lambda_6_RS
plain-language theorem explainer
The RS cosh Higgs potential predicts a nonzero dimension-six sextic self-coupling λ₆ = m_H²/(720 v⁴); the SM renormalizable potential has λ₆ = 0. Anyone citing the A28 cosh-vs-Mexican-hat mismatch or the HL-LHC/FCC-hh BSM falsifier uses this constant. It is a one-line closed-form definition extracted from the even Taylor series of cosh.
Claim. For Higgs mass $m_H$ and electroweak VEV $v$, the Recognition Science prediction for the sextic Higgs vertex is $\lambda_6^{\mathrm{RS}}(m_H,v) = m_H^2/(720 v^4)$. The Standard Model has vanishing sextic coupling at tree level (potential of degree at most 4).
background
The module resolves attack A28 by comparing two Higgs potentials under the linear field identification of HiggsEFTBridge. The RS form is $V_{\cosh}(h)=\Lambda^4(\cosh(h/v)-1)$ with canonical normalization $\Lambda^4=m_H^2 v^2$. Because cosh is even, every odd Taylor coefficient vanishes and the even tower continues to all orders.
The SM Mexican-hat potential after EWSB is the degree-4 polynomial $V_{\mathrm{SM}}(h)=\tfrac12 m_H^2 h^2+(m_H^2/(2v))h^3+(m_H^2/(8v^2))h^4$. Matching coefficients order by order: quadratic agrees; trilinear is RS 0 vs SM $m_H^2/(2v)$; quartic is RS $m_H^2/(24v^2)$ vs SM $m_H^2/(8v^2)$ (factor $1/3$); sextic is RS $m_H^2/(720v^4)$ vs SM 0. The sextic term is therefore a genuine BSM signature of the cosh potential.
proof idea
Pure definition: the right-hand side is the coefficient of $h^6/6!$ in the Taylor expansion of $\Lambda^4(\cosh(h/v)-1)$ after substituting $\Lambda^4=m_H^2 v^2$. No lemmas are invoked; the body is the closed-form expression $m_H^2/(720 v^4)$.
why it matters
This constant is one of the three quantitative BSM predictions packaged by HiggsCoshBSMFalsifier: $\kappa_{\lambda_3}=0$, $\kappa_{\lambda_4}=1/3$, and $\lambda_6>0$. The companion theorem lambda_6_RS_pos discharges the positivity clause of that falsifier for $m_H>0$, $v>0$.
In the A28 resolution narrative, the nonzero sextic is the cleanest all-orders distinction from the SM: the Mexican-hat stops at degree 4, while the cosh continues. Collider reach is via multi-Higgs final states (tri-Higgs and beyond at FCC-hh). The definition sits downstream of the forcing-chain cost $J$ only indirectly, through the cosh identification $J(e^{\varepsilon})\propto\cosh\varepsilon-1$ that motivates $V_{\cosh}$.
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