IDT vs. Fisher Information Geometry: Demarcation Theorem and Conditions for Coincidence

This work establishes the precise mathematical relationship between Information-Dynamic Theory (IDT) and standard Fisher information geometry. The central result is a demarcation theorem: IDT reduces exactly to Fisher information geometry if and only if M(ω) = 1, which holds if and only if H = G (Hessian equals Fisher metric), which holds if and only if the admissibility weight defines an exponential family (Theorem C46). Fisher information geometry is therefore the M = 1 sector of IDT — a special case, not a competitor. Main results: • The information potential Φ = −ln Z acts as an independent variational object generating the canonical P4-flow. It is not redundant with the Fisher metric G. • The natural-gradient structure in IDT emerges from the interaction between Φ and G, not from Fisher information alone. • When H ≠ G (M(ω) > 1), the system operates beyond Fisher geometry: physical time dt = M(ω)·dτ runs faster than information time, and the spectral depth M(ω) − 1 ≥ 0 is directly measurable. • The bilateral spectral constraint 1 ≤ M(ω) ≤ δ (Theorems C29 + G1) quantifies the degree of departure from Fisher geometry in terms of observable frequencies. Numerical verification: SHNO system of Devolder et al. (arXiv:2604.11438), M = 13.0, δ = 13.9, M − 1 = 12.0 — the system operates deep in the non-Fisher IDT regime. Status: Mathematical clarification · Theoretical consistency analysis · Level 5 canon-derived results. Related IDT records: 10.5281/zenodo.18296688 10.5281/zenodo.18654781 10.5281/zenodo.18995629 10.5281/zenodo.20501056