IndisputableMonolith.CondensedMatter.AndersonLocalizationFromJCost
Module linking Anderson localization to the Recognition Science J-cost: disorder domains carry a nonnegative cost that vanishes only at equilibrium, and localization is certified once that cost exceeds a canonical positive threshold. Condensed-matter theorists working in RS units would cite the certificate and the threshold lemmas. The file is mostly definitions plus elementary nonnegativity and positivity facts built from the Cost import.
claimDefine a domain cost $C$ built from the RS $J$-cost on disorder configurations; prove $C \ge 0$ with $C = 0$ only at equilibrium; fix a canonical threshold $\theta > 0$; and package an Anderson-localization certificate asserting that whenever $C > \theta$ the eigenstates are localized.
background
Recognition Science measures mismatch by the unique cost $J(x) = (x + x^{-1})/2 - 1$ forced by the Recognition Composition Law. The Cost module supplies that functional and its basic calculus; Constants supplies the RS tick $\tau_0$.
In disordered media, Anderson localization is the exponential spatial confinement of eigenstates above a critical disorder strength. This module treats a disorder domain as a configuration whose $J$-cost is the domain cost. Equilibrium configurations (perfectly matched ratios) sit at cost zero; any deviation is strictly positive.
The local setting is condensed-matter phenomenology inside RS-native units ($c = 1$, $\hbar = \varphi^{-5}$), not a microscopic tight-binding derivation. The certificate is a Prop-level witness that the cost has crossed a fixed positive threshold, after which localization is asserted.
proof idea
Definition-heavy module. domainCost is introduced from $J$; domainCost_nonneg and domainCost_at_equilibrium are immediate from nonnegativity and the unique zero of $J$ at $1$. canonicalThreshold is a positive constant (canonicalThreshold_pos). AndersonLocCert bundles the inequality domainCost $> \theta$ into a certificate type; cert and cert_inhabited supply a default inhabited instance. No deep spectral theory appears; the argument is cost comparison plus packaging.
why it matters in Recognition Science
Places Anderson localization on the same $J$-cost footing as the forcing chain (T5 $J$-uniqueness) rather than as an independent condensed-matter axiom. Downstream consumers would import the certificate when deriving transport gaps or mobility edges from RS cost bounds. With no recorded used_by edges yet, the module is a leaf that closes a condensed-matter interface: once domain cost exceeds the canonical threshold, localization is available as a hypothesis-free certificate inside later RS solid-state developments.
scope and limits
- Does not derive the localization length or Lyapunov exponents from a microscopic Hamiltonian.
- Does not prove mobility edges or conductivity vanishing; only packages a cost-threshold certificate.
- Does not treat interacting or many-body localization.
- Does not fix numerical values of the threshold beyond positivity in RS units.
- Does not connect to experimental disorder strengths outside the $J$-cost model.