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 HPAPERBANCHOREQUATION carries status VALIDATED at posterior 0. 956 (geometric mean of components) under ALLGATESOPEN gating policy. The runtime cube state separately reports the parent at posterior 0. 994; both reports agree on five-component VALIDATION at threshold. The HPAPERB₂ evidence base is built on a four-pillar decomposition: operator consistency (test 5₂4, curl identity verification), vorticity realization (test 5₂5, evolved substrate field carries divergence-free rotational structure with curl over divergence ratio at 15 to 16 orders of magnitude), structural correlation (test 5₂6, J-to-V spatial Pearson r approximately 0. 95 and spectral Pearson r approximately 1. 0), and interventional causality (test 5₂7, 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 (λc, λfold, λcold, 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 λc scales linearly with the ratio a/b (λc = 0. 0592· (a/b), Pearson r = 0. 99999 across five a values). In contrast, λc 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 ∂λc/∂ (ξ, α, D, a/b) spans seven orders of magnitude. λfold and λcold 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.