False witnesses for small Shor experiments: a coherent modular multiplication modulo 21

Small demonstrations of Shor's algorithm are easy to over-interpret: circuits are often simplified with the answer, and a broken circuit can pass a naive success test. A generic generator receives only N and the base a, never the factors or the order, and writes coset-encoded arithmetic onto a line of heavy-hex qubits. Verdict rules are fixed before submission, and the true circuit is compared with false witnesses (decohered bits, trivial or wrong constants) from the same generator on the same qubits. On ibm_kingston, a coherent controlled multiplication by 16 modulo 21 (585 CZ) beat every witness in two pre-registered runs, the second an exact replication (≥3.2σ and ≥4.9σ; combined, not pre-registered, S=+0.0202±0.0019): of the twelve units modulo 21, only 16 or its inverse produce this interference. The witnesses exposed a coherent idle-phase error invisible in calibration data, which dynamical decoupling refocused (+4.6σ). Pre-registrations are chained to RFC 3161 signatures, OpenTimestamps and a base drawn from a Bitcoin block hash, so that the blockchain itself orders the rules before the draw. In the blind run the true circuit's pattern was present (6.4σ from zero), but the pre-registered 3σ margin over every witness was not reached; post hoc, a contrast immune to single-bit readout bias leaves the true circuit 4.4σ from zero and every witness within 1σ of zero (margins 1.8–3.1σ), which fixes the design of the next blind test. On hardware, a sine window over the coset index beats a uniform one (+7.0σ). The full N=21 factoring circuit stays at the noise floor, as predicted; no factorization is claimed.