Correction note (September 2026): this document is dated July 2026. Later revisions changed some mixing formulas (θ₂₃ PMNS was reverted to 49/75) and lowered the mediator-13 count to 6 of 7. Read the “zero exceptions” statements and the regime table as of July 2026; the current state is in The Mediator-13 Rule Is Not 7/7.
ArXe — Category and Applied Potential
BC-Attractor Theory of Fermionic Mixing
Independent research · July 2026 · Diego Luis Tentor
Classification and application document
Working document, not peer-reviewed
1. What this work is
1.1 Technical layer
A factorization of fermionic mixing parameters in which each angle is expressed as a ratio of ontological arities plus a stochastic attractor. The attractor identifies the coherence regime of the mixing process.
Seven angles. Four regimes. Zero exceptions.
| Regime | Attractor | Constant | Angles |
|---|---|---|---|
| Complete memory | Golden recursion of T⁻¹ | φ | PMNS all |
| Decreasing memory | Harmonic BC cascade | γ | CKM θ₂₃ |
| No memory | Independent BC_open | e | CKM θ₁₃ |
| Closed BC | Rational ratio | — | CKM θ₁₂, δ |
1.2 Conceptual layer
A theory of memory in mixing processes under confinement. The spectrum φ→γ→e does not only describe angles — it describes how a physical system loses coherence as it moves from the continuous to the discrete medium under increasing confinement.
The organizing principle: confinement degrades memory. Generational distance degrades it further.
1.3 Formal category
BC-attractor theory of quark-lepton mixing
Or, in ArXe language: Grammar of fermionic mixing — mixing as a stochastic process whose attractor is determined by the BC structure of the medium.
The work sits at the intersection of:
- Ontological arity factorization (ArXe core method)
- Stochastic process theory (attractors of BC recursions)
- Fermionic sector physics (CKM/PMNS matrices)
- Information theory (memory structure in physical processes)
2. Derived results with independent value
Before addressing applications, the results that stand on their own:
R1 — Exactly 3 fermionic generations: derived as a necessary consequence of n(T⁻¹)=3 and T³ breaking reversal symmetry in the Cayley graph of S₃. Not postulated. Not fitted. Layer 3 of the graph is excluded — not a fourth generation but the mirror of Gen1 under T³.
R2 — 13 as mediator of long-range generational jumps: derived from BC_closed ≥ 2 requirement for d=2 jumps in the Cayley graph. Originally reported as 7/7 (July 2026); rechecked in August 2026 at 6 of 7, with one open exception (θ₁₂ CKM). The exhaustive search was AI-assisted (DeepSeek); this is not independent verification.
R3 — The φ→γ→e spectrum: emerged without being sought. Complete memory (φ) → decreasing memory (γ) → no memory (e) as confinement and generational distance increase. The spectrum organizes all seven angles under one principle.
R4 — δ_PMNS prediction: when measured with error < 10°, δ_PMNS should carry φ or no constant. If it carries e or γ, the theory is refuted. Falsifiable with DUNE and Hyper-Kamiokande data expected within the next decade.
3. Applied potential
3.1 Materials screening — coherence regime classification
Context: ArXe-based materials screening for critical scaling exponents can, in principle, use arity factorization to produce stability/fit-type scores for candidate materials.
What this work adds:
The spectrum φ→γ→e describes coherence regimes under confinement. In condensed matter physics, the transition between coherent and incoherent regimes determines whether a material is a superconductor, a Mott insulator, or an ordinary metal.
The BC-attractor classification could introduce a fourth screening parameter: coherence regime near the phase transition.
Concretely:
- Materials where order parameter fluctuations have complete memory (correlated, long-range) → φ-regime → candidate for long-range ordered phases, superconductivity
- Materials where fluctuations have decreasing memory (partially correlated) → γ-regime → candidate for glassy, critical, or marginal phases
- Materials where fluctuations are independent (incoherent) → e-regime → candidate for Mott insulators, Anderson localization, strongly confined phases
Practical implementation: the coherence regime could be estimated from the structure of the dominant interactions (BC_open vs BC_closed) without simulation, providing a pre-screening filter before computing full critical exponents.
If confirmed, this would let coherence regime be estimated structurally (from BC_open vs BC_closed patterns of the dominant interactions) rather than only computed numerically from full critical exponents — a potential pre-screening criterion for materials science applications (superconductors, insulators, critical systems).
3.2 Particle physics — falsifiable prediction
The work produces one immediate falsifiable prediction for experimental particle physics:
P1 — δ_PMNS regime: the CP-violating phase of the PMNS matrix should carry φ or no mathematical constant. If it carries e or γ, the BC-attractor framework is refuted.
Current experimental uncertainty on δ_PMNS: ±52°/−25° (PDG 2023). The prediction becomes testable when error < 10°.
Experiments: DUNE (expected first results ~2028), Hyper-Kamiokande (expected ~2027).
P2 — BSM fermionic sector: if a confined BSM sector with a 1↔3 jump exists, its mixing angle should carry e in the denominator. If free, it should carry φ. This is a structural prediction about undiscovered physics.
3.3 Quantum chemistry — perturbative convergence
The e-attractor result — e emerges from n independent BC_open — has a direct reading in quantum perturbation theory.
In Feynman diagram expansions for confined systems, high-order diagrams that contribute incoherently to a process accumulate without interference. If those contributions are independent (no correlation between diagrams of different topology), their accumulation follows the e-process.
Practical implication: the convergence criterion for perturbative series in confined systems may scale as powers of e rather than factorially. This could give a new convergence test for QCD perturbative expansions and for strongly coupled condensed matter models (Hubbard model, t-J model).
This is speculative — it requires formal connection between the BC-attractor framework and the diagrammatic structure of the perturbative expansion. But the structural analogy is direct.
3.4 Information theory and thermodynamics
The φ→γ→e spectrum maps onto a spectrum of information retention:
- φ-regime: system retains complete history → maximum information, maximum correlation length
- γ-regime: system retains partial history with harmonic decay → intermediate information, logarithmic correlation
- e-regime: system retains no history → minimum information, exponential decay
This maps onto thermodynamic regimes:
- φ: ordered phase (long-range order, low entropy)
- γ: critical phase (scale-free correlations, maximal susceptibility)
- e: disordered phase (short-range order, high entropy)
If this correspondence holds, the BC-attractor classification provides a thermodynamic classification of phases without computing partition functions — purely from the ontological structure of the system’s interactions.
Applied potential: thermodynamic phase classification from structural BC analysis, without simulation.
4. Roadmap toward application
Near term (0–2 years)
Close the formal gap: derive the connection between BC_closed(T⁻³) and memory degradation in purely BC terms.✅ Done — arxe_formal_gap_closure_en.md closes this fully: BC_closed(T⁻³)=2 with BC_open=1 forces route serialization, uniform d=2 routes give e, and harmonic-weighted d=1 routes give γ (unit amplitude cost per channel use, derived from BC_open=1 canonical normalization). Status upgraded from E4 partial to E4 complete. The next open item toward E5 is uniqueness of the underlying mixing-spiral object (§8 point 5 of the synthesis) and the δ_PMNS falsification test (§8 point 6).- Test the coherence-regime classification: apply BC-attractor classification as a pre-screening filter against known materials with established phase diagrams (cuprates for φ-regime, Mott insulators for e-regime).
- Monitor δ_PMNS measurements: track Hyper-Kamiokande and DUNE results. The prediction is falsifiable within this window.
Medium term (2–5 years)
- Extend to PMNS matrix elements: verify that |V_ij| elements of CKM and PMNS matrices have ArXe factorization consistent with the BC-attractor framework.
- Test perturbative convergence hypothesis: compare convergence rates of QCD perturbative series with e-scaling prediction for incoherent processes.
- Develop thermodynamic classification: formalize the φ/γ/e → ordered/critical/disordered mapping and test against known phase diagrams.
Long term (5+ years)
- Derive the full mixing structure: if δ_PMNS is confirmed to carry φ or no constant, extend the derivation to cover all 8 mixing parameters (including δ_PMNS) from BC axioms alone.
- BSM prediction: if new physics discovers a fourth fermionic generation or a new confined sector, verify the 13-mediation and e/φ-regime predictions.
5. Summary
| Result | Category | Applied domain | Timescale |
|---|---|---|---|
| 3 generations derived | Theoretical — E4 | Particle physics | — |
| 13 = long-range mediator | Theoretical — E4 | Particle physics | — |
| φ→γ→e spectrum | Theoretical — E4 | All domains | — |
| δ_PMNS prediction | Falsifiable — E4 | Particle physics experiment | 2–5 years |
| Coherence regime screening | Applied — E3+ | Materials science | 0–2 years |
| Perturbative convergence | Speculative | Quantum chemistry | 2–5 years |
| Thermodynamic classification | Speculative | Statistical physics | 2–5 years |
Independent research — July 2026
Diego Luis Tentor
“The spectrum φ→γ→e is not a description of mixing angles.
It is a description of how physical systems lose memory
as they move from the continuous to the discrete.
That is a statement about structure — not about particles.”