The Random Lattice: Anderson's Model Postulate, Priced
Year: 2026
Type: preprint
Abstract: In 1958, P. W. Anderson founded the theory of localization on a model: a tight-binding electron on a lattice whose site energies are random, an environment of zero — no correlations between sites, no motion in the disorder, no self-consistent field, no action of the electron on the lattice it moves through — and the model's finding, that strong enough randomness localizes the state, is a discovery I credit at full strength: my bench re-computes it from nothing but a declared draw, and the finding stands, with the band-centre participation ratio falling from 7.333333 toward 1 as the box widens. What I evaluate is the model's content. The phenomenon of the world — a real crystal's actual environment — is evaluated nowhere inside the model it appoints: the assumption is one correlation excluded (step-to-step correlation, measurable in fact: 0.355000 W between my declared uncorrelated neighbours against 0.111111 W with 9 of 9 resonant pairs for a declared correlated draw of the same width), and at W/V = 8.000000 the band-centre participation ratio reads 1.586593 in the uncorrelated model against 4.220743 under the correlated draw — a factor of 2.660783 between the model's world and a world with one excluded thing restored, at identical width, identical hopping, identical chain. Counted plainly: 1 model postulate (2 clauses: the box draw; the excluded environment), 1 named mechanism in the model (discreteness) against 2 in Mott's rival 1949–56 picture at the same class of address (discreteness, discreteness-driven self-consistent field), 0 derivations in 1958 of the static box's correspondence to any real disorder. Grade for this census entry, printed at full size: REASON FOUND / STILL TURNING — PREMISE-ONLY, explicitly not a kill and not a falsification. The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0. Keywords: Anderson 1958; localization; tight-binding model; random site energies; participation ratio; Mott; model postulates.
Author: Aldred, William Alexander
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225419818