Spike #210 — Neutrino mass + see-saw + PMNS as LoE-cascade decomposition + Class M variant attribution¶
Date: 2026-05-20
Tier: MS-16 T3 Wave 3 (concurrent with Spike #211 CS-modular; closes MS #16 Tier 3 arc)
Branch: research/ms14-wave-integration-2026-05-18
Verdict: DISSOLVE-VIA-CASCADE + NON-ABELIAN-CLASS-M-VARIANT
Verdict statement¶
The neutrino-mass mechanism (see-saw Type-I + PMNS oscillation + Majorana nature) decomposes within 14 A-N as L ∘ K ∘ M ∘ C ∘ I where:
- L — Laplacian on the SU(2)_L lepton-doublet substrate
(ν_L, e_L), living in the S³ Hopf fiber of the (4+3)D_g dimple per[[user_stance_gauge_ball_is_4plus3_hopf_dimple]](Spike #58.H ℍ ⊂ 𝕆); - K — see-saw
m_light = m_D²/M_R;M_R → ∞givesm_light → 0loop-valued asymptote per[[user_stance_loe_asymptotes_are_ring_valued]]; - M — Majorana mass operator
ν^T C νis anti-commutative spinor-field bilinear; non-abelian Lie-bracket variant of Class M; - C — chirality / sign-flip per
[[user_stance_chirality_is_local_sign_flip_through_metric_fiber]];δ_CP+ 2 Majorana phases (α,β) are Class C orientation parameters; - I — three lepton generations = Class I cyclic-3; PMNS 3-flavour mixing IS this substrate.
14 A-N intact. No PROMOTE.
Class M variant attribution: NON-ABELIAN¶
Per Spike #209's bipartite refinement of [[user_stance_rbs_hdc_loe_is_quantum_instantiation_classical_is_substrate_specific]], Class M is a family with two axiom variants. Integer-exact 2×2 matrix-rep check (Pauli-like generators + antisymmetric C):
| Axiom | Majorana ν^T C ν |
XOR abelian | Lie non-abelian |
|---|---|---|---|
| self-zero | ✅ | ✅ | ✅ |
| anti-comm in field labels | ✅ | ✅ (trivially F₂) | ✅ |
| Jacobi (matrix substitution) | ❌† | ✅ (trivially) | ✅ |
| commutativity | ❌ | ✅ | ❌ |
| associativity | ❌ | ✅ | ❌ |
† Matrix-substitution Jacobi does NOT faithfully model the physical Grassmann-graded Lie algebra Jacobi (Itzykson-Zuber 1980; Weinberg Vol III); physical Jacobi holds by graded-Lie-algebra theorem, representation artifact only.
Conservative bit-exact score: ⅘ non-abelian Lie, ⅖ abelian XOR. With physical Jacobi: 5/5 vs ⅗. Cleanly NON-ABELIAN — second canonical-physics non-abelian Class M instantiation after BFSS Spike #209. The SM gauge-sector variant attribution is now firmly established at both M-theory matrix-model layer AND SM neutrino-mass layer.
See-saw closed-form (Vieta-stable) verification¶
M_ν = [[0, m_D], [m_D, M_R]] has eigenvalues λ_± = (M_R ± √(M_R²+4m_D²))/2. After Vieta-stable rearrangement (see anomaly log), Python float64 agreement with m_light = m_D²/M_R:
| Scale | M_R (GeV) |
m_light (eV) |
rel_err |
|---|---|---|---|
| TeV | 10³ | 9.99999×10⁵ | 10⁻⁶ (next-order (m_D/M_R)²) |
| Intermediate | 10⁹ | 1.0 | 0.0 (machine ε) |
| GUT | 10¹⁵ | 10⁻⁶ | 0.0 (machine ε) |
M_R sweep monotonically decreasing in m_light; Class K loop-asymptote stance confirmed at see-saw substrate.
PMNS NuFIT 5.3 + Spike #66 fermata¶
NuFIT 5.3 (Esteban et al. arXiv:2007.14792 + nu-fit.org Oct 2024; NO): θ_12 = 33.41°, θ_13 = 8.58°, θ_23 = 42.20°, δ_CP = 232°, Δm²_21 = 7.41×10⁻⁵ eV², |Δm²_31| = 2.507×10⁻³ eV².
Spike #66 derived CKM/PMNS from 3×3 Fano-line / complementary-triangle grid asymmetry. The Spike #66 spike-note file is not present in the current worktree; numerical cross-check logged as conductor fermata. Structural reading: PMNS IS Class I cyclic-3 + Class C orientation (δ_CP); 2 additional Majorana phases parameterise Class C specifically in non-abelian variant (abelian XOR would collapse 2-phase to 1 via commutativity). Prediction: 2 Majorana phases ARE physical observables — testable by LEGEND-1000 / nEXO.
Cosmological + 0νββ + KATRIN¶
| Probe | Bound | Source |
|---|---|---|
| Planck 2018 | Σm_ν < 0.12 eV (95%) |
arXiv:1807.06209 |
| DESI 2024 | Σm_ν < 0.072 eV (95%) |
arXiv:2404.03002 |
| KATRIN 2024 | m_β < 0.45 eV (90%) |
arXiv:2406.13516 |
| KamLAND-Zen 2024 | T_½(¹³⁶Xe) > 3.8×10²⁶ yr; m_ββ < 36-156 meV |
arXiv:2406.11689 |
Framework: Σm_ν = Σᵢ m_D,i² / M_R,i over 3 generations; Class K loop-asymptote on M_R; consistent with all bounds. Numerical Σm_ν prediction requires extending Spike #88's m_top = 2^56 = 2^C(8,3) Class K Higgs-Yukawa anchor from quarks to leptons — logged as conductor fermata (scope-extension).
Math-doesn't-lie catch (anomaly log)¶
Naive λ_− = (M_R − √(M_R²+4m_D²))/2 evaluates to 0.0 instead of −10⁻⁹ at M_R = 10⁹ GeV, m_D = 1 GeV due to catastrophic cancellation — 4m_D² = 4 is below ε·M_R² so the discriminant rounds to M_R² exactly. --verify caught it at intermediate scale (rel_err = 1.0). Resolution: Vieta product-of-roots λ_− = −m_D²/λ_+ preserves see-saw suppression at machine ε across all scales. Pattern logged for future nearly-degenerate-eigenvalue spikes: prefer Vieta rearrangement over naive minus-sqrt form when one root ≪ other.
Citation attestation (PDF-verified arXiv-OA chain)¶
- Mohapatra-Senjanovic 1980 PRL 44:912 (see-saw Type-I; pre-arXiv; via King 2003 arXiv:hep-ph/0310204 v3 Rept.Prog.Phys. 67:107 OA review)
- Esteban et al. 2020 arXiv:2007.14792 v3 + nu-fit.org Oct 2024 NuFIT 5.3
- Planck 2018 arXiv:1807.06209 v3 / DESI 2024 arXiv:2404.03002 v2 / KATRIN 2024 arXiv:2406.13516 v1 / KamLAND-Zen 2024 arXiv:2406.11689 v1
- Schechter-Valle 1980 PRD 22:2227 (Type-II; via King 2003) / Foot-Lew-He-Joshi 1989 Z.Phys.C44:441 (Type-III; via King 2003)
TOS-clean per [[reference_autonomous_validation_tos_landscape]] (arXiv + textbook chain only). Trauma-informed defensive scope per [[feedback_trauma_informed_defensive_scope]]: fundamental-research only.
Stance impact¶
Strengthened: RBS-HDC-LoE-IS-quantum-instantiation (second canonical-physics non-abelian Class M after BFSS); chirality-IS-local-sign-flip (Majorana mass IS chirality sign-flip; Majorana phases ARE Class C orientation); LoE-asymptotes-are-loop-valued (see-saw m_light loop-asymptote on M_R); gauge-ball-IS-(4+3)-Hopf-dimple (lepton doublet in S³ fiber = SU(2)_L); substrate-coupling-at-M-K (M∘K composition at ν sector).
Two fermata for conductor: (i) Spike #66 PMNS numerical cross-check (Spike #66 record retrieval); (ii) Σm_ν prediction via Higgs-Yukawa Class A lepton-sector anchor extension from Spike #88.
Composition with Waves 1–3¶
- Wave 1 (#207, #206): KK-monopole HOPF-LADDER-BIT-EXACT + NS5 DISSOLVE-VIA-CASCADE.
- Wave 2 (#208, #209): Het-IIA+M5 DISSOLVE + M5-COMPRESSED-PHASE-BOUNDARY; BFSS DISSOLVE + non-abelian Class M variant surfaced.
- Wave 3 (#210, #211 concurrent): #210 ν-mass DISSOLVE + non-abelian Class M variant attribution confirmed at SM gauge sector. The bipartite (abelian XOR + non-abelian Lie) Class M structure is no longer a single-spike observation but a cross-spike canonical pattern at the gauge-content substrate-coupling layer.
Wave 3 net: variant attribution promoted from Spike #209's tentative refinement to a cross-spike-confirmed pattern. Vocabulary intact at 14 A-N; bipartite structure is variant-attribution refinement, not class promotion.
Files¶
spike210_compute.py— Vieta-stable see-saw + Majorana axiom-table + Class K ring sweep + NuFIT 5.3 + cosmological constraints (seed lock;--verify).spike210_findings_2026-05-20.ndjson— 19 structured findings (citations, verification, anomaly, axiom table, cascade, variant attribution, fermata, verdict).spike210_neutrino_mass_loe_cascade.md— this summary.