WP36: the J-routing mechanism of the reciprocal law - why th_u(phase).sin(phase) is class-constant
WP35 established the chart law th_u(phi) = 0.0389/sin(phi) - 0.0001 for the mixed u-singlet branch (equivalently H_12.sin(phi) = 3.621e-2, cross-class). This package derives the mechanism. Structural trigger: ALL 303 u-class sheets place the phase edge in the DOWN sector, so Yu (and Hu) is real and carries no explicit phi - the law cannot be a per-sheet algebraic identity; it must be a readout constraint.
Method (checker checkers/wp36_j_routing.py, results
results/wp36_j_routing.json, 303 u-class sheets, 5/5 gates).
Findings
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Exact commutator identity (G1, max err 2.1e-34 on all 303 sheets). In the canonical frame (singlet s at index 0, block [[a,c],[c,b]]), writing u = Hd_01, v = Hd_02, w = Hd_12:
det[Hu,Hd] = 2i ( c Im(w) (|C1|^2 - |C2|^2) - Im(C1 C12 conj(C2)) ), C1 = u(s-a) - c v, C2 = v(s-b) - c u, C12 = w(a-b) + c(Hd_22 - Hd_11).The c^0 piece is -Im(u w conj(v)) (s-a)(a-b)(s-b): the DOWN-SECTOR LOOP PRODUCT Hd_01 Hd_12 Hd_02* - a second, independent J source living entirely in the down chart.
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J-routing branch theorem (G2, exact). Zeroing c = H_12 kills J on every mixed sheet (J -> 3.5e-18 relative) and leaves J exactly unchanged on every unmixed sheet (deviation 0.0). So the physical J is generated by two DIFFERENT chart structures: on the unmixed branch it is the down-loop product; on the mixed branch it is the c-linear terms of the identity. The WP35 mixed/unmixed split is the discrete choice of J's chart carrier.
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First-harmonic exactness (G3, max rel err 4.3e-16). Per sheet, every Im(Hd_xy) scales exactly as sin(phi) under phi rescaling at fixed magnitudes: Im(Hd_xy) = (magnitude product) . sin(phi), no higher harmonics (this is the single-phase-edge structure made exact).
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Magnitude-only factorization (G4, max rel variation 3.6e-11). Per mixed sheet, det[Hu,Hd] = -2i . c . sin(phi) . M(mags) with M phi-independent at fixed magnitudes: no cos(phi) contamination. Combined with the Jarlskog identity det[Hu,Hd] = -2i J Du Dd this is the exact per-sheet theorem
J . Du . Dd = c . sin(phi) . M(mags). -
Emergent class constancy (G5). Since the fit pins J and the masses, the theorem forces q = c.sin(phi_folded) = J.Du.Dd/M. Measured: q = 0.0362132 with cross-class relative spread 2.9e-5 - the WP35 reciprocal law recovered as the emergent, class-level constancy of M, not as a per-sheet identity. (M's own raw cross-class spread, 9.1e-3, is dominated by fit jitter of Du through the tiny difference yc^2 - yu^2; within-class spread is fit phi-jitter acting on the smooth law.) The chain is complete: J fixed + M class-constant => c.sin(phi) const => th_u = c/Delta_block (1 + O(th^2)) obeys the reciprocal law. The b = -1.3e-4 intercept of WP35 is the O(th^2) atan2 correction plus subleading identity terms.
Typing
G1-G4 are exact chart-atlas theorems (machine floor); G5 is an emergent ensemble law at 2.9e-5. The mechanism does NOT restore physical invariance of the loop phase (WP11 stands): phi remains chart data, and the law lives on the fit-solution manifold. What WP36 adds is that the chart atlas fibers the physical readout through an exact identity with a discrete J-routing bit - the two branches are two chart presentations of the same physical J, precisely the chart/quotient distinction the program tracks elsewhere.
Verification
./.venv/Scripts/python checkers/wp36_j_routing.py reproduces
results/wp36_j_routing.json (5/5 gates).