A Minimal Static Synthesis of Topological Matter, Substrate Depletion, and Eective Conformal Geometry Paper III of the VFT Admissibility Programme

We construct the minimal static eective-eld synthesis suggested by the two preceding ad- missibility audits of the Verarmungsfeldtheorie (VFT) programme. A real scalar substrate eld χdenes an exponential conformal factor A (χ) = e−α (χ−χ0) on a xed three-dimensional bookkeeping metric δij. A topological Skyrme eld U: R3 → SU (2) propagates on the ˜ substrate-built spatial metric hij= A2 (χ) δij and acts as a source for χ through the confor- mal coupling alone, with no explicit depletion term. The static energy functional couples a quadratic substrate sector to the two Skyrme energy densities under the sectoral weighting E2 →AE2, E4 →A−1E4. We report four results obtained on the spherical B = 1 hedgehog sector at χ0 = 1, mχ ∈0. 5, 1, 2. (i) A at-anchor limit at α→0 reproduces the Paper I Stage-B Skyrme minimiser with χ≡χ0, virial residual below 5×10−4, and topological charge preserved to numerical precision. (ii) A perturbative coupled branch exists for α≤0. 1 across the three substrate masses, with linear core response φ (0) ∝−αand slopes that depend mono- tonically on mχ but do not collapse onto a single eective coupling. (iii) For the central case mχ = 1, mesh-inherited continuation and multi-guess probes verify a single smooth solution branch up to α= 0. 3, with |∆E|spread∼10−10 over four distinct initial proles at four tested couplings. The coupled branch internalises the energy-side sectoral imbalance identied in Paper I as a dynamical Derrick-balancing mechanism: at the admissibility edge α= 0. 28, the bare sectoral imbalance (E4−E2) / (E4 +E2) =0. 115 is reduced under the conformal weighting to 0. 022, with the residual carried by E∇χ+ 3EV as required by the coupled Derrick identity to a relative accuracy of 3 ×10−4. This does not resolve the Paper I inertia obstruction: the moduli inertia functional GXX on the self-consistent prole is not computed here and remains the object of a subsequent time-dependent audit. (iv) The branch reaches a total-depletion cuto at αdep = 0. 282 (±10−3), dened by χ (0) = 0. This cuto is not a metric singularity the exponential conformal factor remains nite, A (0; αdep) ≃1. 32 and the dierential sys- tem continues smoothly past it. It is the point at which the positive-substrate interpretation of the harmonic minimal eective theory exhausts the natural scale set by χ0. Two structural diagnostics quantify the regime: the metric nonlinearity NA=max |A−1|rises only to ≃0. 32 at the cuto, while the substrate nonlinearity Nχ=max |χ−χ0|/χ0 reaches ≃0. 99. The min- imal exponential coupling thus operates as a nonlinear lter that converts strong substrate deformation into a moderate metric response. The construction demonstrates static structural closure of topological matter, substrate depletion, and eective conformal geometry; inertial closure is not claimed, and is reserved for a subsequent time-dependent audit.