The Slope at the Right Estimator: An Errors-in-Variables Fit of the Baryonic Tully–Fisher Relation, and the Death of the Exact-Law Line

The baryonic Tully–Fisher relation (BTFR), log M_b = a + b·log V_f, carries a standing conjecture in this program's galaxy work: that its slope is exactly 4 — an exact law at galaxy scale, of the kind the program's wider search keeps failing to find produced anywhere. The October 27, 2026 scour of the full SPARC database graded that conjecture UNTESTED rather than dead, for a stated reason: the available grade was a forward ordinary-least-squares fit, slope 3.592562 ± 0.090545 on the published 118-galaxy table, and OLS attenuates — errors in log V_f bias its slope toward zero — so a 4.50σ OLS miss is not an adjudication. This paper runs the adjudication at the estimator the question requires: a bivariate errors-in-variables maximum-likelihood fit of the same table, using the table's own per-point errors in both coordinates, with the intrinsic scatter fitted rather than assumed. The EIV slope is 3.660773 ± 0.088828 (bootstrap standard error, seed 137, 1,000 resamples; 95% CI [3.487133, 3.838556]), with fitted intrinsic scatter 0.122540 dex. The slope excludes 4 at 3.82 standard errors. The pre-registered rule for this door, fixed before the fit ran, reads: an EIV slope excluding 4 at ≥ 3 SE kills the exact-law line outright; an EIV slope including 4 with intrinsic scatter below 0.05 dex stands it. The first clause fires, and the second fails independently on both of its terms — the slope does not include 4, and the fitted scatter is two and a half times the stand threshold (even forcing the slope to exactly 4, the refitted scatter is 0.1331 dex). The exact-law line is dead. The death is reported at full size with its scope attached: it is a death on this table, at this table's own shipped errors, under this program's pre-registered rule — and the fitted non-integer slope and large likelihood-weighted scatter are themselves the informative residue, examined in §5. The BTFR remains an excellent empirical relation. What it is not, on the record's own terms, is an exact law with slope 4.