HopfionSeries XIV - The Thick-Torus Profile Correction to the Density-Feedback Hopfion

The density-feedback Faddeev–Niemi Hopfion framework derives vEW from a single input TCMB with a 0.69% residual, identified in Paper IV as an O(R0−2) correction from the finite toroidal geometry at the physical torus radius R0 = 3. This paper sets up and solves the exact perturbation theory for this correction in six stages, all completed, reducing the residual from +0.69% to 0.012%. Stage 1. The O(1/R0) correction is proved to vanish identically: the leading curvature term in the toroidal Laplacian is proportional to cos χ, whose azimuthal average ⟨cos χ⟩ = 0 kills every O(1/R0) contribution to every observable. The leading correction is therefore O(R0−2), confirming the identification of Papers I and IV. Stage 2. The metric (volume) correction at R0 = 3 is proved: the toroidal Jacobian (R0 + ρ cos χ)/R0, after averaging ⟨cos2 χ⟩ = 1/2, shifts the profile-integral ratio by δ(I4/I2a) = −0.37%. Stage 3. The profile correction f2(ρ) is proved to satisfy a fully determined Sturm–Liouville BVP ℒ0[f2] = S2(ρ), f2(0) = f2(∞) = 0, with Jacobi potential V″(f0) = C*2 cos(2f0)/ρ2 and source S2 = (C*/2R02) sin f0(1 − C* cos f0), derived in closed form from the modified BPS ansatz with no free parameters. The BVP is solved numerically (N = 8000 log-spaced points): peak |f2|max ≈ 0.212 at ρ ≈ 1, sign change at ρ ≈ 2.5, and asymptotics f2 ~ ρ3.97 near 0 and f2 ~ ρ−1.43 at large ρ, both verified to 0.1%. Stage 4. The full 3D Faddeev–Niemi profile integrals are evaluated in exact closed form via three azimuthal averaging identities (⟨r⟩ = R0, ⟨1/r⟩ = 1/√(R02 − ρ2), ⟨1/r2⟩ = R0/(R02 − ρ2)3/2), giving the closed-form toroidal correction formula J43D = R0 I4 + (φ8+1) I2ator, J2a3D = R0 I2a + (φ8+1) I4tor, where the toroidal winding parameter Qt2 = φ8+1 = 47.979 equals the tangential effective-mass numerator of Paper XIII. The cross-coupling structure — azimuthal winding feeding J4 and Hopf density feeding J2a, swapped from the 1D assignment — resolves the factor-of-4.2 discrepancy between thin-torus and full 3D computations. At C* = 3.4318, R0 = 3 the formula gives J43D/J2a3D = 0.22717 versus WZW target 0.22721 (0.018% residual). Combined with the Paper I WZW relation d ln ye/dC* = 1/(2k) = 1/6, this gives a Yukawa correction δye/ye = +0.68%, reducing δvEW/vEW from +0.69% to 0.015% and δΛQCD/ΛQCD from +0.97% to 0.015%. Stage 5. The condensate's Q = 0, J = 0 sector (Paper XIII spin tower J = Q/4) is the scalar Higgs sector. The quartic self-coupling is derived as λ = (k+2)/2 × rWZW2 = 5 · 25/3/φ10 = 0.12907, where rWZW = J4/J2a = 24/3/φ5 is the WZW profile ratio and (k+2)/2 = 5/2 is the torus four-point normalization. The normalization factor 5/2 is proved via the KZB monodromy (Falceto–Gawędzki 1996): on the torus, the b-cycle monodromy of the conformal blocks is the modular S-matrix, which converts plane OPE weights (Cj)2 dj to torus modular weights dj2 sin2(π/(k+2)). These sum to (k+2)/2 by the pentagon Parseval identity ∑n=1k+1 sin2(nπ/(k+2)) = (k+2)/2, a consequence of the (k+2)-th roots of unity summing to zero. The resulting Higgs mass is mH = √(k+2) × rWZW × vEW = √5 × (24/3/φ5) × vEW = 125.10 GeV (measured vEW), 125.11 GeV (framework vEW) 0.13σ and 0.04σ respectively from the ATLAS combined Run 1+2 measurement 125.11 ± 0.11 GeV. Stage 6. The O(R0−4) profile correction from the BVP solution f2(ρ) cos(2χ)/R02 is computed by full 2D Gauss–Legendre quadrature. The first-order f2 correction vanishes identically (⟨cos(2χ)⟩ = 0); the leading contribution is second-order, from ⟨cos2(2χ)⟩ = 1/2 and the geometry–profile cross-term ⟨cos(2χ) cos2 χ⟩ = 1/4. The corrected 3D ratio r3D(2) = 0.227164 (gap −0.022% from rWZW), with scaling gap × R04 = 1.8% confirming the O(R0−4) identification. The vEW residual reduces from +0.015% to +0.012%, a 20% improvement. The convergence hierarchy is: 0.69% → 0.015% (46× reduction at O(R0−2)) → 0.012% (1.25× at O(R0−4)). The remaining +0.012% is consistent with O(R0−6) corrections at ~0.001%, below the 0.013% me residual from the MS-bar-to-pole scheme conversion. The same profile ratio rWZW governs the Yukawa coupling, the electroweak VEV, and the Higgs self-coupling: all three close to 0.015% via the identical thick-torus correction, refined to 0.012% at second order.