Burdick Crag Mass: A Substrate Wave Model for Galactic Rotation Curves Without Dark Matter — v7 with Anchor Partition Regime Map, ROOTREENTRY Anatomy, and Macro-Torsion Operator - Tully-Fisher Exponent Stability

Version 7. 0: Paper A of the Burdick Crag Mass (BCM) framework. All results from v6 are retained. Version 7. 0 adds the Anchor Partition Regime Map, ROOTREENTRY galaxy anatomy, and a structurally regime-safe Macro-Torsion operator tested against the complete SPARC175 sample. Core BCM performance (retained from v6): BCM beats Newtonian baryonic fits on 130 of 175 SPARC galaxies (74. 3%) and beats MOND on 172 of 175 (98. 3%). Mean reduced chi-squared: BCM 63. 1, Newton 157. 6, MOND 1122. 6. Four frozen global parameters plus one amplitude coefficient per galaxy. Tully-Fisher exponent alpha = 3. 42 (95% CI 3. 03, 3. 88), with bracket-stratified breakdown at the massive velocity bracket (alpha = -0. 29, CI fully below classical) identifying a scale-dependent substrate coupling transition. Version 7. 0 additions: The Anchor Partition Ratio (APR) is applied to all 175 SPARC galaxies as a direct decomposition of observed rotation curves against Newtonian baryonic predictions, requiring no BCM solver execution. APR = (Vobs squared minus Vₙewton squared) divided by Vobs squared, computed point-by-point from published SPARC data. Across 175 galaxies, the APR median is 0. 693 and 152 of 175 galaxies (86. 9%) sit on the substrate-dominant side of the decomposition. Six empirical APR regimes are identified: MASSFLOOR (23 galaxies, Newton explains the curve), DWARFINTERMEDIATE (18 galaxies), SUBSTRATEPLATEAU (62 galaxies, vmax 50-150 km/s, APRₒuter median 0. 77, BCM wins 91. 9%), MIXEDTRANSITION (29 galaxies), SUPPRESSIONVALLEY (37 galaxies, vmax 150-300 km/s, APRₒuter median 0. 47, Newton wins 75. 7%), and ROOTREENTRY (6 galaxies, vmax above 300 km/s, APRₒuter median 0. 60, BCM underfit). The regime assignment is determined by vmax and APRₒuter without solver execution. Joining the regime labels with archived BCM solver records confirms the regime map predicts BCM performance: SUBSTRATEPLATEAU recovers 91. 9% BCM wins and SUPPRESSIONVALLEY recovers 75. 7% Newton wins. A sharp APR discontinuity is confirmed at 125 km/s: mean APRₒuter below 125 km/s is 0. 6179 versus 0. 4248 above, a delta of +0. 193. This boundary aligns independently with the Tully-Fisher exponent transition reported in v6 (peak bracket alpha at mid-velocity before collapse at massive bracket), constituting two independent measurements identifying the same substrate transition in the SPARC data. The six ROOTREENTRY galaxies (ESO563-G021, NGC2841, NGC5985, UGC02487, UGC02885, UGC02953) carry APRₒuter between 0. 52 and 0. 76, indicating substantial outer-disk substrate signal. However, the current BCM solver underperforms Newton on these systems (mean BCM outer RMS 94. 0 versus Newton outer RMS 78. 7). Anatomical analysis of each galaxy confirms the substrate signal is outer-disk concentrated in 5 of 6 systems, and the solver failure is largest in the outer radial zone. UGC02487 (vmax = 383 km/s) is the sole ROOTREENTRY BCM win and is identified as the primary calibration target. A Macro-Torsion operator is proposed to address the ROOTREENTRY outer-disk underfit: T = eta times Theta (vmax minus 300 km/s) times the absolute value of (partial vₚhi over partial r minus vₚhi over r). The Heaviside function at vcrit = 300 km/s gates the operator: it fires only for ROOTREENTRY systems and produces zero effect on all other regimes. Tested against all six APR regimes, the operator shows 53% RMS sensitivity for ROOTREENTRY and exactly 0. 000% sensitivity for every other regime across the full eta sweep, confirming structural isolation. The operator reduces ROOTREENTRY mean outer RMS from 113. 5 to 58. 4 at first-probe parameters. It is not yet integrated into the BCM solver; it is presented as a confirmed directional extension with a falsifiable prediction: the six ROOTREENTRY galaxies should show improved outer-disk residuals when the solver is extended with this term. A companion Volume Dilatant operator was tested and rejected as formulated: it degraded control regime performance catastrophically (SUPPRESSIONVALLEY outer RMS increased 305%) due to per-galaxy normalization that removed mass discrimination. A physically anchored reformulation is reserved for a future version. All theoretical concepts, dimensional ontology, experimental design, and originating insights belong solely to Stephen Justin Burdick Sr. Computational processing was performed at SJB's direction using AI tools. Emerald Entities LLC — GIBUSH Systems. Version 3. 0: A substrate-based phenomenological alternative to cold dark matter in which the dark matter signal is modeled as the energy maintenance budget of a pre-existing 2D spatial substrate. Using four global parameters (lambda=0. 1, kappa=2. 0, grid=256, layers=8) and zero per-galaxy tuning, we classify 175 SPARC rotation curve galaxies into six physically distinct substrate interaction states. On 29 representative galaxies, BCM achieves a mean reduced chi-squared of 42. 1 compared to NFW's 92. 1 (two fitted parameters per galaxy) and MOND's 1630. 4. The framework generates falsifiable predictions for IceCube/KM3NeT neutrino flavor gradients, Betelgeuse tachocline recovery, and binary mass transfer rates. All code and data: https: //github. com/Joy4joy4all/Burdick-Crag-Mass Version 4. 0: Preprint presenting a substrate wave model for galactic rotation curves using four frozen global parameters and zero per-galaxy tuning. Reduced chi-squared comparison against NFW and MOND on 29 SPARC galaxies: BCM wins 13/29 (mean χ²=42. 1 vs NFW 92. 1), dominates massive bracket 7/12 (avg χ²=62. 7 vs NFW 241. 8). MOND wins 0/29. The substrate maintenance luminosity scales as L_ν ~ vₘax³. 9, independently recovering the baryonic Tully-Fisher relation (L ~ v⁴; Tully & Fisher 1977) from the same frozen parameters — this scaling was not an input. All 29 galaxies remain sub-Eddington (max L_ν/LEdd 200 km/s, alpha = -0. 29, CI fully below classical). This scale-dependent transition motivates a theoretical extension in a forthcoming companion paper. Paper A's claims do not depend on that extension. Content retained from v5: six-class substrate taxonomy, Einstein coupling with stress-energy tensor Tₘuₙu, null pump proof (sigma to zero without SMBH injection, 5 of 5 essential), boundary dynamics with sigmacrit saturation clamp, three perturbation regimes, Pi = sigmaₑdge over sigmacrit control parameter, binary stellar dynamics on Alpha Centauri, HR 1099, and Spica, and falsifiable predictions for IceCube/KM3NeT flavor gradients, Betelgeuse tachocline recovery (2030-2031), and a 225 dollar benchtop device. Production run: 175 galaxies at grid=256 with 25000 integration steps per galaxy, 148. 6 minutes wall time on a single NVIDIA GPU. Source data: BCMᵥ26TullyFisherAmplitudeScalingAblation (Test 2 revision 6) and BCMᵥ26TullyFisherExponentStability (Test 4), both completed 2026-04-21 and referenced by timestamp in the paper.