pages/conclusion/page.md

# conclusion


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# 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.