Local Completion Is Not Transportability: Dynamic Quotients, Distinction Budgets, and an Adversarial Replication Protocol

We investigate a finite representation problem in which a representation can be exactly sufficient for one operational architecture while failing to preserve distinctions required under another. The study separates generative truth, operational completion, and transportability without assuming that history is fundamentally irreducible. Using the complete binary history space H = {0,1}^12, containing 4096 histories, we define a fixed present quotient, three independent historical residues, and architecture-specific continuation functions. We prove that a one-bit auxiliary representation can have zero marginal mutual information with a designated historical residue while providing one bit of conditional mutual information relative to the present quotient. This representation completes architecture A exactly but leaves architectures B and C unresolved. We establish an exact cell-cardinality theorem for the minimum deterministic fixed-length re-identification budget and show that two additional bits are necessary and sufficient to recover the joint held-out responses in the declared construction. A further counterexample demonstrates that increasing retained conditional information from one to three bits need not improve transportability. The finite mathematical results were independently reconstructed by both investigators. An adversarial comparison protocol preserved negative findings and exposed provenance differences between historical learner implementations, including a gradient-path defect in a historical GRU experiment. These learner results are retained as provenance evidence rather than promoted into the independently replicated mathematical core. The conclusions are deliberately restricted: local operational sufficiency does not imply transportability, and additional retained information does not necessarily preserve distinctions relevant to changed operational requirements. The work does not establish a universal limitation of neural networks, predictive-state representations, or sufficient-state approaches.