The Counting Theorem: Eight Pre-Registered Roads to (411, 400), All Computed, and the Price of the Missing Integer

The integers 411 and 400 (unpaired residue 11; density ratio 411/400 = 1.027500) are the author's declaration, premise P-13, adopted and never derived. This paper attacks them with eight counting rules pre-registered before computation: Diophantine control, flux quantization, local mode counting, the census machinery, Axiom H counting, scar/grain combinatorics, closure self-consistency, and energy balance. None produces the pair with zero free parameters. The census family is eliminated permanently by its own theorem (its conversion map conserves the count identically); the pairing involution constrains the pair's parity and names 11 as the fixed-point count of the sever law's pairing operator. The structural finding: every count the frame owns is local and O(1), or global and 10^36 to 10^122; 411 is mesoscopic and the frame has no mesoscopic generator. The gap is priced four ways (a sever-law fixed-point count of 11; S_sev = 284.8835 nats plus the sever degeneracy; a tube core radius of 28.67 crossing lengths; or Omega_Lambda measured to 8.4 ppm, a factor ~1,222 beyond Planck). Verdict: PRICED FAILURE. The vacuum catastrophe is NOT SOLVED.