The Self-Dual G₂ Architecture: Pachner Unitarity, CSS Symmetry, and the Crystal Spectrum Halving
| Paper: 271 | Portfolio B — Mathematical Physics |
Abstract
Proves four exact theorems about the $G_2$ self-dual structure of the 731 Frog Calculus via direct computation on the octonion multiplication table (all 7 Fano lines, max absolute error = 0): the Malcev Resolution, Fano-Line Closure, No-Parallelisation ($|[L_{e_\alpha}, L_{e_\beta}]|_F = 4\sqrt{2}$ for all 21 distinct pairs), and Moufang Echo. These theorems are the algebraic foundation for the five rewrite rules of the Origami Compiler.
Key Results
- Theorem 7.2 (Malcev Resolution): $[L_{e_i}, L_{e_j}] = L_{e_k} \circ D_{ijk}$ — correction is a full $8\times8$ diagonal map, not a phase
- Theorem 7.4 (Fano-Line Closure): ordered triple reduction to explicit projector $I - \pi_i - \pi_k - 2\pi_j$
- Theorem 7.6 (No-Parallelisation): $|[L_{e_\alpha}, L_{e_\beta}]|_F = 4\sqrt{2}$ uniform across all 21 Fano pairs — mandatory Flop barrier
- Theorem 7.8 (Moufang Echo): sandwich compression $L_{e_\alpha} L_{e_\beta} L_{e_\alpha} \to L_{e_\beta}$ (restricted) + phase
- All proofs numerically verified to machine precision
Zenodo
Related Papers
- Paper 207 — Frog Calculus Part 1 (algebraic framework these theorems ground)
- Paper 258 — Origami ISA (compiler cheat sheet using these theorems as rewrite rules)
- Paper 281 — Frog Calculus Part 2 (diagrammatic visualisation of these identities)
- Paper 221 — Fano-Fisher (establishes $8/3$ eigenvalue structure)