V_SM_difference_not_zero
plain-language theorem explainer
The SM Mexican-hat Higgs potential is not even: its values at field points +1 and −1 differ by m_H²/v, nonzero whenever m_H ≠ 0 and v > 0. Anyone separating the RS cosh potential from the SM quartic cites this as the odd-parity witness. The proof rewrites the two point evaluations and closes by ring algebra.
Claim. For real parameters $m_H \neq 0$ and $v > 0$, the SM Mexican-hat potential after EWSB satisfies $V_{\mathrm{SM}}(m_H,v,1) - V_{\mathrm{SM}}(m_H,v,-1) = m_H^2/v$.
background
This module resolves attack A28 (cosh vs Mexican-hat self-coupling mismatch) by making the two Higgs parameterizations theorem-grade and structurally distinct. The RS form is $V_{\cosh}(h) = \Lambda^4(\cosh(h/v)-1)$, even in $h$. The SM form after EWSB in unitary gauge is the degree-4 polynomial $V_{\mathrm{SM}}(m_H,v,h) = \tfrac12 m_H^2 h^2 + \tfrac{m_H^2}{2v}h^3 + \tfrac{m_H^2}{8v^2}h^4$, with a nonzero trilinear.
Under the bridge identification $\Lambda^4 = m_H^2 v^2$, the quadratic terms match, but the trilinear (RS 0 vs SM $m_H^2/(2v)$) and quartic (factor $1/3$) disagree, and the cosh carries a genuine sextic BSM term. The present identity isolates the parity failure of the Mexican-hat: evaluating at opposite field points yields a pure odd contribution proportional to $m_H^2/v$.
proof idea
One-line algebraic proof. Rewrite the left-hand side by the two point-evaluation lemmas for the SM potential at $h=1$ and $h=-1$, then finish with ring, which cancels the even (quadratic and quartic) pieces and leaves the cubic contribution $m_H^2/v$.
why it matters
Feeds the parent theorem that the cosh and SM potentials are unequal as functions: because cosh is even, $V_{\cosh}(1)=V_{\cosh}(-1)$, while this identity shows $V_{\mathrm{SM}}(1)-V_{\mathrm{SM}}(-1)=m_H^2/v\neq 0$, so at least one of $\pm 1$ is a disagreement witness.
That non-equality is the structural core of the A28 resolution in the Higgs EFT bridge: the mismatch is not a bug but a falsifiable BSM signature (di-Higgs probes the trilinear; tri-Higgs and precision probes the quartic/sextic). Within RS, the cosh form is the natural potential descending from the J-cost geometry; this lemma pins the concrete numerical gap that collider measurements can test.
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