18.5. Summary
This chapter carries the proofs the physics chapters import. The Mosco-convergence theorem is stated with its seven assumptions A1–A7, the structural ones verified and the equidistribution condition A3 supplied by P3’s maximum-entropy sublimation (§18.1). The -uniform rate is proved for the i.i.d. surrogate by a five-step Berry–Esseen argument with three closures, from which equicoercivity A7 follows by the reduction lemma (§18.1.6). Dirac-operator convergence, which the spectral-triple framework requires beyond the Laplacian’s, is reduced to a single analytic estimate (§18.3) and carried through the L_round round-trip and the L_comm lemma chain under the relaxed norm-equivalence criterion (§18.4); the uniform bound’s sharp value and its limit law are derived in Chapter 19’s account of the frontier. No reader needs this chapter to follow the physics; the rest of the book needs it to exist.