Complete Identifiability and Cross-Model Rigidity of Relative-Norm Observations: Conformal Quotients, Gauge Selection, Markov Flexibility, and the Fisher–Finsler Boundary

This paper treats the relative-norm observation S_{F,ξ}(v) = F(v)/F(ξ) as a genuine observation map, rather than interpreting it in advance as a normalization formula for some known metric. The central questions are: what exactly does this observation identify, what are all of its unidentifiable degrees of freedom, and under what conditions can statistical naturality eliminate some or all of these freedoms? We establish a rigorous chain from observational identifiability to statistical rigidity. First, for general smooth absolutely positively one-homogeneous fiber norms, we prove a complete observation-quotient theorem: two norms produce the same relative observation if and only if they differ by a unique positive smooth pointwise factor; hence the observation space is in bijection, in the set-theoretic sense, with the quotient by positive conformal rescalings. This conclusion uses neither the parallelogram law nor any Fisher structure. Second, under the parallelogram law, the Jordan–von Neumann polarization theorem shows that every admissible observation uniquely determines a smooth Riemannian metric normalized along the reference direction, and we give an exact description of the scale coordinate and of all set-theoretic sections. Third, on the finite probability simplex, we prove a symmetry no-go theorem for intrinsic selection of the reference direction: there is no everywhere-nonzero tangent reference field that is natural under all permutations, showing that the reference-direction gauge cannot be selected intrinsically from global symmetry alone. Fourth, on the fixed binary simplex, we completely classify all positive one-dimensional Riemannian metrics that are monotone under all interior-preserving binary Markov affine maps, obtaining w(1−p) = w(p) and p²w(p) nondecreasing, and equivalently derive a differential inequality for the Fisher conformal factor. This yields an infinite-dimensional flexibility cone and an explicit non-Fisher continuous family, rather than reducing the non-uniqueness of the binary single-model case to a single counterexample. Fifth, within the Fisher conformal orbit, we establish a general propagation mechanism for the conformal factor along Fisher-isometric embeddings admitting Markov left inverses, and prove that a strictly reduced Markov family generated by one fixed binary anchor already forces all pointwise conformal factors to collapse to the same global constant. Sixth, we extend the Markov contraction construction on the Finsler side from the single L⁴ example to the even-integer L^{2k} family and sharpen the dimensional threshold for non-Hilbert counterexamples to n = 3; when n = 2, no non-Hilbert norm exists because the tangent space is one-dimensional. Finally, we place the existing Fisher-uniqueness results of Čencov, Nagaoka, Ay–Jost–Lê–Schwachhöfer, and Lê at explicitly declared external interfaces, rather than presenting external uniqueness statements as internal proofs of this paper. The resulting structure is the unified chain “observation quotient—gauge selection—single-model flexibility—cross-model rigidity—Fisher representative—Finsler boundary.” 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.