Non-Associative Information Geometry: The Fano-Fisher Metric Decomposition Theorem on $G_2$

Paper: 221 Portfolio C (AI) — AI & Deep Learning

DOI: 10.5281/zenodo.20076498

Abstract

Develops the Fisher information geometry of the exceptional Lie group $G_2 = \mathrm{Aut}(\mathbb{O})$ from first principles. The central result is the Fano-Fisher Decomposition Theorem: the Hessian $\Psi(\theta_\mathrm{ref}) = 2V^\top V$ of the associator energy functional has rank exactly 4, with all four non-zero eigenvalues universally pinned at $8/3$ (derived from the $G_2$ Casimir), and global average $(32/49) \cdot I_{14}$ (derived from the Fano incidence count). The active 4-dimensional friction subspace rotates between configurations (the crystalline turnstile) while the eigenvalue remains constant — pure rotational anisotropy with no scale variation. All four claims proved to machine precision ($\lvert\varepsilon\rvert < 2 \times 10^{-16}$) via exact analytical Jacobian $\Psi = 2V^\top V$.

Theorem (Fano-Fisher Decomposition)

The Hessian $\Psi(\theta_\mathrm{ref})$ satisfies:

  1. $\mathrm{rank}(\Psi) = 4$ universally
  2. All four nonzero eigenvalues equal $8/3$ exactly
  3. Global average: $\frac{1}{49}\sum_{\theta, e_A} \Psi = \frac{32}{49} I_{14}$
  4. The active 4D friction subspace rotates (crystalline turnstile)

Zenodo

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Code

Code supplement — empirical proof via exact analytical Jacobian; verifies all four claims to machine precision.