Papers · William Alexander Aldred

One Scale, Two Tables: Nine Forms, Three Rescues, No Closure — The Closure Identity for Algebraic Acceleration Laws Completed on SPARC

by William Alexander Aldred — Independent Researcher

Published October 31, 2026 · Zenodo · 24 pp · DOI: 10.5281/zenodo.23843493

Abstract

Cycle 1 of this program built the Closure Identity — for any algebraic one-scale law g_obs = g_bar·ν(g_bar/g†), the corrected BTFR ordinate Y_i = log M_b + log[y_f·ν(y_f)²] must sit on Y_i = 4·log V_f − log(G·g†), slope exactly 4, intercept frozen by the RAR's own scale — and ran it on an eight-form space. No form closed both locks. But the space had a hole in it, and the kill's own anatomy suggested three rescues. This paper closes the hole and runs the rescues. The completed space stands at nine named forms — McGaugh, simple, standard as registered (ν_1), standard as named in prose (ν_std*, the literal inversion of μ(x) = x/√(1+x²)), and the ν_q grid at q = 0.50, 0.75, 1.00, 1.50, 2.00 — and no form in it closes both locks on SPARC's two reductions. The ninth form, run blind under Cycle 1's identical protocol with the audit numbers barred from the run, passes the consistency and scatter locks and fails closure (Design 2, §3). Exact enclosed-mass bookkeeping, computed per galaxy from its own rotation-curve decomposition with zero free parameters, closes nothing: the correction is large (a 0.707 dex swing, above 1 for 101 of 116 galaxies) and it changes no verdict (Arm A, §4). The identity does not close regime-locally either: with each regime's intercept frozen by that regime's own scale, the low-V_f regime fails the scatter lock for all nine forms and both regimes fail closure for all nine (Arm C, §5). The measured closure-defect curve is the same curve on both halves of the sample — the halves agree in all four form × coordinate units — and it still does not transfer as a closure, missing the frozen intercept in seven of eight directions (Arm B, §6). What survives is the phenomenon, not a law for it: one scale pays both tables' normalization — frozen-line means sit within ±0.10 dex across the space, with the two exceptions named in the body — while the corrected slope refuses 4 by an amount that depends on regime, ≈3.5–3.6 at low V_f and ≈3.2–3.3 at high V_f, for every form tested. One registered check, the within-regime cross-reduction consistency (Lock 0_R), could not be executed: the published RAR table carries no galaxy identifiers. The two-regime verdict rests on the scatter and closure locks, which do not route through it. Every claim in this abstract is sized to the named nine-form space, to SPARC's two reductions, and to the registered machinery — and to nothing wider.

The result, in plain language

Cycle 1 wrote the bill for paying both of SPARC's tables at once — one algebraic law, one scale, slope exactly 4, the line's starting point frozen by the other table's scale — and the bill came back unpaid: no form in the registered eight-form space closed both locks. But that kill had a hole in it. One named form had never actually been run, and the failure's shape — the scale paying, the slope refusing, differently in small galaxies than in big ones — suggested three honest ways the class might still be saved.

This paper runs all of it. The ninth form, run blind under Cycle 1's identical protocol, passes the consistency and scatter locks and fails closure, like the eight before it. Exact bookkeeping — each galaxy's own enclosed mass, computed from its own data, no free parameters — is large, and it changes no verdict. Freezing each regime's intercept with that regime's own scale closes nothing: the low-V_f regime fails the scatter lock for all nine forms, and both regimes fail closure for all nine. And the measured defect curve is the same curve on both halves of the sample — the halves agree — yet it still does not transfer as a closure.

What survives is the phenomenon, not a law for it. One scale genuinely pays both tables' normalization, for every form tested. The corrected slope refuses 4 by an amount that depends on regime — about 3.5–3.6 in the small galaxies, about 3.2–3.3 in the big ones. The failure of the algebraic one-scale class on SPARC is a slope failure with a regime structure. It is not a normalization failure, and it is not a bookkeeping artifact.

Program note

This paper is Cycle 2 of the Square campaign — the completion of the Closure Identity test opened by One Scale, Two Tables (paper 82 on this register). It closes the hole Cycle 1 left in its own form space and gives each of the three rescues its registered day, in the order the registration adopted. Nothing in this paper touches the vacuum catastrophe; that problem stands open on this same register, graded not solved. The paper's grades are printed on its face: the identity is derived; the constitutive postulate it tests is assumed and owned as such; each scale is fitted to one reduction and spent, frozen, on the other; every verdict quantity is measured. One registered check is disclosed at full size in the paper: the within-regime cross-reduction consistency (Lock 0_R) could not be executed, because the published RAR table carries no galaxy identifiers — the two-regime verdict rests on the scatter and closure locks, which do not route through it. What stands open after Cycle 2 is the phenomenon itself: why the slope runs with regime, measured and unexplained.

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