The Unity Clock: Effective Dimensional Collapse of the Addition-Multiplication Commutator on Coprime Lattices

We prove computationally that the commutator C = Dₛym, P_τ — measuring the tension between additive distance and multiplicative inversion on coprime residue lattices — undergoes universal effective dimensional collapse: although its numerical rank grows with lattice size, exactly 4 singular values capture ≥98% of the Frobenius energy at every primorial level from m = 30 (n = 8) through m = 510, 510 (n = 92, 160). KEY RESULTS: - Effective Rank Theorem: The top 4 modes capture 98. 6% of ‖C‖² at m = 30, 030 (n = 5, 760) and 99. 77% at m = 510, 510 (n = 92, 160), verified via matrix-free randomized SVD in 15 seconds on commodity hardware. - Subspace Alignment: The coupling Hamiltonian Hcoup = cos (2πd/m), governing the Rabi phase transition at αc = √ (135/88), projects ≥99. 99% onto C's top 4 left singular vectors despite having zero Pearson correlation with C. - Universality: The effective rank = 4 holds for all squarefree products of ≥3 distinct primes, not just primorials. Ablation studies removing primes 2, 3, or both have no effect on the mode count. MECHANISM: The 1/k² Fourier decay of the tent distance function selects 2 dominant harmonics; each splits into 2 modes via τ-parity (±1 eigenspaces), yielding 2 × 2 = 4 modes universally. COMPUTATIONAL IMPLICATION: The n-dimensional eigenproblem reduces to a 4×4 effective system — a compression factor of 5×10⁸ at n = 92, 160. Companion papers: 1 "Active Transport on the Prime Gas" — DOI: 10. 5281/zenodo. 192432582 "The Arithmetic Black Hole" — DOI: 10. 5281/zenodo. 194420063 "The Universal Two-Prime Formula" — DOI: 10. 5281/zenodo. 19210625