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

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+x^2)), 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.