kappa_lambda_3_RS
plain-language theorem explainer
The RS Higgs trilinear modifier is fixed at zero: the cosh potential is even, so its cubic Taylor coefficient vanishes while the SM Mexican-hat value does not. Collider phenomenologists and anyone auditing the A28 cosh-vs-Mexican-hat resolution cite this constant. It is a one-line definition, not a derived equality.
Claim. The Recognition Science Higgs trilinear self-coupling modifier is $\kappa_{\lambda_3}^{\mathrm{RS}} := 0$, i.e. the ratio of the $h^3$ coefficient in $V_{\cosh}(h)=\Lambda^4(\cosh(h/v)-1)$ to the SM coefficient $m_H^2/(2v)$ is identically zero because $\cosh$ is even.
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 and falsifiable. Under the bridge identification that takes the collider scalar $h$ linear in the RS log coordinate $\varepsilon=\ln x$, with canonical normalization $\Lambda^4=m_H^2 v^2$, the two potentials are
$$V_{\cosh}(h)=\Lambda^4(\cosh(h/v)-1),\qquad V_{\mathrm{SM}}(h)=\tfrac12 m_H^2 h^2+\tfrac{m_H^2}{2v}h^3+\tfrac{m_H^2}{8v^2}h^4.$$
Cosh is even, so every odd Taylor coefficient of $V_{\cosh}$ vanishes. The SM trilinear $m_H^2/(2v)$ is nonzero. The modifier $\kappa_{\lambda_3}$ is the ratio of those $h^3$ coefficients; the RS side contributes zero. Quadratic terms match; the quartic ratio is $1/3$; a genuine sextic appears only on the RS side.
proof idea
No proof: the declaration is the constant definition $\kappa_{\lambda_3}^{\mathrm{RS}}:=0$. The mathematical content is the structural fact that $\cosh$ is even (hence all odd powers in the Taylor series of $V_{\cosh}$ are zero) while the SM cubic coefficient is nonzero, so the ratio is defined to be zero. The companion theorem kappa_lambda_3_RS_eq_zero is then rfl.
why it matters
This constant is the first bullet of the module falsifier structure: HL-LHC di-Higgs is predicted to measure $\kappa_{\lambda_3}=0$ (target $\pm 0.5$ at $1\sigma$ in $3,\mathrm{ab}^{-1}$), not the SM value near 1. Downstream, HiggsCoshBSMFalsifier packages $\kappa_{\lambda_3}=0$, $\kappa_{\lambda_4}=1/3$, and $\lambda_6>0$ as the concrete BSM signature of the cosh parameterization under the linear-in-$\varepsilon$ field map of HiggsEFTBridge. The mismatch is intentional structure, not a bug: no analytic field redefinition equates cosh and Mexican-hat at all orders. It closes the A28 self-coupling attack by turning the odd-sector vanishing into a sharp collider prediction.
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