Papers · William Alexander Aldred

The Defect Is Not in the Shape: A Derived Curve for the Closure Defect, Tested on SPARC

by William Alexander Aldred — Independent Researcher

Published October 31, 2026 · Zenodo · 6 pp · DOI: 10.5281/zenodo.23851420

Abstract

Cycle 2 of this program (paper #83) closed the nine-form space and left one object standing: one scale pays both tables' normalization, and the corrected BTFR slope refuses 4 by an amount that depends on regime, 3.498363–3.633395 in the low-V_f half and 3.205598–3.269601 in the high-V_f half, for every form tested. This paper asks if that defect can be derived, not fitted. The route is registered before the run. First, an exact decomposition: the closure defect of each galaxy splits, by algebra alone, into a disk-geometry term, a flat-to-edge velocity term, and twice the form's RAR residual at the galaxy's outermost point. The identity checks numerically for all nine forms (largest deviation 0.000098). Second, the derivation under test: replace the outer-point residual by the form's own RAR residual curve, measured on the full cut RAR (93 galaxies, 2,060 points) in deciles of the form's own variable, and read the predicted corrected slopes off the predicted ordinate with Cycle 1's own estimator. The residual curves are flat. Across all nine forms their decile means stay within −0.102 to +0.084 dex, and they carry almost no slope in either regime. The predicted slopes land at 3.7220–3.8336 in the low regime and 3.6304–3.7748 in the high regime. The sealed slopes sit at 3.498363–3.633395 and 3.205598–3.269601. Every form misses the registered ±0.20 agreement band in the high regime, by +0.4219 to +0.5052, and every form misses or touches its edge in the low regime, by +0.2002 to +0.2236. Zero of nine forms agree in both regimes. The verdict is OUTSIDE, at the registered size: the defect is not the ν families' shapes folded through the sample's y_f geometry. The diagnostic partition says where it lives instead. The shortfall is carried by the outer-point RAR residuals' own trend with V_f, a slope part of −0.498 to −0.601 in the low regime and −0.747 to −0.836 in the high regime, and inside each regime those residuals do not trend with y_f (correlations −0.266 to +0.245 across forms and regimes, centered on zero). The defect is real, it is at the rim of each galaxy's measured curve, and no function of the law's own variable carries it. Grades print on the face: the decomposition is derived; the residual curves and the partition are measured; each scale is fitted to one reduction and spent frozen; the constitutive postulate stays assumed.

The result, in plain language

Cycle 2 left a slope that refuses 4 by regime, for all nine forms at once. We asked if the forms themselves wrote that refusal. Registered the derivation first, built it from the forms’ own residual curves, and spent it against the sealed numbers. It misses for all nine. The defect is not in the shape.

It lives at the rim of the rotation curves, in residuals that trend with how fast a galaxy turns and not with the acceleration they sit at. Measured, located, unexplained.

Program note

Cube Campaign Batch 1, target C-04 — companion to papers 82 and 83. Cycle 2’s kill left a measured shape: a closure defect that runs with regime for all nine forms at once. This paper derives the defect’s exact split, registers shape-derived predictions before opening the sealed values, and spends them. Its grades are printed on its face: the split is derived and verified (max deviation 9.75×10⁻⁵); the derivation of the defect from the forms’ shapes is refuted at registered size; one narrowing is disclosed — the published regime ranges from paper 83 were known from orientation, and no per-form sealed value chose the model. What stands open is the defect’s carrier: the outer-point residual term, located and unexplained.

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