The Fall Relation as the Observable Projection of an Angular-Momentum Retention Law in Disc Galaxies

The baryonic Fall relation (jbar ∝ Mα bar) is commonly treated as a universal scaling law. We show it is the projection of a two-dimensional angular-momentum retention law, log fj = aM(state) logMbar + aR(state) logRd+c, whose topology changes across the transition from evolutionary (X2) to morphologically fossil (X1) disc galaxies. Using a Gaussian mixture model (GMM) in the (Δj, n) plane applied to N = 447 MaNGA Σ-A+B disc galaxies (H26a combined catalogue), we establish eight results. (i) Two regimes: ΔBIC = +244 stable over 50 seeds. (ii) Fall variance: σ2 within = 82.4% ± 2.9%, σ2 between = 17.6% ± 2.9% (bootstrap IC95%). (iii) Fall slopes: αX2 = 0.783, αX1 = 0.969, Δα = 0.180 (IC95%: +0.077, +0.282; P(Δα > 0) = 0.9993; robust to five estimators). (iv) BTFR size coefficient: βX2 = −0.143, βX1 = −0.453 (p < 10−4 both). (v) (Δj, n) is minimal: adding Dn(4000) raises BIC. (vi) Δj ⊥ Dn(4000): discriminates the regime in 5/5 Dn(4000) quintiles (Bonferroni). (vii) PCA separates a stellar-quenching axis (PC1: 37.7%) from a kinematic axis (PC2: 16.8%). (viii) TNG100: live and fossil progenitors indistinguishable at z = 2 (ΔΔj = 0.004 dex), diverging by 0.134 dex by z = 0 (ρ = 0.548, p = 3 × 10−28). The fj law for X2 achieves R2 = 0.908, σ = 0.090 dex (−51% vs mixed Fall). The topology change is the suppression of the mass dependence in X1 (aX1 M ≈ 0, ΔaM = +0.313, IC95%: +0.164, +0.478). The identities α = 2/3 + ∂ log fj/∂ logMbar and 1 + β = ∂ log fj/∂ logRd anchor these results to the Mo, Mao & White (1998) framework.