math.IT — Information Theory
Finite Alphabet Communication with Noisy Forward Memory in a Normalized Loop
We study one-use communication of a finite classical message through a stipulated normalized loop containing a backward binary symmetric channel and one noisy forward memory carrier. The carrier is either a classical bit or a qubit; no shared random seed, retained side register, or entanglement is supplied. For every finite message count, we characterize the qubit optimum by a constructive scalar feasibility test. For rational noise parameters and a rational target success, the test uses O(M²) constant-size rational arithmetic checks. The proof combines an exact decoder reduction, minimum-trace allocation for constant encoders, and a concave geometric cap with a matching polygon construction. For three messages, the qubit and classical-bit optima coincide throughout the allowed noise range; an explicit Bernstein certificate proves the remaining polynomial inequality. For four messages, an exact rational qubit protocol exceeds a bound valid for every classical-bit protocol. These are conditional mathematical results for the declared resource model, not a construction of a physical backward channel. Independent human review and originality assessment remain pending.