lambda_6_RS_pos
plain-language theorem explainer
Under positive Higgs mass and vev, the RS cosh potential forces a strictly positive sextic self-coupling λ₆ = m_H²/(720 v⁴). Phenomenologists packaging the HiggsCosh BSM falsifier cite this as the genuine higher-order signature against the SM Mexican-hat (which has λ₆ = 0). The proof unfolds the closed form and applies elementary positivity of squares and products.
Claim. For all real $m_H > 0$ and $v > 0$, the RS-predicted Higgs sextic coupling satisfies $\lambda_6^{\mathrm{RS}}(m_H,v) > 0$, where $\lambda_6^{\mathrm{RS}}(m_H,v) = m_H^2/(720\, v^4)$ under the canonical bridge normalization $\Lambda^4 = m_H^2 v^2$.
background
This module resolves attack A28 by making the structural mismatch between the RS cosh Higgs potential and the SM Mexican-hat explicit and theorem-grade. The RS form is $V_{\cosh}(h) = \Lambda^4(\cosh(h/v)-1)$, which is even in $h$; all odd Taylor coefficients vanish and every even order is nonzero. The SM form after EWSB is the degree-4 polynomial $\tfrac12 m_H^2 h^2 + (m_H^2/(2v))h^3 + (m_H^2/(8v^2))h^4$.
Under the linear field identification of HiggsEFTBridge (canonical collider scalar $h$ with $v\cdot\varepsilon$, $\varepsilon=\ln x$) and normalization $\Lambda^4 = m_H^2 v^2$, the Taylor match is: quadratic OK; trilinear RS$=0$ vs SM$\neq 0$; quartic RS is $1/3$ of SM; sextic RS $= m_H^2/(720 v^4)$ while SM has none. The sibling definition of the RS sextic is exactly that closed form.
Positivity of $\lambda_6$ is therefore the clean BSM claim at order $h^6$: the cosh potential necessarily generates a positive sextic vertex once $m_H$ and $v$ are physical.
proof idea
One short tactic proof. Unfold the definition of the RS sextic to the ratio $m_H^2/(720 v^4)$. The numerator is positive by positivity from $m_H>0$. The denominator is positive by positivity from $v>0$ (constant $720>0$). Finish with div_pos. No framework lemmas beyond the definition are required.
why it matters
This is one of the three quantitative BSM predictions packaged by the master falsifier higgsCoshBSMFalsifier: $\kappa_{\lambda_3}=0$, $\kappa_{\lambda_4}=1/3$, and $\lambda_6>0$, together with structural inequality of the two potentials. The downstream constructor wires this theorem directly as the lambda_6_pos field.
In the Recognition framework the mismatch is not a bug: it is the collider-facing signature of the cosh potential forced by the J-cost geometry (T5) once the Higgs sector is identified through the EFT bridge. HL-LHC di-Higgs primarily stresses the trilinear prediction; the sextic is a longer-horizon FCC-hh / precision multi-Higgs target, but its strict positivity is already theorem-grade and closes the A28 attack at the algebraic level.
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