Modular Phase Geometry of Semiprime Residue States: Exact Character Laws and Virtual Physics Assays
Abstract
Abstract We introduce Modular Phase Geometry, a finite arithmetic phase construction for semiprime residue states. The central object is the residue-square phase embedding WM(r)=(r+i(r2 mod M))/MW_M(r)=(r+i(r^2 \bmod M))/M, defined on the unit group GM=(Z/MZ)×G_M=(\mathbb{Z}/M\mathbb{Z})^\times. For M=360M=360, the unit group has ϕ(360)=96\phi(360)=96 states, while the square map has six image states, a sixteen-element kernel, and image exponent six. The product law for semiprimes is exact: if N=pqN=pq, then rN≡rprq(modM)r_N \equiv r_p r_q \pmod M, and rN2≡rp2rq2(modM)r_N^2 \equiv r_p^2 r_q^2 \pmod M. By finite abelian character decomposition, every product observable F(ab)F(ab) over GMG_M admits a separable character expansion. For the M=360M=360 embedding, the audited rank hierarchy is rank(y)=6\mathrm{rank}(y)=6, rank(x)=44\mathrm{rank}(x)=44, and rank(W)=49\mathrm{rank}(W)=49. We complement the exact arithmetic theory with virtual physics assays. An expanded coherent-scattering experiment over eleven moduli, four axial scalings, 64 probe wavevectors, 96 pages, and 127 null replicas per setting found strongest suppression at M=420M=420, z=1.00z=1.00, with actual/null intensity ratio 0.4361, z-score -3.659, and lower-tail plus-one p=0.007812p=0.007812. The theorem baseline M=360M=360 also showed strong suppression, with best actual/null ratio 0.4894, z-score -3.164, and p=0.007812p=0.007812. Additional tight-binding, wavepacket, heat-kernel, electrostatic, phonon, and…
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