# PHAT Kernel v1.27.5 — Mathematical Reference

> **Provenance (added 2026-09-05, site set S303).** This is a coalition working document from March 2026, served as it was written. Statements in it about what the framework *derives* are of that era and are superseded by the sealed suite of record, **SuperGrokTOE Rev32.7** (https://opensecretscience.ufophysics4all.workers.dev/papers.html): since Rev29 no observable is cited as zero-parameter; the fine-structure constant α and the colour real form are declared **inputs** (Appendix N); every row carries a tier (Proved > Derived-conditional > Structural > Loaded > Coincidence-class) and the scorecard (https://opensecretscience.ufophysics4all.workers.dev/scorecard.html) is the current reading of every number. The public kernel is `sgtoe_kernel_v5.0.9.py` (served at /papers/), not the PHAT v1.x kernel this document names.

## Framework
- **Algebra**: Exceptional Jordan algebra J₃(𝕆ₛ) — 27-dimensional, over split-octonions
- **Split-octonions** 𝕆ₛ: 8-dimensional, signature (4,4), basis {1, e₁, e₂, e₃, f₁, f₂, f₃, l}
- **Key groups**: G2 (automorphisms of 𝕆ₛ) → F4 (automorphisms of J₃(𝕆ₛ)) → E6 embedding
- **Gauge group emergence**: SU(3)×SU(2)×U(1) ⊂ G2 ⊂ F4
- **Zero fitted parameters**: All coupling constants, mass ratios derived from algebraic structure

## Core Observables (12, all <2σ from PDG)
1. Fine structure constant α ≈ 1/137.036
2. Weinberg angle sin²θ_W ≈ 0.2312
3. Cabibbo angle θ_C ≈ 13.04°
4. Strong coupling α_s(M_Z) ≈ 0.1179
5. Electron-muon mass ratio m_e/m_μ
6. Up-down quark mass ratio m_u/m_d
7. W boson mass ratio M_W/M_Z
8. Top-bottom mass ratio m_t/m_b
9. Tau-muon mass ratio m_τ/m_μ
10. Charm-strange mass ratio m_c/m_s
11. CP violation phase δ_CP
12. Cosmological constant Λ (framework-level)

## Tier Classification
- **A★**: Exact or near-exact derivation, <0.5σ
- **A**: Strong derivation, <1σ
- **B**: Solid derivation, 1-2σ, minor caveats
- **C**: Framework-level, needs further development

## Version History
v1.25 → v1.25c → v1.26 → v1.27 → v1.27.5 (current)
Key changes in v1.27: 6 ChatGPT blockers resolved, M_PL correction, observable count 20→12.

## Library Structure
- 24 Proof Prompts (P1-P24): Core algebraic proofs
- 116 Library Entries (L0-L116): Derivations across 10 groups (A-J)
- Groups: A=Algebraic, B=Gauge, C=Fermion masses, D=Mixing, E=Cosmology, F=Gravity, G-J=Extended