Burdick Crag Mass Paper B v2.0: Five-Component Compositional Validation of the Anchor Equation with Four-Pillar Vorticity Evidence and 175 SPARC Galaxies

Version 2.0: Burdick Crag Mass (BCM) Paper B. Establishes the Anchor Equation E equals (M times Phi of sigma) times c squared plus the closed contour integral of J dot dOmega over the Aleph-Null contour, as a compositional theorem through five-component validation in the BCM Bayesian hypothesis engine. All five sub-hypotheses carry status VALIDATED at v27 cycle 3 close (2026-05-03), in the publication-reproducible consolidator: Phi-sigmoid form fits substrate response (posterior 0.939, 30 R3-independent events), J carries vorticity in the dual-pump topology (posterior 0.981, 103 R3-independent events from a four-pillar evidence chain), the Aleph-Null loop integral converges (posterior 0.938, 30 events), the recovery limit holds with Phi approaching 1 and J approaching 0 at large sigma (posterior 0.936, 28 events, GATE component), and the M-sigma inversion mapping is geometrically continuous (posterior 0.989, 176 events). The compositional parent H_PAPER_B_ANCHOR_EQUATION carries status VALIDATED at posterior 0.956 (geometric mean of components) under ALL_GATES_OPEN gating policy. The runtime cube state separately reports the parent at posterior 0.994; both reports agree on five-component VALIDATION at threshold. The H_PAPER_B_2 evidence base is built on a four-pillar decomposition: operator consistency (test 5_24, curl identity verification), vorticity realization (test 5_25, evolved substrate field carries divergence-free rotational structure with curl over divergence ratio at 15 to 16 orders of magnitude), structural correlation (test 5_26, J-to-V spatial Pearson r approximately 0.95 and spectral Pearson r approximately 1.0), and interventional causality (test 5_27, perturbation propagation Pearson r equals 0.96 with null-field collapse to 17 percent of baseline vorticity). The 176-event H5 evidence base derives from a SPARC scale-up across 175 galaxies (Lelli et al. 2016c rotation-curve catalog, V-component scaling by sqrt of 1 plus or minus 0.01, strict local monotonicity test, one R3-clean event per galaxy). The validation is presented at v1.0 Layer A spectral invariant commitment level. A Layer B open question on the bridge-projection chain from Cube 2 into Cubes 7, 8, and 9 (mean absolute discrepancy 0.485 in v19 corpus subset, structurally diagnosed as non-injective state-vector underdimensioning) is documented in Section 5 as the primary open question for Paper B v3. Builds on Paper B v1.0 (DOI 10.5281/zenodo.19700387) and the Paper A SPARC fits (DOI 10.5281/zenodo.19680280). Version 1.0: Six numerical probe experiments on the Burdick Crag Mass (BCM) substrate wave PDE characterize the control operators governing the four-field state tensor (λ_c, λ_fold, λ_cold, H). Perturbations along four axes — the nonlinear coupling coefficient ξ ∈ [0.005, 0.07], the Taylor source coefficient a with saturation b held fixed, the memory regularization α ∈ [0.0, 0.95], and the diffusion coefficient D ∈ [0.1, 2.0] — resolve a clear operator hierarchy. The spectral eigenvalue λ_c scales linearly with the ratio a/b (λ_c = 0.0592·(a/b), Pearson r = 0.99999 across five a values). In contrast, λ_c is invariant under ξ to within 10^-9 relative range across 14× variation, invariant under α to within 10^-4 relative range across the converged α subset, and exhibits only a weak shear under D (~1.9% relative range across 20× variation). The Jacobian ∂λ_c/∂(ξ,α,D,a/b) spans seven orders of magnitude. λ_fold and λ_cold are invariant under all four axes. The hysteresis loop H ≈ 0.05 is not reduced by any tested axis and is attributed by elimination to absorbing-state asymmetry of the cold-torus terminal phase. The framework is presented as a non-equilibrium effective-field reconstruction with emergent invariant spectrum; no phase transition or saddle-node fold bifurcation is claimed.