KL-Geometric Structure of Observer Entropy: A Minimal Information-Theoretic Framework

This paper establishes a finite information-geometric formulation of observer entropy defined via Kullback–Leibler divergence under coarse-graining. The main result (Bridge Theorem) proves the local expansion S_obs(p_θ, ε) = ½ ε² vᵀ I(θ*) v + O(ε³), showing that observer entropy is governed at leading order by the Fisher information matrix. All results are derived in a fully finite setting with explicit regularity assumptions and complete cubic remainder estimates. v2.0 introduces the Riemannian-geometric formulation, additional numerical verification, and sufficient conditions under which the parametric deformation assumption becomes a theorem. v3.0 extends the numerically verified scope of the Bridge Theorem to off-center base points: twelve randomly sampled parameters θ* ≠ 0 (six per worked example) with random within-class perturbation directions confirm quadratic scaling and O(ε³) remainder control uniformly away from the symmetric reference. A quantitative validation criterion (rel.err(ε) ≤ 0.8ε for ε ≤ 0.03) is introduced and empirically calibrated across all nine worked-example configurations, separating balanced (O(ε²) relative error) from unbalanced (O(ε) relative error) regimes. The verification script is extended with Fisher-matrix agreement checks, closed-form consistency checks, and Landauer energetic cost tables at T = 300 K. v3.24 is a substantive release introducing a partition-free, coordinate-free reformulation of observer entropy in the language of fibre bundles (new Section 11). The equivalence relation ∼ε is encoded as a surjection π:Ω↠X with fibres Ωx=π−1(x), and the uniform lift Πεp becomes the pull–push composition π∗π∗. The main new result is the exact closed-form identity Sobsfib(μ)=logN−Hw(μ) (Theorem 11.8), valid for every strictly positive measure μ on the fibred ground space, with N the common fibre cardinality and Hw the within-fibre conditional entropy. This identity is non-perturbative and complements — rather than replaces — the local Fisher–Rao expansion of the Bridge Theorem, providing a kinematic global expression alongside the geometric local one. An unbalanced-fibres extension (§11.6) covers fibres of unequal cardinality, including the structurally distinct configuration of Example 2. The verification script is extended to v3.24 with six new tests (T11.1–T11.6) verifying push-pull identities, idempotence of π∗π∗, the closed-form identity for both balanced and unbalanced fibres, the range 0≤Sobsfib≤logN, and equivalence with the partition-threshold definition. Three new supplementary diagnostic figures are added (Fisher eigenvalue structure, Landauer energy budget at T=300 K, and closed-form fibre identity). No change to the mathematical content of the Bridge Theorem, the Sufficient Conditions Theorem, the Riemannian-geometric formulation, the Cognitive Uncertainty Principle, the Landauer bound, the empirical C∗ analysis, the off-center robustness checks, or Figures 1–7.