Papers · William Alexander Aldred

One Scale, Two Tables: The Second Variable Fails Too — The Σ_bar-Running Scale and the Closure Identity on SPARC

by William Alexander Aldred — Independent Researcher

Published October 31, 2026 · Zenodo · 14 pp · DOI: 10.5281/zenodo.23851076

Abstract

Cycle 2 of this program closed the named space of algebraic one-scale acceleration laws on SPARC: nine forms, three rescues, no closure. It left one door open in print. A law with a second variable could still stand, and the named candidate in the literature, the external field effect, could not be tested here, because no per-galaxy environment field ships in SPARC's tables. But one second variable does ship in the tables: the baryonic surface density Σ_bar. This paper registers one law against that door and runs it. The law is the McGaugh form with a scale that runs with surface density, g†(Σ) = g†_0 · (Σ_bar/Σ_0)^γ, with Σ_bar defined exactly from the SPARC main table (Σ_bar = M_b/(2π R_eff²), M_b built from the tabulated 3.6 µm luminosity at Υ = 0.5 plus 1.33 times the tabulated HI mass), the pivot Σ_0 = 156.77 M☉/pc² set at the cut sample's median, and both parameters, g†_0 and γ, fitted on the radial acceleration relation alone and spent frozen on the baryonic Tully–Fisher side. The RAR returns γ = +0.076 ± 0.057: a running consistent with zero at its measured precision. The scatter lock passes (0.154855 dex, a hair under the one-scale incumbent's 0.155523). Closure still fails. The corrected slope reaches 3.904191, the closest approach to 4 on this program's McGaugh line (z = −0.946566, the slope term passes), but the intercept offset is +0.182157 dex against the ±0.10 band, replication holds on one of four withheld halves, and the regime anatomy keeps its shape (slopes 3.584089 low, 3.292609 high). The law also fails the consistency lock in its adapted form, Lock 0-Σ: the full-set RAR fit returns a different running (γ = +0.203668, pivot-scale ratio 0.844294 against the registered [0.98, 1.02] band), so the running is not a stable property of the RAR reduction itself. One inherited check could not run in its original form: Lock 0 compares the published RAR table against the constructed pairs, and the published table names no galaxies, so no per-galaxy scale exists for its rows. The adapted check is named as adapted wherever the locks are listed, here included. The verdict is a kill, sized to one form, one surface-density definition, one dataset: the second variable SPARC already carries does not close the identity either. What survives is the same phenomenon the first two cycles left: one scale pays both tables' normalization (the frozen line centers at −0.014468 dex), and the slope refuses 4 by an amount that depends on regime.

The result, in plain language

Cycle 2 left one door open: a law with a second variable. The external field effect can’t be tested on SPARC — the data isn’t in the tables. But one second variable is: the baryonic surface density, Σ_bar. So we registered one law — the McGaugh form with a scale that runs with Σ_bar, both parameters fitted on the RAR alone, spent frozen on the BTFR — and ran it under the same locks that killed nine forms.

The RAR’s answer: γ = +0.076 ± 0.057. Consistent with zero. Spent anyway, the point estimate pulls the slope to 3.904, the closest the McGaugh line has come to 4 — and still fails: offset +0.182 dex against a ±0.10 band, replication on one of four halves, and the full-set RAR fit returns a different γ than the cut fit. The second variable fails too. The normalization still never fails. The slope still refuses, by regime. That phenomenon is still standing, and it still has no law.

Program note

Cube Campaign Batch 1, target C-03 — Row 6 of the Square program, after paper 82 (Cycle 1) and paper 83 (Cycle 2). Cycle 2 left one honest door open: a law carrying a second variable. This paper registers the one second variable SPARC’s tables actually ship — baryonic surface density — defends its definition against three rivals on facts in the registration, and spends it under the same locks. Its grades are printed on its face: measured, against a reproduction gate passed at max |Δ| = 0.0 against the sealed Cycle 1 record. The phenomenon it leaves standing is the program’s: the normalization never fails; the slope refuses, by regime; and no law yet named — nine forms, three rescues, one second variable — closes the identity.

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