Rank, Fibers, and Falsifiability: A Rank-Aware Possibilistic Cramér–Rao Bound for Evidence-Driven Contraction

The Possibilistic Cramér–Rao Bound (PCRB) of the Theory of Epistemic Abductive Geometry (TEAG) floors the rate at which evidence may delete admissible microstates: \(H_π^{+} \geq H_π^{-} + \tfrac{n}{2}\log(1-I_k)\), with \(n\) the state dimension and \(I_k\) the Choquet information content of the observation. This paper corrects and strengthens the bound by exploiting a structural fact the earlier formulation ignored: possibilistic entropy lives on state space, but evidence deforms the impossibility field through the state-to-measurement map, which is not injective — many states produce the same evidence. For a measurement of rank m ≤ n, the surprisal field is the pullback of an \(m\)-dimensional field and is constant on the (n-m)-dimensional fibers of the measurement map. Three results follow. (1) Fiber conservation: a single observation leaves the fiber-direction geometry of every admissible \(\alpha\)-cut exactly invariant; the entire entropy change is the admissibility-integral of the log base-mass retained, an identity we prove by exact Fubini factorization. Ignorance is conserved along every direction the evidence cannot compare — the Principle of Comparative Information as a conservation law. (2) The rank-aware PCRB: under an explicit base-marginal coupling condition that formalizes the innovation–state isotropy assumption in its correct (\(m\)-dimensional) home, \(H_π^{+} \geq H_π^{-} + \tfrac{m}{2}\log(1-I_k)\) — a strictly stronger floor than the state-dimension version whenever \(m < n\), recovering it at full rank. (3) Falsifiability–observability: over \(\ell\) observations with dynamics, a direction of state space is falsifiable if and only if it lies in the accumulated pulled-back row space; the state is totally falsifiable if and only if the system is observable; and evidence-versus-evidence falsification — the detection of inconsistency, bias, and model stress through joint total falsification — is possible if and only if the stacked system is overdetermined, \(\ell m > n\) after accounting for rank. Consequences for the PCRB-admissible basin, for the ESPF reference implementation, and a falsifiable prediction about previously reported over-pruning in rank-deficient tracking are derived. Tightness of the corrected floor remains open and is inherited, not resolved, by the rank refinement.