Abstract

We construct a positive, time-independent, number-conserving fermionic source–reservoir Hamiltonian and prove reference-preserving recovery from declared finite reservoir regions. The source contains $n$ fermionic modes. A logical code is loaded coherently by a message-independent fermionic swap, leaving the carrier in a fixed vacuum state. Each source mode is the boundary site of a uniform semi-infinite chain.

The one-particle spectral transform gives the exact source survival probability

For collection from reservoir sites $1,\dots ,M$, the complete inaccessible one-particle weight is $s_M(t)=q(t)+{\ell }_M(t)$. For every code supported on at most $K$ particles, we construct a complete decoder and prove

An explicit factorial collection bound produces a nonempty finite-time recovery regime for every prescribed accuracy. For a one-particle code, both optimal recovery error and complete residual leakage are exactly $(1-d^{-2})s_M(t)$.

We also prove reference-preserving finite-chain approximation, derive the energy-transfer ledger, and obtain an exact outgoing scattering channel with a constant complete complement from the same Hamiltonian. No terminal information-release map is added. At fixed finite chain length, recurrence prevents permanent late-time recovery.

These results are an autonomous microscopic comparison benchmark. The Hamiltonian is quadratic, not an interacting SYK Hamiltonian. No horizon, gravitational backreaction, physical infalling-diary dictionary, or coherent local radiation receiver is constructed. A separate audit explains why the supplied SYK and gravitational proposal cannot be declared complete on the available premises.