IndisputableMonolith.Physics.Anderson_Localization4_FromJCost
Module defining a J-cost domain functional, a positive canonical threshold, and an inhabited Anderson-localization certificate in the RS physics layer. Condensed-matter or RS auditors tracing disorder localization to the unique cost J would land here. Content is largely definitional: cost construction, nonnegativity, threshold positivity, and a bundled cert type.
claimIntroduce a domain cost $C$ built from the Recognition cost $J$, a canonical threshold $\theta>0$, and a certificate type asserting Anderson-type localization bounds controlled by $J$-cost on domains.
background
Recognition Science forces a unique nonnegative cost $J(x)=\frac{x+x^{-1}}{2}-1$ (forcing step T5 / J-uniqueness), also written $\cosh(\log x)-1$. The imported Cost module supplies that functional; Constants supplies the RS-native tick $\tau_0$.
Anderson localization is the pinning of eigenstates by strong disorder. This module rephrases a localization threshold in RS-native language: cost is assigned to spatial domains via $J$, and a canonical positive cutoff separates localized from extended regimes.
Sibling names indicate domainCost with an evaluation identity and a nonnegativity lemma, canonicalThreshold with positivity, and an AndersonLoc4Cert bundle together with an inhabited witness.
proof idea
Definition-and-certificate module, not a deep existence proof. Domain cost is assembled from $J$; nonnegativity is inherited from $J\ge 0$. The canonical threshold is a fixed positive real (positivity lemma). The certificate type packages the cost/threshold inequalities; inhabitation is a one-line or constructive witness that the cert structure is nonempty. No tactic-heavy localization analysis appears at module scope.
why it matters in Recognition Science
Places Anderson-type localization inside the RS physics layer so disorder bounds speak the same language as the unique cost $J$ from the forcing chain. Imports only Constants and Cost, so it sits directly on the J-cost foundation rather than on phenomenological disorder parameters. No used_by edges are recorded yet; parent consumers are not wired in this snapshot. Indirectly supports the program of deriving condensed-matter thresholds from T5 J-uniqueness without new free parameters. Does not itself invoke T7 (eight-tick) or T8 ($D=3$).
scope and limits
- Does not prove Anderson localization for concrete disordered Hamiltonians.
- Does not derive mobility edges or critical exponents numerically.
- Does not encode spatial dimension $D=3$ or eight-tick structure here.
- Does not connect domain cost to the mass ladder or $\alpha$ band.
- Does not supply downstream consumers in the current graph snapshot.