Papers · William Alexander Aldred

One Scale, Running: A Logarithmically Running-Scale Form, and No Closure — A Non-Algebraic Test of the Closure Identity on SPARC

by William Alexander Aldred — Independent Researcher

Published October 31, 2026 · Zenodo · 11 pp · DOI: 10.5281/zenodo.23850980

Abstract

Papers #82 and #83 ran the Closure Identity on a named space of nine algebraic one-scale laws, g_obs = g_bar ν(g_bar/g_†). No form in that space closed both of SPARC's reductions. Paper #83 named the door it did not walk through: a law that is not algebraic at all. This paper walks through it with one registered shape. The shape takes the standard form ν_1 and lets its scale run: g_†(y) = g_†_0 [1 + α (1+y)], with y = g_bar/g_†_0. The logarithm makes the law non-algebraic. It carries one extra constant, α. Both constants were fitted on the radial acceleration relation alone and frozen before the baryonic Tully–Fisher side was touched. The fit chose α = +0.513699 and g_†_0 = 1.351082×10⁻¹⁰ m/s². The form passes the consistency lock (scale ratio 1.009886, band [0.98, 1.02]) and the scatter lock (0.156501 dex against the 0.161523 line). It fails closure. Its corrected slope, b = 3.768984, passes the registered slope condition at widened errors (z = -2.2695). Its intercept offset is +0.462276 dex against a ±0.10 band, and replication fails on all four withheld halves. The anatomy is the same anatomy as the nine algebraic forms: measured along the frozen line, the relation centers at −0.019073 dex, inside the band. The offset is the slope shortfall counted through the lever arm. Split by regime, the corrected slope runs 3.530920 at low V_f and 3.231734 at high V_f — the same regime structure #83 measured for the algebraic space. One named non-algebraic form now stands tested beside the nine algebraic ones, and it does not close. That is the claim, at that size: this form, this dataset, this machinery. No sentence here grades non-algebraic laws in general, or running scales in general.

The result, in plain language

Papers 82 and 83 killed nine algebraic forms. This one runs the first non-algebraic shape: the standard form with its acceleration scale running logarithmically, its running constant fitted on the RAR alone and frozen before the BTFR side was touched. It passes consistency and scatter, passes the slope term — and fails closure on the intercept, in all four replication halves.

The regime slope that killed the algebraic space is still standing: 3.53 low, 3.23 high. The fence grows by one honest form.

Program note

Cube Campaign Batch 1, target C-02 — the Closure Identity program’s first step outside the algebraic class: a non-algebraic, logarithmically running-scale form, registered before computation, its running constant fitted on the RAR alone and frozen. Its grades are printed on its face: the nine-form baseline was reproduced digit-for-digit before the new form ran (max |Δ| = 0.0); the execution ran twice, byte-identical; one deviation is disclosed — D2, the scatter lock is partly in-sample for the running constant. The fence this paper closes is exactly one form wide, and it prints itself that way.

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