Local Information Geometry of Statistical Representations: Controlled Random Probes, Relative-Entropy Information Measures at Threshold Boundaries, and Nonregular Resolution Laws

This paper develops a local information-geometric theory for statistical representations and structural boundaries. The central objective is not to enumerate yet another collection of mutually isolated identifiability conditions, but to establish a mathematical chain that can be audited layer by layer: how structural differences generate statistical differences, how relative entropy near a boundary determines the natural local scale, and when parameter-independent representation compression changes that scale. We first write the composite representation from structure to invariant measure to observation to statistic as T = S ∘ K ∘ ν, and, under quantitative separation moduli and local Fréchet structure, derive layerwise criteria for blind spots. We then introduce a controlled random probe: an external, parameter-independent, and controllable random perturbation ρE is injected before a nonlinear statistical functional, with V_ρ(P) = E_{X∼P,E}[ψ(X + ρE)]. Under finite smoothness and moment conditions, we prove an explicit finite-order expansion of the probe difference. Its first nonzero derivative–moment coefficient exactly determines the order at which the probe becomes visible. This yields a deterministic criterion for “lifting” a blind spot to a higher order and provides a fully analytic exact example showing that a statistic that is strictly invisible at ρ = 0 can acquire a second-order visible signal under a controlled random probe. This result is distinct from ex post noise addition: the probe acts before the nonlinear statistic and therefore changes the observation experiment itself, rather than attempting to recover information already lost. For a two-regime threshold model, we further generalize local boundary information from a scalar density to a boundary relative-entropy information measure defined on the threshold axis. Under general heterogeneous design, we define the matrix-valued measure B_∂(A) = E[ZZ^⊤ 1{S ∈ A}], and prove that the candidate-threshold relative entropy in the Gaussian threshold experiment is exactly the boundary information measure over the misclassification interval. When G has a density g, the measure has the explicit density ι_∂(s) = g(s)/(2σ²) Δθ^⊤ M(s) Δθ. At the same time, under general heterogeneous design we obtain an exact matrix representation of the population least-squares loss, L(τ) − L(τ*) = Δθ^⊤ Ω_±(τ) Δθ, where Ω_± is the matrix parallel sum of the misclassification information matrix and the background information matrix. We prove 0 ⪯ Ω_± ⪯ B_± and, under local nondegeneracy, obtain first-order equivalence between population loss and KL divergence without requiring completely homogeneous design. To unify regular and nonregular local scales, we define the local relative-entropy order β: if KL(P_h ‖ P_0) = c|h|^β + o(|h|^β), then the relative-entropy budget of n independent observations reaches constant order at h_n ≍ n^{−1/β}. If the output representation is generated by a parameter-independent Markov channel, the data processing inequality implies that the output local relative-entropy order cannot be smaller than the input order. If a strict order jump β_U > β_X occurs, then on the natural local scale of the input experiment the output experiment has vanishing total variation, yielding a strict sense of “resolution degradation.” When the orders coincide, one can define the asymptotic relative-entropy fidelity coefficient η = c_U/c_X ∈ [0,1]. This chain places identifiability, information fidelity, and local resolution on a common local experimental scale. Finally, we define the statistical visibility order α and establish a finite-order Fréchet transfer theorem together with a local Hölder composition law, explicitly distinguishing the visibility order α from the relative-entropy order β. The Hopf normal form is used only as an analytic validation example, showing that different statistical functionals can generate different visibility orders; no claim is made of being the first to discover Hopf observable singularity. Fisher information, Gaussian common-design calibration, three-regime global masking, and sample consistency are included as calibration, counterexample, and interface results. The paper is purely theoretical: it uses no empirical data, numerical simulations, parameter fitting, or plots. The principal results are derived analytically from probability measures, matrix projections, and statistical-experiment theory. 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.