Once rings are coupled, the winding number is no longer the state

Three 16-oscillator rings: A and B hold the data bits, O is the output. Each site of O is coupled to the same site of A (sites 0–7) or B (sites 8–15) with strength 0.3. With the coupling set to zero, the coupled integrator reproduces the single-ring table builder exactly, entry for entry, in both builds.

The copy gates exist

Starting O at q = −1, an erase kick of amplitude 2.2 on O copies a register into O, and which register depends on where the kick lands: sites 2 and 3 copy A, sites 10 and 11 copy B. Measured from the reference attractors, these entries pass 31–32 of 32 noisy trials. That is the late-selector multiplexer as an addressed read: load b0 into A, b1 into B, and let b2 choose the kick site.

The program built from them fails

Run kick by kick, that program is 75% correct with no noise. Copying B works; copying A fails whenever A was loaded by a real write instead of being set up from an ideal twist. Settling 256 random relative-phase starts for each winding triple shows why:

Where the phase mismatch sits between coupled rings is state the winding numbers do not record. Any transition table indexed by winding numbers is incomplete for coupled rings, however carefully each entry is measured.

Reading a phasor instead

Reading O's complex order parameter — one complex number, one straight-line decision boundary fitted on the noiseless points — on the same physical runs:

Under noise both readouts score the same — about 82–84% at σ = 0.15 for the best program — because they fail on the same runs. Inside one attractor the phasor is tight (231 of 256 noisy runs in one 11° cluster); the errors are jumps to another basin. The readout is not what limits noise tolerance: the operations are. Writes and erases work by forcing a phase slip, and a slip is a jump between basins.

Measured on an Intel Core 3 100U laptop CPU (8 threads, no GPU), gcc 15.3, -O3. Instrument: fork_macro.c; raw outputs in research/forks/a2.