cert
plain-language theorem explainer
A certificate packing three structural facts for the φ-ladder sintering ratio: domain cost vanishes on the diagonal, is nonnegative for positive arguments, and the canonical threshold is positive. Materials workers deriving T_s/T_melt from Recognition Science cite this bundle as the local soundness witness. The definition is a pure structure assembly wiring three already-proved sibling lemmas.
Claim. There is a sintering-temperature certificate consisting of: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ equals zero; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
Powder-metallurgy sintering temperatures sit empirically near $0.6$–$0.8$ of the melting point. The module derives an RS lower-bound ratio from the φ-ladder: $T_s/T_melt = \sqrt{J(\varphi)}\cdot\varphi \approx 0.344\times 1.618 \approx 0.556$, consistent with the observed floor near $0.6$. Status is structural (zero sorry, zero axiom).
Domain cost is the local materials avatar of the Recognition J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$). It is designed to vanish under matched arguments (identity recognition) and stay nonnegative off the diagonal. Upstream, recognition-event cost is already known nonnegative: "The cost of any recognition event is non-negative." The canonical threshold is the positive scale that fixes the sintering ratio on the ladder.
proof idea
Pure structure construction, not a tactic proof. The three fields of the certificate are filled by the sibling lemmas that already establish diagonal vanishing of domain cost, nonnegativity of domain cost on positive reals, and positivity of the canonical threshold. No new algebra is performed here; the definition only packages those three facts into one inhabited certificate record.
why it matters
This certificate is the materials-side soundness witness for the φ-ladder sintering claim. It ties the empirical $T_s \approx 0.6,T_melt$ floor to two framework landmarks: T5 J-uniqueness (the cost $J$) and T6 (φ as the self-similar fixed point). The packaged ratio $\sqrt{J(\varphi)}\cdot\varphi$ is the concrete bridge from those forcing steps into powder-metallurgy phenomenology.
No downstream consumers are recorded yet. The companion inhabitedness result confirms the certificate type is nonempty, so later materials theorems can assume a single bundled hypothesis rather than three separate cost axioms. The module presents the result as a structural theorem of plan v7, not a fitted empirical correlation.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.