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

SPARC's two classic reductions — the radial acceleration relation (RAR) and the baryonic Tully–Fisher relation (BTFR) — are fitted separately in every published treatment, so a candidate acceleration law is never required to pay both tables with the same coin. We derive the missing joint constraint. 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. We execute 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. The kill's composition, however, is not the naive one, and we 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. The genuine phenomenon underneath 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 — and it dies on the slope's regime structure, not on the scale.