Domain-First KL Rigidity of the Higgs Potential: A Positive-Domain Log Barrier and the Prediction κ_λ^vac = −1/3

抄録

Higgs vacuum phenomenology is usually organized by vacuum-local polynomial ansätze—the renormalizable Standard Model form and polynomial effective-field-theory deformations—and then tacitly interpreted as if they characterized the global vacuum sector. Such truncations are locally valid, but they do not by themselves determine the completion status or infinite-distance character of the gauge-invariant positive radial boundary ρ ≡ 2H†H → 0⁺. We formulate a test-facing, domain-first shape hypothesis on the interior ratio domain ψ ≡ ρ/v² > 0, with the electroweak vacuum anchored at ψ = 1. Admissible interior shapes are restricted by equilibrium anchoring, strict convexity, and a minimal two-type homogeneous marginal closure. The continuous homogeneous marginal types of degrees 0 and −1 are respectively constant and reciprocal, motivating ∆′(ψ) ∈ span{1, ψ⁻¹}. Within this declared class, the generator is fixed up to scale to ∆(ψ) = a(ψ − 1 − ln ψ), with a > 0. The selected generator induces the Hessian scale metric ds²_∆ = a dψ²/ψ² = a d(ln ψ)², under which ψ → 0⁺ lies at infinite distance from every interior point. The same limit also lies at infinite potential excess and infinite anchored Bregman separation. These consequences follow from the selected generator, not from positivity alone. In unitary gauge, ψ = (1 + h/v)², so the finite additive endpoint h → −v⁺ corresponds to ln ψ → −∞. After fixing the normalization with (m_h, v), the selected shape correlates the near-vacuum derivative tower and yields κ_λ^vac ≡ λ₃/λ₃^SM = −1/3. Comparison with data is performed through κ_λ^vac → κ_λ^tmpl → κ_λ^fit, with a declared finite confounder set and matched truncation. The primary falsification handle is the triangle–box interference pattern in gg → hh, tested through interference-sensitive differential information rather than inclusive rates alone. The proposal therefore constitutes a falsifiable global shape hypothesis for the Higgs vacuum sector.