A Theory of Local Visibility for Random Statistical Representations: Bigraded Filtrations, Successive Obstructions, and Fisher–KL Information Transfer

This paper develops a unified local interface from finite-dimensional Itô stochastic dynamical systems to statistical representation spaces and, building on earlier two-scale results, formulates a more complete local visibility structure. Its basic architecture has four mutually linked layers: the first nonzero jet order r along a parameter direction, the corresponding activation exponent γ of the leading coefficient as the dynamical noise scale σ ↓ 0, a nested filtration of invisible directions, and successive obstruction maps generated by recursive lifting of partial parameter jets. This yields an auditable framework linking stochastic dynamics, statistical representation, and local information geometry, without treating any single Fisher correction or Kramers formula as a universal noise law. First, under a common dominating measure and quadratic-integrability regularity, we prove the conditional-score orthogonal projection decomposition between quenched and annealed Fisher information, I_F^q = I_F^a + E[Cov(s_q | Y)] ⪰ I_F^a, and derive the associated KL chain-rule information budget and data-processing contraction under parameter-independent Markov compression. Second, for a stable linear stochastic system with a stationary Gaussian snapshot experiment, we establish an exact Fisher formula, noise monotonicity, and a Fisher–KL–signal identity as a computable benchmark, without extending this benchmark to general stochastic systems. Third, we define the local random visibility signature χ_T(μ*, v) = (r, γ), prove its invariance under regular parameterization and regular noise-scale transformations, and establish the transfer law (r, γ) ↦ (kr, kγ) under critical statistical representation composition. Fourth, we define a nested hierarchy of directional invisible sets Z_k^σ and then consider higher-order jets of parameter curves; the problem of removing k-th order visibility can be recursively reduced to a lifting equation in a linear image space, yielding a successive obstruction tower. In particular, the second-order obstruction is represented by the quotient-space class [D²T_σ(μ*)[v, v]]; when it is nonzero, no local parameter curve with first-order tangent v can lift statistical invisibility to second order. More generally, the k-th order obstruction is defined on the space of liftable (k−1)st-order partial jets, rather than being incorrectly represented as a single invariant depending only on the initial direction. Fifth, under an additional two-parameter polyhomogeneous expansion, we introduce the visibility bigraded spectrum Spec_vis and its associated graded layers, and prove that the joint scaling t = σ^β u projects the bidegree (r, γ) to the single exponent γ + βr. Furthermore, within a finite-dimensional real-analytic statistical-representation subclass, we construct a Koszul filtered complex constrained by representation components and derive a genuine spectral sequence from the 𝔪-adic filtration; its first nontrivial differential reads the linearly visible layer, while later differentials organize higher-order jet constraints on local statistical fibers. Sixth, we establish a Fisher–KL two-scale bridge for general regular statistical readouts: if D_{σ,v}(t) ∼ Cσ^γ|t|^r, then 2KL ∼ J_Q C²σ^{2γ}|t|^{2r}, and repeated observations induce the information-scale resolution law |t|_res ≍ N^{−1/(2r)}σ^{−γ/r}. When r > 1, first-order Fisher information can be exactly degenerate while KL remains positively separated at higher order, so Fisher corresponds precisely to the first-order sector of the local visibility structure. Finally, through regular-parameterized stochastic potential families, critical statistical readouts, finite-observation-window Eyring–Kramers scales, and a zero-noise invariant-measure cluster theorem, we demonstrate the constructibility of the framework, its dynamical interface, and its scope limits. The central theoretical contribution is an independently auditable organizing principle: local structural response in random statistical representation is encoded by the chain two-scale signature — invisible filtration — successive obstruction — bigraded associated layer, with a strict filtered-complex spectral-sequence realization in a finite-dimensional analytic subclass, and Fisher and KL serving as statistical transfer mechanisms. The paper does not re-claim discovery priority for the missing-information principle, the KL chain rule, Eyring–Kramers theory, zero-noise invariant-measure theory, or existing theories of higher-order statistical singularity. The priority claims are restricted to the stochastic-dynamical statistical-representation interface developed here, the structural transfer of the two-scale signature, the construction of successive visibility obstructions, and the unified theorem chain relating these objects. The paper is purely theoretical and uses no empirical data, numerical simulation, or parameter fitting. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently determined by the author. Multiple AI tools assisted with code implementation, data presentation, and text drafting. The author bears full academic responsibility for all research content.