Domain-First KL Rigidity of the Higgs Potential and a Sign-Reversed Trilinear Coupling: From Mexican Hat to Witch’s Hat

抄録

Higgs vacuum phenomenology is usually organized by vacuum-local polynomial ansätze—the renormalizable SM form and polynomial EFT deformations—and then tacitly interpreted as if they characterized the global vacuum sector. Such truncations are locally valid, but they do not encode the gauge-invariant domain of the Higgs radial invariant ρ≡ 2H†H ≥ 0 or its genuine boundary at ρ→ 0. We propose a test-facing, domain-first shape hypothesis on the interior ratio domain ψ ≡ ρ/v^2 > 0 (vacuum at ψ = 1). Admissible interior shapes are restricted by a selected specification: anchoring at the vacuum, strict convexity, and a minimal closed drift span. Within this declared class, the excess-to-equilibrium functional is fixed up to scale to the log-barrier convex-excess form ψ− 1− ln ψ, implying a “Witch’s hat” barrier as ψ → 0+. The boundary point is not completed inside the hypothesis class; validity is tracked only through an explicit operating-envelope protocol. After fixing the normalization with the measured inputs (mh,v), the near-vacuum derivative tower is fully correlated and yields the theory-side target κλvac ≡ λ3/λ3SM =−1/3. Comparison to data is performed through an explicit interpretation layer, κλvac → κλtmpl → κλfit , with a declared minimal confounder set and matched truncation. The primary falsification handle is the triangle–box interference pattern in gg → hh, probed through interference-sensitive differential information rather than inclusive rates alone. The outcome is binary: exclusion of the mapped negative target interval associated with κλvac =−1/3 rules out the hypothesis; otherwise it survives the primary gate and proceeds to post-survival robustness audits.