The Joint Across the Turning Point: Wentzel, Kramers and Brillouin's Connection Formulas, Priced
Year: 2026
Type: preprint
Abstract: In 1926 Gregor Wentzel, Hendrik Kramers and Léon Brillouin independently supplied the missing piece of the semiclassical method: the connection formulas that join the oscillatory and evanescent WKB branches across a classical turning point. I credit those papers at full strength: the WKB method is one of the most successful approximation schemes in physics, the connection formulas are its crown, and their measured accuracy is astonishing for an asymptotic joint — on my own bench the joint's error falls to 0.000207 at n = 4 on the linear bench and 0.001635 at n = 4 on the quartic bench, and is exactly zero on the harmonic-oscillator control. Nothing in this paper falsifies the method or a number it returns. What I evaluate is the joint's derivation status. The π/4 phase and the factor-of-2 amplitude rule are not derived from the true potential at the turning point; they are computed on a linearised substitute problem — the Airy problem — and imported across the region where the expansion itself has broken down, and the import carries no 1926 error bound. My bench evaluates what the substitution leaves behind, on declared inputs only, against exact answers obtained by other means. The joint is free on the harmonic-oscillator control (0.000000 at every n printed), leaves a ground-state relative error of 0.007637 (0.7637%) on the Airy bench where the substitute *is* the true potential (exact 1.855757, WKB 1.841584), and leaves 0.182206 (18.2206%) at the quartic ground state (exact 0.667977, WKB 0.546267) — the same joint, three values, none of them predictable from anything the 1926 papers print. Counted plainly: 1 connection matching against 0 derivations from the true potential in 1926, and 0 error bounds. 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: W7225423166