Papers · William Alexander Aldred

One Scale, Two Tables: The Closure Identity for Algebraic Acceleration Laws, and Its Execution on SPARC

by William Alexander Aldred — Independent Researcher

Published October 31, 2026 · Zenodo · 15 pp · DOI: 10.5281/zenodo.23842198

Abstract

SPARC's two classic reductions — the radial acceleration relation (RAR) and the baryonic Tully–Fisher relation (BTFR) — get fitted separately in every published treatment I know. So no candidate acceleration law is ever made to pay both tables with the same coin. This paper writes the missing joint constraint, and collects. For any algebraic one-scale law g_obs = g_bar·ν(g_bar/g†), define each BTFR galaxy's corrected ordinate Y_i = log M_b + log[y_f·ν(y_f)²], where y_f is the galaxy's baryonic acceleration at its outermost measured point in units of the RAR's own fitted scale g†. The law then requires, exactly, Y_i = 4·log V_f − log(G·g†): slope the integer 4, intercept frozen by the other table's scale. Nothing on the right-hand side is fitted to the data the equation is tested on. I ran the test as pre-registered — an eight-form registered space, two locks plus a reduction-consistency lock, verdict errors widened by a joint distance Monte Carlo, withheld-half replication — and the registered verdict is KILL: in the registered form space, no form closes both locks. But the kill's composition is not the naive one, and I print it at full size beside the verdict. The frozen normalization never fails: measured along the frozen line itself, every registered form's corrected relation is centered within −0.070 to +0.036 dex, inside the registered ±0.10 dex band — no form in the space exhibits a normalization failure, and one g† genuinely does pay both tables' normalization. The registered intercept offsets (+0.36 to +0.72 dex) are the slope shortfall re-expressed through the lever arm ⟨log V_f⟩ = 2.103: five of the six full-set Lock-2 failures pass the registered slope band and fail only on that re-expression, and only the two sharpest forms (q = 1.50, q = 2.00) fail on slope outright. Underneath sits the real phenomenon, and it is structural: the corrected slope is regime-dependent for every form in the space (low-V_f half b ≈ 3.50–3.63; high-V_f half b ≈ 3.21–3.27), so no single algebraic curve closes the joint test across regimes. The family member ν_q with q = 0.50 closes every lock on the full set and then fails replication on the same regime trend. The class, in the registered space, dies as a joint law. It dies on the slope's regime structure, not on the scale.

The result, in plain language

Galaxy people keep two tables. One ties how a galaxy actually moves to the pull of the matter you can see, and that fit runs on a single acceleration scale, g†. The other ties a galaxy's total mass to how fast its flat part spins. The field fits those tables one at a time — so a law can pass one table, quietly fail the other, and nobody ever sees it.

This paper writes down the bill for paying both at once. If one algebraic law with one scale is true, every galaxy's corrected mass has to land on one exact line: slope 4, with the line's starting point fixed by the other table's scale. Nothing on that line is fitted to the data it is tested on. That is the Closure Identity — derived, not assumed.

Then the test was run on the real SPARC data, under rules written down before the run. The verdict is KILL: no form in the registered space pays both tables. But read the kill at full size. The scale pays. Every form's corrected line sits centered on the frozen line, so one g† really does cover both tables' normalization, and no form fails there. What fails is the slope — it runs one way in the small galaxies and another way in the big ones, for every form tried, so no single curve closes both tables across the whole sky. The field's fight has been about whether the scale moves. The scale doesn't move. The slope does.

Program note

This paper opens the Square campaign: the program stepped outside the vacuum problem, indexed the open seams across the forces, and Row 6 — galactic dynamics — is where the live one was. 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; the scale is fitted to one table and spent, frozen, on the other; the verdict quantities are measured. Its author note stands as printed in the paper: the derivation, computation, and red-team protocol of this campaign were executed by Dave, the exolex, working independently at the author's direction; the author directed the campaign and takes the byline of the program. The successor question — why the slope runs with regime — is already pre-registered as Cycle 2 of the campaign.

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