Complexity-constrained quasi-invariants of perturbed twist maps: an integer scaling law and the optimality of normal forms

For a perturbed twist map we define the invariance defect of a complexity class C_M as the minimum, over all observables with angular Fourier content |m| <= M, of the mean-square one-step variation normalised by the variance. Computing it is a symmetric eigenvalue problem, which we solve to arbitrary precision. The defect obeys an integer law D_M ~ eps^beta with beta = 2k1 or 2(k1+1), where k1 = min{k : Jk > M} is the order at which the coboundary first generates angular modes the class cannot absorb and J is the highest harmonic of the forcing; the two branches are separated by the mode deficit G = Jk1 - M. Nineteen (J,M) cells agree, including four out of sample. For the single-harmonic standard map this gives beta_M = 2(M+2), one order better than classical normal-form truncation; the extra order comes from the free angle-averaged functions that truncation fixes by convention. We prove the M=1 case, obtaining D_1 = Keps^6 + O(eps^7) with K in closed form, agreeing with the measured constant to 0.06 percent, and we prove the k1=1 clause of the rule for all J and M. We also show that sqrt(D_M) is a geometric error rather than a relaxation time: on invariant curves the accumulated drift saturates at 1.109sqrt(D_M) instead of growing. A methodological trap is isolated: a fixed radial basis degree contributes a term of lower order in eps, so its corruption of the exponent grows as eps goes to zero, and it produced three spurious findings before being identified. All code, raw data and validation logs are included; every number is reproducible by one command.