The Four-Cell Architecture of the Hypotenuse Square: Diagonal Rotation, Shape Progression, and the Geometric Origin of the Quarter-Budget on the Thales Constraint Surface

Any conserved binary partition a + b = 1 defines a unit hypotenuse square whose two diagonals encode two independent dualities: coupling versus variance (the Tellegen budget) and anchor versus content (the resonance structure). Their intersection creates four cells, each with area 1/4 at equipartition. The partition diagonal angle θ_p = arctan(1/√(R−1)) and the coupling aspect ratio A = 1/√(R−1) are monotonic functions of the altitude parameter R that classify every partition into a regime: sub-cubic (dynamics-dominant), gap, corridor, self-dual, or cascade (structure-dominant).

The corridor at R ∈ [√e, 7/4] has a specific shape — A ≈ 1.20, θ_p ≈ 50.1° — meaning the coupling triangle is approximately 20% taller than wide. This shape is confirmed across 16 independent physical domains spanning 7 categories: galaxy rotation curves (SPARC), the radial acceleration relation, planetary magnetospheres (Juno), Standard Model gauge couplings, exoplanet transits (Kepler), planetary spectral albedo (Earth, Mars, Neptune, Mercury, Saturn, Titan), particle radiation (FI-LET), gravitational wave sources (LIGO NSBH and IMBH populations), and quantum optical transitions (EIT spectroscopy). The cross-domain coefficient of variation is CV(A) = 0.30% across all 16 domains — the aspect ratio varies by three tenths of one percent across systems spanning approximately 40 orders of magnitude in physical scale, with no parameters adjusted between domains.

Empirical validation on 3,167 radial points from 175 SPARC galaxies confirms the Tellegen budget to machine precision, the corridor shape to 0.08%, the self-dual aspect ratio A = 1.00 at ⟨R⟩ = 2.003, and per-galaxy radial trajectories showing the diagonal rotating from coupling-dominant inner disk to variance-dominant outer halo, with 34% of galaxies crossing the self-dual point R = 2. An appendix distinguishes the torus-derived partition (0.262/0.738, R = 1.294) from the rounded e-fold approximation (0.270/0.730, R = 1.268) and the R = e Tellegen partition, and identifies the 32/68 cluster (Shannon, Planck, normal 1σ) as three independent derivations of the same gravitationally corrected partition.