that’s one of many equally good candidates is weak evidence; a formula that’s the only one passing both filters is strong evidence.
- When candidates tie: resolved by asking which practical measurement choice (which fit, which convention) each candidate is native to — not by inventing a new rule to break the tie after the fact.
Where to go for more
- CKM, full technical status: CKM Angles: Revised Epistemic Status
- PMNS, full technical status: PMNS Angles: Verification and Specificity
- The mediator-13 rule, correction and both open questions: The Mediator-13 Rule Is Not 7/7: Correction and Its Two Causes
- The e/γ derivation route (why these constants, not others): Derivation: 13 as Mediator of the 1↔3 Generational Jump, Derivation: e as Attractor of Independent BC_open, Derivation: γ as Attractor of BC_open with Decreasing Memory, Formal Gap Closure
- The δ_CKM tie-break in detail (three candidates, one adopted): δ_CKM: Axiomatic Archaeology of the Three Candidates (Applying ALO-Inverse)
What this is not
This is not a claim to have derived the Standard Model’s mixing angles from first principles, and not a claim that the framework behind it (ArXe) is established physics. It’s a restricted, falsifiable pattern search with explicit rules, an open exception that isn’t hidden, and a documented history of at least one correction (7/7 → 6/7) made after the fact, not before. Readers are encouraged to check the numbers themselves — every formula above is verifiable against public PDG data.
ArXe Research — August 2026
Diego Luis Tentor