Inside-Out Disc Growth as the Physical Origin of the Extended Baryonic Tully-Fisher Relation

The extended baryonic Tully-Fisher relation (BTFR), log Vflat = α log Mbar + β log Rd + c, with β = -0.136 (IC90%: [-0.198, -0.072]) measured over N = 696 disc-dominated galaxies in a companion paper [H26a; 3], encodes a systematic residual dependence of rotation velocity on disc size at fixed baryonic mass. Here we identify the physical mechanism behind β. We show that β is algebraically equivalent to the coefficient of log Rd in the Fall angular-momentum mass relation: since jbar ≡ 2VflatRd, the coefficient of log Rd in any OLS regression of log jbar on (log Mbar, log Rd) must equal 1+β exactly by algebraic construction. Six results follow. (i) At fixed mass, incorporating Rd reduces the scatter in jbar by 28%, versus 6% when incorporating fgas alone; Rd provides the dominant structural information. (ii) |β| increases from 0.138 at z = 0 to 0.265 at z = 2 in TNG100, following |β| ∝ Σbar0.214 as a cosmological imprint of inside-out disc growth. (iii) After simultaneous control of halo concentration c200 and spin λ, 80% of |β| remains unexplained by halo structural parameters; EAGLE independently confirms β < 0 (βEAGLE = -0.023, IC90%: [-0.036, -0.010]). (iv) The Σ-A/Σ-B/Σ-C bifurcation is dated to z = 3.0-3.5 within the resolution of the progenitor reconstruction (Cohen |d| = 0.025 ns at z = 3.5; |d| = 0.41, p = 0.010 at z = 3.0). (v) The temporal ordering of divergences reveals: stellar mass first (z ≥ 5), then Σbar (z ≈ 3.5), Rd (z ≈ 2.4), and fgas last (z ≈ 1.5); the mechanism driving this ordering remains to be established. (vi) EAGLE independently confirms that |β| increases with z via direct OLS regressions, and the temporal ordering is consistent with M☆ and Σbar diverging no later than Rd, establishing generic behaviour across different AGN feedback prescriptions.