# conclusion --- # Conclusion All six official claims are supported by measurements on real, solved optimisation problems and on the paper's own construction, at $0 on local CPU. The load-bearing result is Claim 2. Theorem 5.1's upper bound cannot be confirmed by computation, but the paper states a matching Omega(p d log Delta_f) lower bound and gives the construction explicitly, and a lower bound is a finite object: a shattered set. Building it and running it shattered 64 instances at native scale with explicit hyperparameter witnesses -- exhaustively over all 2^N bit patterns at five smaller configurations -- and regressing the measured shattered-set size on p*d*floor(log2(Delta_f/2)) across 18 configurations gave slope 1.000000 with R^2 1.000000. The Theorem 5.1 upper bound then sits 2.70x-7.32x above that measured value at every configuration, so the theorem is not just valid on this family but tight to a small constant. The other five claims are supported by testing what each theorem actually *assumes* on real solver output, each with a control in which the assumption fails: the quantifier-elimination chain and a calibrated shattering instrument (Claim 1), the genuine f != g bi-level structure and the measured richness gap against a single-level control (Claim 3), piecewise rationality of the ElasticNet path against an equal-degree polynomial fit and a non-rational group-LASSO control (Claim 4), the semi-algebraic auxiliary-variable encoding against the failure of piecewise polynomiality and a non-semi-algebraic control (Claim 5), and the mp-QP dual with its piecewise-affine path against a rank-deficient control (Claim 6). In every case the paper's upper bound held against the measured pseudo-dimension of the corresponding real tuning class. No claim was falsified. The honest limits -- one-sidedness of lower-bound measurements, sampling above N=12, small alpha pools where each evaluation needs an iterative solve, and the float64 wall in the bit-extracting polynomial -- are set out on the failure-boundaries page.