AndersonLocCert
plain-language theorem explainer
Certificate packing three structural facts for the RS Anderson-localization threshold: domain cost vanishes on equal ratios, is nonnegative for positive arguments, and the canonical threshold is positive. Condensed-matter workers of the disorder-driven metal-insulator transition at J-cost = J(φ) cite it. Plain structure definition; inhabitation is discharged by the companion cert construction.
Claim. An Anderson-localization certificate from J-cost is a triple of facts: (i) the domain cost of equal nonzero ratios vanishes, $\mathrm{domainCost}(r,r)=0$ for all $r\neq 0$; (ii) $\mathrm{domainCost}(m,e)\ge 0$ whenever $m>0$ and $e>0$; (iii) the canonical threshold is strictly positive.
background
The module treats the disorder-driven metal-insulator (Anderson) transition as a threshold on the Recognition Science J-cost of a conductance ratio. J-cost is the unique nonnegative cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$, minimized at equilibrium $x=1$. The module status line records a structural theorem with zero sorry and zero axioms.
domainCost is the local cost functional on a pair of positive real parameters (mass- or energy-like ratios in the conductance setting). Equilibrium vanishing means equal arguments cost nothing; nonnegativity is the condensed-matter shadow of the foundation fact that every recognition event has nonnegative cost. canonicalThreshold is the positive RS cut used for the transition, identified in the module header with $J(\varphi)$ on the conductance ratio.
Upstream, ObserverForcing already proves cost nonnegativity for recognition events via Jcost_nonneg. The present structure only packages the three condensed-matter-facing properties needed downstream.
proof idea
No proof body: this is a structure declaration. The three fields are Prop-valued hypotheses (equilibrium vanishing of domain cost, nonnegativity on positive arguments, positivity of the canonical threshold). Inhabitation is not claimed here; the sibling definition cert fills the fields with domainCost_at_equilibrium, domainCost_nonneg, and canonicalThreshold_pos, and cert_inhabited then exhibits Nonempty AndersonLocCert.
why it matters
Gives the typed interface for the Plan-v7 Anderson-localization-from-J-cost pass. Downstream, cert builds a concrete inhabitant and cert_inhabited records nonemptiness, so later lemmas can assume a certificate rather than re-prove the three facts. The module header pins the physics claim: disorder-driven metal-insulator transition at J-cost equal to $J(\varphi)$ on the conductance ratio, tying the threshold to the T5/T6 landmarks (unique J-cost and the golden fixed point $\varphi$). Structural falsifier named in the module is empirical data outside the RS prediction band. Does not itself compute localization lengths or critical exponents; it only certifies the cost-threshold scaffolding those arguments need.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.