The Perfect Crystal: Bloch's Band Theorem, Priced
Year: 2026
Type: preprint
Abstract: In 1928 Felix Bloch proved that in a potential of exact periodicity the eigenstates of an electron take the Bloch form — a plane wave modulated by a function with the lattice's own periodicity — and with that single theorem founded the band theory of solids. I credit that paper at full strength: the theorem is airtight, my own bench re-returns its band edges exactly at the declared potential, and the band theory built on it underlies the modern understanding of metals, semiconductors and insulators and every device built on them. Nothing in this paper falsifies the theorem or a result derived inside it. What I evaluate is the world the theorem is a theorem about. Exact, infinite periodicity — an unbounded, defect-free, motionless lattice, one electron at a time — is assumed, not derived: no step in 1928 derives how much imperfection the Bloch form survives. My bench evaluates the assumption's first failure on declared inputs only. The perfect lattice's lowest gap is 1.000000, with all states extended (maximum inverse participation ratio 0.014883). One defect site in 200, at declared strength D = −1.0, already inserts a localised non-Bloch state at +0.306063 inside that gap (inverse participation ratio 0.148607, 0.876365 of its weight within 5 sites of the defect, size-stable at N = 100, 200 and 400), splits a second state off below the band at −2.209275, and cuts the gap's smallest clean interval to 0.195006 — and at D = −0.5, to 0.059239. The first failure is not a perturbation of the perfect number; it is a new kind of state inside it. Counted plainly: 1 perfect-periodicity assumption against 0 derivations in 1928. 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.**
Author: Aldred, William Alexander
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225379253