The structural equations below are geometry-first consequences of the BCT ontology. They follow from boundary structure, admissibility constraints, and closure geometry rather than fitted particle parameters, force laws, or field postulates.
In \(a = GM/r^{2}\), the combination \(GM\) is not an acceleration. It has units \([GM] = \mathrm{m^{3}\,s^{-2}}\): volume per ordering-parameter squared.
Ordinary acceleration appears only after this volumetric quantity is divided by spherical area, \(4\pi r^{2}\). BCT starts with the volumetric acceleration directly; Newton reaches it indirectly by introducing kilograms, then cancelling them through \(G\).
Choose a volumetric closure geometry. Write enclosed volume as a function of the ordering parameter. Differentiate twice. Solve for boundary acceleration.
BCT does not reject Einstein. It reinterprets GR as a powerful projection compression. The closure laws above are the native BCT statement. In compact weak-field systems, the same throughput structure can be compressed with extraordinary success into metric language.
That is why Einstein's tested compact-regime predictions remain. But the metric is the shadow, not the source: BCT's native object is boundary throughput, not curved spacetime as ontology.
The same closure law has two gravitational readings. The sign of the volumetric term sets the regime; the geometry sets the falloff.
Compact systems — solar systems, planets, stars, and other localized bodies — sit in the compact regime, so their R³ projection is the familiar Newton/GR limit. Extended systems — especially galaxies — use the same closure law in the saturated regime: once tangential routing saturates across the disk, the volumetric contribution reverses regime, while the kinematic contribution remains inward and follows the slower extended-geometry falloff.
A spiral galaxy is a mixed closure object: the bulge is read through the spherical law, while the flattened disk is read through the cylindrical law. The equation above is a simplified overlap picture, not a full sourced-field solve; real galaxies require morphology-weighted superposition.
In BCT, matter does not generate gravity as a source force. Persistent matter fixes where coherence must be processed: dualcores anchor curvature inventory; monocores provide the compact-regime \(V \to T\) throughput route. Newtonian mass is the compact-regime projection of that combined architecture.
In compact systems this compresses to \(GM/r^2\). In extended systems, the same closure architecture enters a saturated routing environment, so the gravitational reading changes regime toward \(T \to V\) recirculation instead of requiring a new force or dark-matter source term.
Selected SPARC galaxies fitted with the BCT stars-only model. No dark matter halo, no modified gravity parameter — only the stellar mass-to-light ratio Υd (and Υb where a bulge is present) is fitted per galaxy. All 20 achieve χ²/dof < 2.1. Median χ²/dof: 1.11.
Every SPARC galaxy with ≥5 data points, fitted identically. The representative result is the full 165-galaxy scan rather than the selected demonstrator: 73/165 galaxies fall below χ²/dof < 2, the median χ²/dof is 2.39, and the same-data BCT stars-only ansatz wins against MOND on 123/165 galaxies.
Mass-to-light ratios remain in a normal range: median Υd = 0.61, with 97 galaxies in Υd ∈ [0.2, 0.8].
Paper 3 also reports a morphology-resolved companion benchmark using three fixed-shape kernels derived from the closure-law architecture and the full baryonic profile, including gas. This is a downstream validation layer, not a replacement for the closed-form stars-only ansatz. On the same 165-galaxy SPARC sample, it improves the median χ²/dof to 1.44, places 97/165 galaxies below χ²/dof < 2.0, wins against MOND on 121/165 galaxies, and wins against the stars-only ansatz on 118/165 galaxies. The same comparison also widens the MOND margin, with median Δχ² increasing from 0.16 to 0.67.
Gas sensitivity is deliberately separated from the headline ansatz. Full gas weighting in the stars-only ansatz worsens the median χ²/dof from 2.39 to 3.07, while a modest ~25% gas contribution improves the gas-rich tail to a median of 2.00 without changing the stellar-dominated majority.
The Baryonic Tully-Fisher Relation is presented in Paper 3 as a boundary-matching result at the compact/extended transition. Velocity continuity forces the scaling vflat4 ∝ Mb, with the asymptotic BCT expression vflat4 = (4/3) G Mb atrans.
Head-to-head comparison on the full SPARC sample using identical data, identical stellar input, and identical Υ fitting. The BCT stars-only ansatz fits better on 123 galaxies; MOND fits better on the remaining 42.
Scope note: these results support the reduced asymptotic-matching ansatz and the morphology-resolved supplement extension. They do not close the full sourced-field, lensing, cluster, or cosmology programmes.
Every entry below is derived from the 2T+1V boundary architecture and the upstream commitments established in Papers 1 and 2. No new ontological primitives are introduced. Quantitative results shown in earlier sections are not repeated here.