The Grain, Self-Set: The State-Varying Species, Its Collapses, and the Close of the Grain Question
Year: 2026
Type: preprint
Abstract: AF-33 executed the position-varying species of the non-uniform grain and closed with one species unbuilt: a grain that varies with the field's own state is a nonlinear construction no linear bench in the line could carry. This paper builds it. The grain field takes the ladder's two rungs (fine τ_b, coarse G2 = 7.75τ_b) according to the local response amplitude against a declared threshold a = 0.5 — half the drive's peak — in two zonings: SV-A, fine where |a| < a and coarse where |a| ≥ a; SV-B, the reverse. The operator is AF-33's time-domain form of AF-1's H, solved by fixed-point iteration from the uniform fine start, the changed-cell count printed at every step. Calibration is AF-33's bench ported verbatim: the spectral prints reproduce digit-for-digit (7.868703×10⁻² at τ_cell; 6.981292×10⁻¹ at τ_b) before any state-varying number is read. All four graded runs converge — in 1, 3, 2, and 3 iterations respectively — and their fixed points are the result: SV-A at the narrow pulse is the uniform fine field identically (the response never reaches the threshold, at a = 0.25, 0.5, or 0.75); SV-B at the narrow pulse is the uniform coarse field identically (verified digit-for-digit against the spectral solve at the ladder's coarse grain run in-bench); at the wide pulse SV-A settles on a coarse island covering 1.3% of the domain and SV-B on a fine island, returning E_early 7.887×10⁻² and 7.909×10⁻² — strictly between the uniform coarse print (7.851×10⁻²) and the uniform fine print at that width (7.966×10⁻²). Impulse probes driven through the state-varying operator itself are bounded and decaying in both zonings: no growing mode. F3's pass is missed by a factor of at least 79, as it was missed by position in AF-33 — the escape does not exist in the nonlinear species either.
Author: Aldred, William Alexander
Language: English
Topic: Complex Systems and Dynamics
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225464246