IndisputableMonolith.Materials.TrainedNeuralNetworkFromJCost
Materials module that packages the J-cost as a domain cost for neural-network weights and states a canonical positive threshold together with an overparameterization certificate. Materials and ML theorists working in the RS stack would cite it when linking trained nets to the Recognition cost. The file is mostly definitions and elementary nonnegativity or positivity lemmas, plus an inhabited certificate record.
claimOn a real domain for network parameters, define a domain cost $C$ induced by the Recognition $J$-cost, prove $C\ge 0$ and evaluate it at equality cases, fix a canonical threshold $\theta>0$, and package an overparameterization certificate $\mathrm{Cert}$ that is inhabited.
background
Recognition Science builds physics from a single cost functional $J$. In the forcing chain, $J$ is uniquely fixed as $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), and the golden ratio $\varphi$ appears as the self-similar fixed point. The Cost import supplies that $J$-cost; Constants supplies the RS-native units and time quantum.
This materials module lifts $J$ to a domain cost on parameter space suitable for trained networks. Sibling definitions introduce the domain cost, its value at distinguished points, nonnegativity, a canonical positive threshold, and a record-type overparameterization certificate together with a proof that the certificate type is inhabited.
The local setting is therefore not a full learning-theory development, but a thin RS-native interface: cost, threshold, and a certificate object that later materials or ML layers can assume.
proof idea
Definition-heavy module rather than a deep theorem chain. Domain cost is introduced from the imported $J$-cost; equality and nonnegativity facts are short algebraic or order lemmas. The canonical threshold is a positive constant definition with a positivity proof. The overparameterization certificate is a structure (or similar record) bundled with an inhabitation witness, so downstream code can obtain a concrete cert without reconstructing fields.
why it matters in Recognition Science
Places trained neural nets inside the Materials slice of the RS monolith by tying their cost landscape to the same $J$ that forces $\varphi$, the eight-tick octave, and $D=3$ upstream. No downstream used-by edges are recorded yet, so the module is a leaf interface: it exposes domain cost, threshold, and an inhabited overparameterization certificate for later materials or certification theorems to import. It does not itself close a forcing-chain step (T5--T8), but keeps NN-side arguments on the same cost footing as the rest of Recognition Science.
scope and limits
- Does not prove generalization bounds or PAC-Bayes statements for real trained nets.
- Does not derive network architecture, depth, or width from the forcing chain.
- Does not claim empirical training dynamics equal $J$-gradient flow.
- Does not connect the certificate to measured materials constants or the alpha band.
- Does not discharge any sorry outside its own elementary lemmas.