Papers · William Alexander Aldred

Every Form Has the Same Seam: The Intercept Accounting of the Closure Family

by William Alexander Aldred — Independent Researcher

Publicly deposited November 1, 2026 · Zenodo · 6 pp · DOI: 10.5281/zenodo.23869681

Abstract

Paper #152 took one interpolation form of the radial acceleration relation — the sharp ν_q form at q = 0.50 — and showed that its whole advantage over the standard McGaugh form is carried by construction: a correction computed from the baryons alone reproduces the form's fitted intercept galaxy by galaxy, and what is left after the correction is a tilt, not a shift. That left an obvious question standing. Was q = 0.50 special, or does every form in the family carry the same seam? This paper runs the decomposition form by form across all seven distinct forms of the family — McGaugh, simple, standard, and ν_q at q = 0.50, 0.75, 1.50, and 2.00 — on the same matched sample of 116 SPARC galaxies, with no refits and no new fits of any kind. The accounting identity at the heart of #152 holds for all seven forms, exactly: the combined predictor built from each form's own correction term and the consensus of the other six forms reproduces that form's fitted intercept galaxy by galaxy to within 3.1×10⁻¹⁵ dex, against a registered bar of 10⁻⁹. The simpler two-term version of the seam — the correction term minus the intercept difference — also closes for all seven forms, with residual scatter between 0.0009 and 0.0030 dex against a bar of 0.005. And in every one of the seven forms, what survives the accounting is a tilt: the per-galaxy difference changes sign across the sample's depth bands in all seven. No form in the family buys its intercept honestly and no form is exempt. The seam is not a property of one sharp form. It is a property of the family.

The result, in plain language

Paper #152 took one interpolation form of the radial acceleration relation — the sharp ν_q form at q = 0.50 — and showed that its whole advantage over the standard McGaugh form is carried by construction: a correction computed from the baryons alone reproduces the form's fitted intercept galaxy by galaxy, and what is left after the correction is a tilt, not a shift. That left an obvious question standing. Was q = 0.50 special, or does every form in the family carry the same seam? This paper runs the decomposition form by form across all seven distinct forms of the family — McGaugh, simple, standard, and ν_q at q = 0.50, 0.75, 1.50, and 2.00 — on the same matched sample of 116 SPARC galaxies, with no refits and no new fits of any kind. The accounting identity at the heart of #152 holds for all seven forms, exactly: the combined predictor built from each form's own correction term and the consensus of the other six forms reproduces that form's fitted intercept galaxy by galaxy to within 3.1×10⁻¹⁵ dex, against a registered bar of 10⁻⁹.

The simpler two-term version of the seam — the correction term minus the intercept difference — also closes for all seven forms, with residual scatter between 0.0009 and 0.0030 dex against a bar of 0.005. And in every one of the seven forms, what survives the accounting is a tilt: the per-galaxy difference changes sign across the sample's depth bands in all seven. No form in the family buys its intercept honestly and no form is exempt. The seam is not a property of one sharp form. It is a property of the family.

Program note

Paper 157 of the November 1, 2026 day charge — batch 3 of the writing bench, deposited the day it was written. Its verdict, as printed on its face: STAND — the combined accounting identity holds for all seven distinct forms (max residual 3.1e-15 dex vs the 1e-9 bar); the two-term seam closes for 7 of 7 forms; every form's remainder is a tilt

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