IMM Paper 23: Conditional Transfer Bounds, Collider Identities, and Spectral Dynamics in Finite-Record Observer Chains

We study information transfer in finite-record observer chains: pairs of finite-state observers that evolve by private probes and coin-gated deterministic meetings, with one observer's symbol stream as the only read-out channel. Three results are established with explicit epistemic status. Theorem 1 (proven): the cumulative recovery _n = I(V; Y_ 1:n ) of one observer's initial state from the other's observation record obeys the conditional transfer bound _n I_0 + _ k n ^*_k , where ^*_k is the history-conditional per-meeting transfer; the proof is an elementary data-processing chain, and no novelty is claimed for the technique. Corollary 1 (proven, conditional): the computable capacity form _n I_0 + n( ( ) + c 2) under a c -bit meeting-write hypothesis that holds by construction ( c=1 ) on the census family. Proposition 1 (proven, exact algebra): the collider identity _ = _ + I( _ ; _ _ ) , separating conditional from measured transfer by a non-negative redundancy surplus. An exact, certified census over six meeting mechanisms at full coupling then shows the conditional transfer is nearly universal ( _ 2 on every mechanism) while the surplus spans four orders of magnitude ( 4 10^ -4 to 0.627 ): transfer conditional on the receiver is uniform; what varies is redundancy. The empirically supported tighter bound _n I_0 + n\, _ (Conjecture 1; certified positive margin on all 24 tested cells) is shown not to follow from Theorem 1, and to be equivalent to a non-recyclability statement: the collider surplus never converts into recovery. Finally, a spectral analysis refutes exactly the hypothesis that an observed 12 convergence rate of the recovery gap is a state-chain eigenvalue --- the characteristic polynomial of the =1 kernel has p( 12) = -183/2^ 36 0 , with = 1 its only rational root --- relocating the rate to the observation filter. The filter's transfer structure is then constructed exactly (Appendix C): it reproduces the observed ratio sequence six-for-six at four decimals, but the ray automaton is not finite, no finite truncation block carries eigenvalue 12 (exact determinants on 33 , 692 blocks of a 200 , 000-ray truncation), and an elementary reconciliation bound leaves the 12 -limit open against a measured live-mass decay rate near 0.47 . A replicated descriptive computation of the theorem budget against actual recovery is reported in Appendix B. A closing discussion (Section sec:composite ) proposes a structural correspondence between the two-observer channel and composite biological observers --- hemispheric integration across the corpus callosum --- explicitly graded as analogy: it motivates hypotheses and makes, as yet, no distinguishing empirical prediction. All quantities are computed exactly (rational arithmetic; certified enclosures for census verdicts); no sampling occurs anywhere in the paper.