kappa_lambda_4_RS_eq_one_third
plain-language theorem explainer
Under the RS cosh Higgs potential the quartic self-coupling modifier equals exactly 1/3 of the SM Mexican-hat value. Collider phenomenologists citing the A28 cosh-vs-hat resolution or the packaged BSM falsifier use this identity. The proof is pure definitional reflexivity: the constant is defined to be 1/3.
Claim. The Recognition Science kappa modifier for the Higgs quartic equals $1/3$. Equivalently, the ratio of the $h^4$ Taylor coefficients $m_H^2/(24 v^2)$ (cosh potential) over $m_H^2/(8 v^2)$ (SM Mexican-hat) is exactly one third.
background
This module resolves attack A28 by making the structural mismatch between the RS cosh Higgs potential and the SM Mexican-hat potential theorem-grade. With the bridge identification $\Lambda^4 = m_H^2 v^2$ and linear map $h = v,\varepsilon$, the cosh form $V_{\mathrm{cosh}}(h) = \Lambda^4(\cosh(h/v)-1)$ is even in $h$, so all odd Taylor coefficients vanish. Its quartic coefficient is $m_H^2/(24 v^2)$.
The SM unitary-gauge potential after EWSB is the degree-4 polynomial $V_{\mathrm{SM}}(h) = \tfrac12 m_H^2 h^2 + (m_H^2/(2v)) h^3 + (m_H^2/(8 v^2)) h^4$. The ratio of quartic coefficients is therefore $(1/24)/(1/8) = 1/3$. The named constant kappa_lambda_4_RS is defined to be that ratio.
Sibling facts record the companion trilinear modifier (exactly zero under cosh) and the genuine dim-6 sextic $\lambda_6 = m_H^2/(720 v^4)$ absent from the renormalizable SM potential.
proof idea
One-line term proof by rfl. The constant is defined as the real number $1/3$, so equality to $1/3$ is definitional and needs no lemmas.
why it matters
Feeds the packaged falsifier higgsCoshBSMFalsifier, which bundles four structural BSM signatures of the cosh sector: vanishing kappa-3, kappa-4 equal to one third, positive sextic $\lambda_6$, and pointwise inequality of the two potentials. That package is the theorem-grade discharge of attack A28 (cosh vs Mexican-hat self-coupling mismatch).
In the framework the mismatch is not a bug: it is the collider-facing signature of the even cosh potential forced by the RS cost geometry. Di-Higgs at HL-LHC probes the trilinear (predicted zero here); tri-Higgs and precision quartic measurements at FCC-hh probe this factor-of-three suppression. The identity therefore sits on the experimental edge of the Standard Model sector rather than in the T0–T8 forcing chain itself.
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