One Scale, Running: A Logarithmically Running-Scale Form, and No Closure — A Non-Algebraic Test of the Closure Identity on SPARC

Papers #82 and #83 ran the Closure Identity on a named space of nine algebraic one-scale laws, g_obs = g_bar ν(g_bar/g†). No form in that space closed both of SPARC's reductions. Paper #83 named the door it did not walk through: a law that is not algebraic at all. This paper walks through it with one registered shape. The shape takes the standard form ν_1 and lets its scale run: g†(y) = g†_0 [1 + α ln(1+y)], with y = g_bar/g†_0. The logarithm makes the law non-algebraic. It carries one extra constant, α. Both constants were fitted on the radial acceleration relation alone and frozen before the baryonic Tully–Fisher side was touched. The fit chose α = +0.513699 and g†_0 = 1.351082×10⁻¹⁰ m/s². The form passes the consistency lock (scale ratio 1.009886, band [0.98, 1.02]) and the scatter lock (0.156501 dex against the 0.161523 line). It fails closure. Its corrected slope, b = 3.768984, passes the registered slope condition at widened errors (z = −2.2695). Its intercept offset is +0.462276 dex against a ±0.10 band, and replication fails on all four withheld halves. The anatomy is the same anatomy as the nine algebraic forms: measured along the frozen line, the relation centers at −0.019073 dex, inside the band. The offset is the slope shortfall counted through the lever arm. Split by regime, the corrected slope runs 3.530920 at low V_f and 3.231734 at high V_f — the same regime structure #83 measured for the algebraic space. One named non-algebraic form now stands tested beside the nine algebraic ones, and it does not close. That is the claim, at that size: this form, this dataset, this machinery. No sentence here grades non-algebraic laws in general, or running scales in general.