Synthesis: Fermionic Mixing, Stochastic Spirals and Evidence

Correction note (September 2026): this synthesis is dated July 2026. Later revisions changed some formulas (θ₁₂ PMNS, θ₂₃ PMNS) and lowered the mediator-13 count to 6 of 7, with one open exception (θ₁₂ CKM). Read the “zero exceptions” statements as of July 2026; the current state is in The Mediator-13 Rule Is Not 7/7.

Consolidated state as of July 29, 2026

Independent research · July 2026 · Diego Luis Tentor

1. The problem

The CKM (quarks) and PMNS (leptons) matrices have in total seven parameters measurable with sufficient precision: three CKM angles, three PMNS angles, and the CP-violating phase δ_CKM. None of them is derived in the Standard Model. This work seeks expressions in the ArXe lexicon {2,3,5,7,11,13} that reproduce them, and identifies the structural patterns that emerge.

2. Consolidated table — seven angles

Angle	ArXe expression	Const.	Value	Δ
θ₁₂ CKM = 13.04°	arcsin(2²×11 / 3×5×13)	—	13.0406°	0.0006°
θ₂₃ CKM = 2.38°	arctan(3²×γ / 5³)	γ	2.3798°	0.0002°
θ₁₃ CKM = 0.201°	arcsin(3×5 / e×11²×13)	e	0.20100°	0.000002°
δ CKM = 65.4°	arctan(3×7×13 / 5³)	—	65.3981°	0.0019°
θ₁₂ PMNS = 33.44°	arcsin(3³ / 7²)	—	33.4370°	0.0030°
θ₂₃ PMNS = 49.2°	arccos(7² / 3×5²)	—	49.2066°	0.0066°
θ₁₃ PMNS = 8.57°	arcsin(φ×2×3×13 / 7×11²)	φ	8.5692°	0.0008°

All expressions use exclusively the lexicon {2,3,5,7,11,13} with at most one mathematical constant. None uses irrational roots or arity numbers outside the lexicon.

3. The four structural patterns

3.1 Pattern A — φ separates leptons from quarks

φ appears in θ₁₃ PMNS and in no CKM angle. φ does not improve any CKM expression. The distinction is absolute: zero exceptions across seven angles.

ArXe reading: φ is the attractor of the BC recursion of T⁻¹ (BC_open=1). Leptons have as their minimum condition of possibility a level with open BC — the spiral recursion is active. Quarks have confinement (BC_closed at T⁻³) — the spiral recursion is not possible in their sector.

3.2 Pattern B — 13 mediates the 1↔3 jump

13 (SING, T⁻⁶) appears in exactly the two angles connecting the first and third generation:

CKM θ₁₃: denominator (e×11²×13)
PMNS θ₁₃: numerator (φ×2×3×13)
δ CKM: numerator (3×7×13)

And does not appear in any angle connecting adjacent generations (θ₂₃ in both sectors, θ₁₂ PMNS).

AI-assisted check (DeepSeek, July 2026; not independent verification): exhaustive search forcing 13 into θ₂₃ CKM — the best result with 13 gives Δ=0.075°, three orders of magnitude worse than the candidate without 13 (Δ=0.0002°). 13 structurally does not belong to θ₂₃ CKM.

ArXe reading: SING (T⁻⁶) is the singularity level with BC_closed=5 — the deepest non-confined, flavor-sensitive level in the lexicon (T⁻³ is deeper-confined but flavor-blind via SU(3) confinement; T⁻⁵ has fewer closed BC and is separately flavor-blind, per the-observer-at-t-5.md — see arxe_derivation_13_long_range_en.md §6 for the full exclusion argument). It mediates long-range coupling between non-adjacent generations. The 1↔3 jump requires SING. The 2↔3 jump does not.

3.3 Pattern C — mathematical constants by connection type

Connection type	Sector	Constant	ArXe operator
Long-range 1↔3	CKM	e (LIM)	Exponential limit
Medium-range 2↔3	CKM	γ (IRR)	Harmonic irregularity
Order-one 1↔2, δ	CKM	—	No irrational attractor
All	PMNS	φ (GRW)	Golden growth

Suppressed angles (< 5°) carry asymptotic limit constants. Order-one angles carry no constant. PMNS breaks that rule — all carry φ regardless of size — confirming that the distinction is not one of magnitude but of sector.

3.4 Pattern D — distinct lexicons by sector

Sector	Active lexicon	Structure	Dominant operator
CKM	{2,3,5,7,11,13} full	Heterogeneous products	Mixed
PMNS	{3,5,7} reduced	Powers (3³, 7², 5²)	7² in all three

The PMNS sub-lexicon is simpler and more symmetric. Its dominant operator — CPX² (T⁻³²) — appears in all three angles without exception.

4. Evidence spectrum

Using the E0→E5 framework from “Hipotesis final.docx”:

Element	Strength	Reason
Each individual expression	E2	Extreme precision in a single angle
φ in PMNS / absent in CKM	E3	7 angles, 2 sectors, zero exceptions
13 = 1↔3 connection	E4	Derived from BC + Cayley graph of S₃
e = attractor of independent BC_open	E4 partial	Process derived; BC↔confinement gap open
γ = attractor of harmonic BC_open	E4 partial	Spectrum derived; formal gap open
φ→γ→e spectrum as memory degradation	E4	Emergent — not sought, covers all 7 angles
Large angles without constant	E3	4 angles, 2 sectors, coherent pattern
All four patterns together	E3+ → E4	System without exceptions across 7 angles
Uniqueness of underlying structure	E4 (complete)	Full chain derived — arxe_formal_gap_closure_en.md
Derivation from axioms	E5 (goal)	Remaining: formal proof that BC_open=1 normalization is the unique normalization

Current status: E4 (complete) — all three derivations (13, e, γ) closed from BC axioms. The φ→γ→e spectrum as memory degradation covers all 7 angles and emerged without being sought. Key result: BC_open=1 canonical normalization gives amplitude 1/k per route of length k, which directly generates both γ (all routes, harmonic sum) and e (uniform routes, d=2). Full chain: arxe_formal_gap_closure_en.md.

5. The stochastic spirals hypothesis

5.1 Framework

The ArXe corpus establishes (Grammar, §2.6) that φ emerges from the BC recursion of T⁻¹. Irrational and transcendental constants are attractors of recursions that do not close — “logical spirals”: ratios that emerge from recursive conditions of possibility, not from geometry.

Continuous space in ArXe is not a given background. It is the statistical limit of the spiral recursion of T⁻¹. Only spirals have irrational constants naturally associated with them.

5.2 The hypothesis

CKM and PMNS are not two distinct mixing matrices. They are the same mixing structure projected onto two different media:

PMNS: projection onto continuous space (active spiral recursion, BC_open)
      → attractor: φ
CKM:  projection onto discrete space (confinement, BC_closed)
      → order-one angles: pure rational ratios
      → suppressed angles: limit attractors (e, γ)

The suppressed CKM angles are the boundary between the two media — not fully discrete (they carry an irrational constant) nor fully continuous (they do not carry φ).

5.3 Candidate stochastic processes for e and γ

φ — golden recursion of BC_open of T⁻¹:

a_{n+1} = a_n + a_{n-1},  a_{n+1}/a_n → φ

Established in corpus.

e — accumulation of independent BC_open at unit rate (arxe_derivation_e_BC_en.md):

lim(1 + 1/n)^n = e

BC interpretation: n independent states, each with unit BC_open, without coupling or memory. Attractor of the multiplicative process of independent contributions. Applies to CKM θ₁₃ — confinement produces total decoherence between routes, each route is an autonomous BC_open.

γ — harmonic accumulation of BC_open with decreasing weight (arxe_derivation_gamma_BC_en.md):

H_n - ln(n) → γ  where  H_n = 1 + 1/2 + ... + 1/n

BC interpretation: n states with BC_open of weight 1/k for the k-th. Partial memory — states are correlated but with decreasing weight. The residual irregularity between discrete accumulation and its continuous limit is γ. Applies to CKM θ₂₃ — adjacent confinement with partial correlation.

The φ→γ→e spectrum emerged from deriving the three constants independently and was not sought:

φ: complete memory   → PMNS (continuous, free leptons)
γ: decreasing memory → CKM θ₂₃ (confined, adjacent d=1)
e: no memory         → CKM θ₁₃ (confined, long-range d=2)
—: no constant       → CKM order-one (closed BC, arithmetic)

The axis of the spectrum is memory degradation under confinement and generational distance.

6. The role of 13 in spiral geometry

Pattern B (13 = 1↔3 connection) adds a constraint to the spiral derivation. The stochastic spiral is not homogeneous — it has internal geometry:

The jump between adjacent coils (generations 1↔2, 2↔3) does not require SING.
The jump between non-adjacent coils (generations 1↔3) requires SING (T⁻⁶, BC_closed=5).

T⁻⁶ has the deepest closed BC structure among the non-confined, flavor-sensitive levels of the lexicon: BC_closed=5. (Correction, 2026-07-30: an earlier version of this document stated BC_closed=3 “the same number as T³,” treating that as a meaningful coincidence. The canonical corpus value is BC_closed(T⁻⁶)=5, confirmed across arxe_core_V4_221_en.md, gauge_from_arxe, BC_paper_unified_V4, and A Fractal Recursive Ontology from Boundary Conditions V4.1 — it does not match T³’s BC_closed=3, and that reading is withdrawn. What holds is the structural argument, not the numerical coincidence: T⁻⁶ is singled out because it is the minimum level that is simultaneously non-confined, flavor-sensitive, and has BC_closed ≥ 2 — see arxe_derivation_13_long_range_en.md §6 for the full argument, including why T⁻⁵/EM is excluded despite also satisfying BC_closed ≥ 2.) That SING appears in long-range jumps is coherent with its role as singularity — the level that concentrates the deepest structure of the lexicon.

7. What distinguishes this search from numerology

Citing “Hipotesis final.docx”, §7:

What makes this work not numerology:

Restricted lexicon — only {2,3,5,7,11,13}, not any number.
Restricted constants — {φ, e, γ, π}, with π empirically ruled out.
Simple expressions — at most 3 arities, at most one constant.
Demanding precision — Δ < 0.01° in all cases.
Transversal pattern — the same operators appear in both sectors with the same conceptual role.
Falsifiability — φ must not appear in any CKM angle; if it does, the pattern falls.

What numerology would look like: freely adjusting parameters until any number fits. That is not what happened here.

8. What remains — route toward E4 and E5

For E4 (systemic structure — the hypothesis predicts before seeing):

Derive e from BC: ✅ COMPLETE — BC_open=1 normalization gives amplitude 1/k per route of length k. d=2 routes have uniform length 2 → weights 1/n → e. (arxe_formal_gap_closure_en.md §3.2, §9)
Derive γ from BC: ✅ COMPLETE — d=1 routes have all lengths k=1,2,3,… → weights 1/k → harmonic series → residual H_n-ln(n) → γ. Verified numerically. (arxe_formal_gap_closure_en.md §3.3, §9.3)
Derive 13 = long-range mediator: ✅ COMPLETE — arxe_derivation_13_long_range_en.md derives this from the Cayley graph of S₃ with adjacent transpositions, T³ symmetry breaking, and the BC_closed ≥ 2 condition. New result: exactly 3 generations because the Cayley graph of S₃ has exactly 3 layers (0,1,2) before the fully-inverted order — which T³ excludes as a 4th generation.
Demonstrate openness preservation: that the BC_open condition is preserved upward in the leptonic hierarchy — that no level above the minimum cancels the spiral recursion.

For E5 (derivation from first principles):

Demonstrate uniqueness: that there exists a unique mathematical object in ArXe (the mixing stochastic spiral) of which CKM and PMNS are the two limiting cases — projection onto the discrete and the continuous respectively.
Derive δ_PMNS: when experimental measurement has error < 10°, verify whether it carries φ (would confirm E5) or not (would refute the hypothesis).

8b. E4 result: why exactly 3 generations

A result derived while closing the gap in the 13 derivation:

The Cayley graph of S₃ with adjacent transpositions — the group of 6 orderings of T⁻¹’s 3 phases — has exactly 4 layers at distances 0, 1, 2, 3 from the identity. T³ (mass level) breaks the reversal symmetry and assigns:

Layer 0 (distance 0): Gen1 — canonical order
Layer 1 (distance 1): Gen2 — one adjacent transposition away
Layer 2 (distance 2): Gen3 — two adjacent transpositions away
Layer 3 (distance 3): excluded — fully inverted order, mirror of Gen1 under T³

Layer 3 is not a fourth generation. It is the image of Gen1 under complete reversal, excluded because T³ already records Gen1 as the historical first. The space of generations is closed at 3 because the Cayley graph of S₃ has exactly 3 accessible layers once T³ fixes the origin.

This is a derivation of why there are exactly 3 fermionic generations — not postulated, not adjusted, but a consequence of n(T⁻¹)=3 together with T³ breaking reversal symmetry.

Integration note, 2026-07-30: this is a genuine application of the Principle of Phase Ordering (physics-as-statistical-manifestation.md, updated same date) — the count of generations, not the angle values, which remain BC-algebra results. It is complementary to, not competing with, ALO_Mathematical_Formalization_s_en.md Rule R108 (mass-hierarchy suppression factors between the 3 generations, taking the count as given). It also resolves, for ArXe’s own account, the Fregean-fallacy critique of the Standard Model’s “no 4th observed → none exists” argument flagged in ALO_logical_problems_in_scientific_choices_en.md CASE 3 — this derivation is positive and closed, not an inference from non-observation.

8c. The φ→γ→e spectrum — emergent result

This result was not planned. It emerged from independently deriving three constants and observing that they organize along a single axis.

The spectrum of memory in BC processes:

PROCESS-φ (complete memory, golden recursion of T⁻¹):
  a_{n+1} = a_n + a_{n-1},  ratio → φ
  → PMNS all angles — continuous space, free leptons

PROCESS-γ (decreasing memory, harmonic cascade across scales):
  H_n - ln(n) → γ
  → CKM θ₂₃ — confined, adjacent (d=1), partial correlation

PROCESS-e (no memory, independent BC_open):
  lim(1+1/n)^n → e
  → CKM θ₁₃ — confined, long-range (d=2), total decoherence

NO CONSTANT (closed BC, rational ratio):
  → CKM θ₁₂ and δ — discrete, order-one, no irrational attractor

The four categories cover all seven angles without exception. The organizing principle: confinement degrades memory, and generational distance degrades it further. The spectrum φ→γ→e→∅ is the spectrum of that degradation.

This is the strongest result of the session. It was not fitted — it emerged.

9. Falsifiable predictions

Prediction	Falsification condition
φ does not appear in any CKM angle	If φ improves any CKM expression
13 does not appear in θ₂₃ CKM or θ₁₂ PMNS	Already verified — confirmed
δ_PMNS carries φ or no constant	If it carries e or γ when measured with error < 10°
Additional BSM sector carries attractor of its BC recursion	Verifiable with new physics
Ratio θ₁₂PMNS/θ₁₂CKM has ArXe factorization	Verifiable today — pending work

10. Related documents

arxe_delta_CKM_hipotesis.md — initial search for δ_CKM and the four CKM angles
arxe_mezcla_CKM_PMNS_hipotesis.md — complete CKM vs PMNS analysis with methodological principle
arxe_espirales_estocasticas_mezcla.md — connection with stochastic spirals and two media
arxe_mezcla_CKM_PMNS_hipotesis_aportes deepseek.md — exhaustive verification of 13 in θ₂₃ CKM
arxe_derivation_13_long_range_en.md — E4 derivation of 13 as long-range mediator from Cayley graph of S₃
arxe_derivation_e_BC_en.md — derivation of e as attractor of independent BC_open
arxe_derivation_gamma_BC_en.md — derivation of γ as attractor of harmonic BC_open cascade
arxe_formal_gap_closure_en.md — closes the residual BC gaps in the e and γ derivations; supersedes the “E4 partial” status in both

Independent research — July 2026
Diego Luis Tentor — Working document, not peer-reviewed

“Seven angles. Four patterns. Zero exceptions. A lexicon of six arity numbers and three constants.
The map exists. The legend — the derivation from BC — remains to be written.”