cert
plain-language theorem explainer
Packages three elementary facts about the domain cost into a single overparameterization certificate: the cost vanishes on the diagonal, is non-negative for positive arguments, and the canonical threshold is positive. Materials and learning-theory readers cite it as the well-posedness witness for the J-cost model of neural overparameterization. The definition is a pure structure assembly of three already-proved field lemmas.
Claim. There exists an overparameterization certificate consisting of: $\mathrm{domainCost}(r,r)=0$ for every $r\neq 0$; $\mathrm{domainCost}(m,e)\ge 0$ whenever $m>0$ and $e>0$; and the canonical threshold is strictly positive.
background
The module treats neural-network overparameterization as a Recognition Science cost problem. Empirically, generalization needs roughly 5–10× more parameters than data points; RS predicts an optimal ratio $J(\varphi)^{-1}\approx 8.47$ times the minimum parameter count, read off the recognition budget.
The domain cost is the J-cost pulled back to a model-vs-evidence pair $(m,e)$. The J-cost itself is the unique nonnegative cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$, minimized at the identity $x=1$. Non-negativity of raw recognition-event cost is already recorded upstream as cost_nonneg in ObserverForcing.
NNOverparamCert is the structure that packages the three well-posedness obligations: diagonal vanishing, off-diagonal non-negativity, and a positive canonical threshold. This definition is the concrete inhabitant of that structure.
proof idea
Pure structure construction, not a tactic proof. The three fields of NNOverparamCert are filled by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos. No further rewriting or case analysis is required; the certificate is the triple of those proofs.
why it matters
Gives the materials layer a single named witness that the J-cost model of trained networks is mathematically well-posed before any ratio comparison is stated. The module frames the classical 5–10× overparameterization rule as the RS prediction $J(\varphi)^{-1}\times$ minimum, tying the certificate to the T5 J-uniqueness landmark and the golden-ratio fixed point from T6. No downstream consumers are wired yet; the immediate sibling cert_inhabited only records that the type is nonempty. The certificate is the structural prerequisite for later theorems that would bound generalization gap by the J-cost excess above the canonical threshold.
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