Affinity Geometry, Transfer Laws, and π Invariants in Normalized Binary and Symmetric Multidimensional Exponential Families

“This work develops and formally audits a normalized exponential-affinity framework for binary and symmetric multidimensional families. Exact and conditional proofs are provided for the affinity susceptibility/Fisher identity, an independent Fisher–Rao π-length representation, balance-count π invariants, a Richardson-accelerated π estimator, one-X-to-another-X transfer laws, cross-invariant transfer, multidimensional Fisher information, dimension-independent Fisher–Rao length π for the stated symmetric d-state family, and cross-dimension transfer. Standard prior art in Fisher information, Jeffreys priors, Wallis/central-binomial limits, Gudermannian identities, and Richardson extrapolation is explicitly separated from project-specific synthesis. No claim is made that the mathematical framework by itself establishes a universal physical law or resolves the microscopic physical origin of QDL λ. Main manuscript: PDF. Supplementary/reproducibility materials: ZIP bundle containing source files, verification code, theorem audit, metadata, and supporting materials.”