From classical mechanics through the Standard Model — every partition function a finite sum, every spectrum a finite set of computable eigenvalues, every topological invariant a computable integer. Ten Parts building physics from first principles on finite foundations, with every claim grounded in experiment.
No laboratory has ever measured a position to infinite decimal places. No computer has ever integrated a differential equation over the continuous real line. No experiment has ever probed an infinite-dimensional Hilbert space. The infinite mathematical apparatus of modern physics — smooth manifolds, Lebesgue integrals, L²(ℝ), the path integral — is theoretical scaffolding from which finite predictions are extracted.
This companion volume works directly with the finite predictions. It constructs the physics on BST's finite foundations and shows that the results match experiment at every precision achieved.
Each Part depends only on earlier Parts and on the AFB paper (Parts III–XIII). The arc follows physics from the simplest systems to the most complex: mechanics → waves → heat → quantum → forces → geometry → gravity → materials → the Standard Model → open problems. Every Part opens with a plain-language orientation grounded in experiment before the formal development begins.
Every prediction verified to the highest precision was computed by finite methods — numerical ODE integration, finite matrix diagonalisation, finite sums of Feynman diagrams, lattice Monte Carlo. BST works directly with the finite predictions.
Voyager 2's Neptune flyby required trajectory accuracy of a few km over 4.4 billion km — computed by Runge-Kutta integration on a finite grid.
GPS satellites correct for time dilation at ~38 μs/day. Without the correction, positions drift ~10 km/day. Computed by evaluating γ = 1/√(1−v²/c²) at specific numerical values.
The Ising model's critical exponents — computed by transfer matrices and Monte Carlo on finite lattices — match experimental measurements in magnetic materials to ~0.1%.
The hydrogen spectrum, computed by diagonalising a finite Hamiltonian matrix, is confirmed spectroscopically to 12 significant figures — one of the most precise agreements in science.
The proton mass (938.3 MeV) computed by lattice QCD — the exact finite-sum partition function this Part constructs — agrees with experiment to sub-percent precision.
Hall conductance quantised to 1 part in 10⁹. The quantisation is explained by the Chern number — an integer topological invariant computed by finite linear algebra.
The waveform from merging black holes matches numerical relativity simulations — discrete Einstein equations on a finite computational grid — to within detector precision.
Bell inequality violation S = 2√2 confirmed by Hensen et al. (2015) with all loopholes closed. In BST: an exact 4×4 matrix computation over ℂ_B(k⁴).
The most precisely measured quantity in physics. The SM prediction from ~12,000 Feynman diagrams — each a finite sum on a finite momentum lattice — agrees to 1 part in 10¹².
Classical physics uses smooth manifolds, infinite-dimensional Hilbert spaces, and functional integrals as theoretical frameworks. But every experimental prediction extracted from these frameworks is a finite number computed by a finite method. BST works with the finite computations directly.
"Nothing in the experimental record requires the infinite real line. What it requires is enough precision — and BST provides that by parameterising precision explicitly." — Bounded Finite Physics, Companion Volume 2026
Ten Parts. From harmonic oscillators to the Standard Model. 88 definitions, 29 theorems, 22 open problems, and 9 Parts grounded in experiment. Every result tagged by recovery type (I–IV) and maturity tier (1–3). The companion imports only from Parts III–XIII of the AFB paper — it does not modify the foundations.
"The physics does not change. The foundations become honest." — Bounded Finite Physics, Companion Volume 2026