Local Information-Geometric Structure of Multi-Observer Aggregation: A KL–Fisher Bridge to Jensen–Shannon Geometry

This paper establishes a local structural bridge between two recently developed information-geometric frameworks for observer-dependent probability representations on finite state spaces. Background. On the parametric side, the partition-adapted framework of Khomyakov (KLEO v3.24.1) defines observer entropy as the Kullback–Leibler divergence between a parametric distribution and the uniform lift of its coarse-graining, and shows that it scales quadratically with the resolution threshold, with a coefficient determined by the Fisher information matrix. On the empirical side, the Interference-First Reality (IFR) framework of Bianchi (2026) identifies multi-observer objectivity with the Fréchet barycenter of observer-induced distributions in the Jensen–Shannon metric, with almost-sure convergence guaranteed by metric barycenter theory. Principal results. The paper derives four main results: 1. Local Metric Bridge Lemma (Section 3): On the open probability simplex, the Jensen–Shannon divergence satisfies the local quadratic expansion D_JS(p ‖ p + εδ) = (ε²/8) I_p(δ) + O(ε³), where the explicit coefficient 1/8 arises from the midpoint structure of the symmetric divergence. 2. Multi-Observer Quadratic Aggregation Theorem (Section 4): For a partition-adapted parametric family satisfying conditions (A1)–(A3) of Khomyakov (KLEO v3.24.1), the empirical Jensen–Shannon Fréchet barycenter of an N-observer ensemble, centered on the within-class neutral locus, exists uniquely for small ε and coincides to leading order with the weighted Fisher–Rao barycenter of the within-class perturbation directions. The minimal Jensen–Shannon dispersion satisfies a quadratic scaling law governed by the within-class block I_ww(θ₀*) of the Fisher information matrix. 3. Leading-order coincidence of estimators (Corollary 6.1): The variational Fréchet minimizer p_F and the closed-form geometric overlap estimator p_G of Bianchi (2026) agree to order O(ε²) in Jensen–Shannon distance within the local regime. 4. Multi-observer Landauer corollary (Corollary 6.6): Under a per-channel irreversibility assumption and an additivity postulate for independent observer channels, the aggregate energetic cost of barycenter formation is bounded below by k_B T times the Jensen–Shannon dispersion. Scope. The bridge is local in ε (the within-class perturbation amplitude) and finite in N (the number of observers). It does not subsume Bianchi's almost-sure N → ∞ consistency theorem nor Khomyakov's ε → 0⁺ quadratic scaling result; it establishes their structural compatibility in the overlapping regime where both apply. The results are specialized to balanced exponential families (softmax), with explicit worked-example coefficients for four- and five-point state spaces. Domain-specific applicability is assessed against the cross-domain distinguishability hierarchy of Bianchi (2026), Table 14, identifying the low-dispersion regime (d_JS ≲ 10⁻² nats) as the primary domain of validity. Keywords: Jensen–Shannon divergence, Kullback–Leibler divergence, Fisher information, Fisher–Rao metric, multi-observer aggregation, Fréchet barycenter, probability simplex, partition-adapted parameterization, observer entropy, information geometry. MSC 2020: 62B10, 94A17, 53B12, 62F12.