The Shape Was Not the Door: At Full Strength the Wider Stencil Rings the Same
Year: 2026
Type: preprint
Abstract: AF-26 exhausted the weight dial: inside the symmetric class's family H_w, quiet dresses exist at both grains, but the dial they turn is the dressed equation's strength — the passing dresses cure equations with radiation constant wτ₀, and at the electron's own strength (w = 1) dress S's pre-echo stands. AF-26's F3 named the one dial left: shape. A wider symmetric stencil at full strength — the five-point symmetric difference on dress S's own half-step, the standard fourth-order centred stencil — might break the trade if the pre-echo were an artifact of the two-point stencil's crudity. This paper runs that hunt on AF-21's bench unchanged (W = 5g Hann pulse, 160g margin, dt = g/16, trapezoid energies), the two-point rows reproducing AF-21/AF-26 digit for digit as calibration before the five-point rows are read. The five-point symbol is a far better derivative — its error against ω is 2000× smaller at the band's low end (2.08×10⁻⁷ against 4.17×10⁻⁴ at x = 0.05), its symbol peaks higher (2.7444/g against 2.0000/g) — and none of it matters: E_early moves from 7.868703×10⁻² to 7.985399×10⁻² at τ_cell and from 6.981292×10⁻¹ to 6.990995×10⁻¹ at τ_b, the energy centroid stays at −2.1 g / −11.9 g, the late energy stays at the 10⁻⁷–10⁻⁸ floor, and the pre-registered pass (E_early < 10⁻³, δ ≥ 0.01, peak |a| ≤ 2) fails on its first clause at both grains while passing the other two. The wide-pulse control (W = 25g) agrees: 2.459511×10⁻⁴ against the two-point's 2.466560×10⁻⁴ at τ_cell, 0.183257 against 0.183187 at τ_b — the shape changes the pre-echo in its third decimal at most, in either direction. The reason is structural and is stated in §2: the pre-echo is the response of the operator (1 − τ₀·d/dt) inverted under a boundedness condition, and a symmetric stencil of any width supplies that inversion with the same one-sided kernel family — the stencil sets the ring's frequency, the operator sets its side. What survives the hunt: AF-21's measurement is now known to be stencil-independent at the percent level, and the symmetric class's full-strength price — anticipation, exchangeable only for strength (AF-26) — stands graded as the class's law on every construction this line has built. The remaining unbuilt shape class (non-uniform grains) is named as this paper's F3, without optimism attached.
Author: Aldred, William Alexander
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225419351