Papers · William Alexander Aldred

Split by the Ladder: The Closure Defect Measured on Good and Bad Distances in SPARC

by William Alexander Aldred — Independent Researcher

Published October 31, 2026 · Zenodo · 10 pp · DOI: 10.5281/zenodo.23851652

Abstract

The Square killed the algebraic one-scale class on SPARC, and the kill had a shape: the corrected slope refuses 4, by an amount that depends on regime, while the frozen-line level holds. Every number behind that shape rides on a distance. A galaxy's mass goes as the square of its distance, and its baryonic acceleration goes as the inverse of it, so a bad distance ladder could in principle wear a physics costume. This paper splits the matched sample of 116 galaxies by the catalog's own distance quality and measures the defect on both sides. The split variable is the fractional distance error, r = e_D/D, from the SPARC properties table, at its median (0.138889): 65 galaxies with good distances, 51 with poor ones. A second split, registered as a companion, goes by method: 57 galaxies with indicator distances against 59 with Hubble-flow distances. Five forms face the test: McGaugh, simple, standard ν_1, and ν_q at q = 0.50 and q = 2.00. Both halves face the same frozen target. The level does not move: frozen-line means differ between the halves by at most 0.037 dex across the five forms, against a difference error of about 0.044 dex. The slope separation points the artifact's way in all five forms — good-half slopes run 3.78 to 4.10, poor-half slopes 3.41 to 3.57 — and it reaches z = +1.54 to +2.22 in the difference, form by form. No form reaches the registered 3σ line on either channel, on either split. Verdict, at the registered size: distance systematics are ACQUITTED as the defect's cause, at a measured resolution of 0.71 to 0.72 slope units and 0.13 to 0.14 dex. The sub-threshold slope tilt prints in this abstract because it is part of the result: a tilt that points the same way in five forms at under 3σ apiece is not a conviction, and it is not nothing. It is the size of what this sample cannot settle. The forms are correlated, so their z values do not combine, and no combined significance is claimed. Cycle 2's distance Monte Carlo widened the slope errors for distance noise. This test does a different job: it measures whether the defect itself is distance-structured. On the level channel, it is not. On the slope channel, it is not at the registered line.

The result, in plain language

The Square’s kill on SPARC had one suspect left with a motive: the distances. Every mass in that test rides on a distance, so we split the sample by the catalog’s own distance quality and measured the defect on both sides.

The level is the same on good distances and bad ones. The slope tilts the artifact’s way at under 2.2σ in every form — printed in the paper at full size, because that’s part of the result — and nothing crosses the registered 3σ line. Verdict: the ladder didn’t do it. Distance systematics, acquitted by measurement, at a resolution the paper prints next to the verdict.

Program note

Cube Campaign Batch 1, target C-06 — the Square’s SPARC sample split by the catalog’s own distance quality, under criteria frozen before the run. Its grades are printed on its face: measured; the reproduction gate passes at max |Δ| = 0.0 against the sealed Cycle 1 record on all five forms; the run’s killed first execution and its completed run agree at every printed digit, and both logs stand in the deposit. The verdict is an acquittal at a printed resolution — the paper names what its test cannot see (a modest distance contribution under the observed tilt) as plainly as what it rules out.

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