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

We study a finite representation problem in which a representation may be exactly sufficient for one operational architecture while failing to preserve distinctions required after a change in the operational equivalence regime. The aim is not to establish that history is intrinsically irreducible, nor to privilege an accessibility-based representation over sufficient-state alternatives, but to separate three notions that are easily conflated: generative truth, operational completion, and transportability. On the complete history space H = {0,1}^12, we define a fixed coarse present quotient Q, three independent historical residues, and three architecture-specific continuation functions. For each architecture we also define the induced operational equivalence relation on histories. In this precise sense, “dynamic quotient” refers to a change in the architecture-induced equivalence partition while the coarse present map Q itself remains fixed. An auxiliary representation z_A is globally uninformative about the privileged residue r_A in the marginal sense, I(z_A;r_A)=0, while becoming fully informative conditional on the quotient, I(z_A;r_A|Q)=1 bit. Consequently, (Q,z_A) completes architecture A exactly while leaving architectures B and C maximally unresolved. The exact finite-domain layer was reconstructed independently by both investigators and agreed on all directly comparable structural quantities. We define an exact family-wise re-identification budget as a minimum over deterministic fixed-length binary codes on the finite domain and prove its cell-cardinality formula. We also show constructively that greater retained conditional information need not increase transportability: additional retained distinctions can increase H(R|Q) without reducing uncertainty about future architecture-specific continuations. The cross-package comparison and its interpretation rules were prospectively frozen under an adversarial no-moving-target protocol. A later provenance and source-level audit revealed that part of the historical learner layer did not constitute a protocol-faithful matched replication, including a gradient-path defect in the historical GRU implementation. Those historical results are preserved as provenance evidence but are not promoted into the exact evidential core. The resulting conclusion is deliberately narrow: local operational sufficiency does not imply transportability, and retained conditional information does not by itself identify which distinctions remain relevant under a changed operational equivalence regime.