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IndisputableMonolith.StandardModel.HiggsCoshBSMPredictions

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Module for the cosh-form Higgs potential under canonical normalization Λ⁴ = m_H² v², and the BSM self-coupling numbers it forces relative to the SM quartic. Collider and EFT theorists tracking RS Higgs predictions would cite it. Content is definitions of V_cosh and V_SM plus elementary parity, evaluation, and inequality lemmas that pin κ_λ₃, κ_λ₄, and λ₆.

claimUnder the canonical normalization $\Lambda^4 = m_H^2 v^2$, the module defines the RS cosh-form Higgs potential $V_{\cosh}$ and the SM quartic $V_{\mathrm{SM}}$, proves $V_{\cosh}$ is even and $V_{\cosh}\neq V_{\mathrm{SM}}$, and records the RS self-coupling predictions $\kappa_{\lambda_3}^{\mathrm{RS}}=0$, $\kappa_{\lambda_4}^{\mathrm{RS}}=1/3$, together with the sextic coefficient $\lambda_6^{\mathrm{RS}}$.

background

Recognition Science forces the unique cost $J(x)=(x+x^{-1})/2-1=\cosh(\log x)-1$ (forcing step T5). On the Higgs sector this becomes a cosh-shaped potential once the dimensionless field coordinate is identified with the canonically normalized collider scalar.

The upstream Higgs EFT bridge formalizes the chain RS cost geometry → effective scalar coordinate → canonical Higgs EFT, with dimensionless coordinate $\varepsilon=h/v$ where $h$ is the mass-dimension-one collider field and $v>0$ is the vacuum expectation value. The present module works under the fixed normalization $\Lambda^4=m_H^2\cdot v^2$ that converts that geometry into a dimensionful potential comparable to the SM quartic.

Sibling objects include $V_{\cosh}$, $V_{\mathrm{SM}}$, parity and pointwise evaluations of both, the inequality $V_{\cosh}\neq V_{\mathrm{SM}}$, and the coupling constants $\kappa_{\lambda_3}^{\mathrm{RS}}$, $\kappa_{\lambda_4}^{\mathrm{RS}}$, $\lambda_6^{\mathrm{RS}}$.

proof idea

Definition-and-evaluation module rather than a deep proof development. $V_{\cosh}$ and $V_{\mathrm{SM}}$ are introduced as explicit real functions of the dimensionless field. Evenness of $V_{\cosh}$ is immediate from the cosh (or $x+x^{-1}$) shape. Point evaluations of $V_{\mathrm{SM}}$ at $\pm 1$ and a nonzero difference lemma separate the two potentials. The coupling lemmas are one-line normalizations: $\kappa_{\lambda_3}^{\mathrm{RS}}=0$ and $\kappa_{\lambda_4}^{\mathrm{RS}}=1/3$ follow by reading off cubic and quartic Taylor coefficients of $V_{\cosh}$ under the stated $\Lambda^4$ convention; $\lambda_6^{\mathrm{RS}}$ is the corresponding sextic coefficient.

why it matters in Recognition Science

Gives the concrete BSM fingerprint of the RS Higgs: vanishing cubic modifier, quartic modifier fixed at $1/3$, and a definite sextic. That fingerprint is what distinguishes the T5-forced cosh potential from the SM Mexican-hat quartic in collider self-coupling measurements and in EFT matching.

It sits directly on the Higgs EFT bridge (RS cost geometry → canonical Higgs EFT) and on the Cost/Constants layers that supply $J$ and the RS unit conventions. No downstream consumers are wired yet in the graph; the natural parents are global SM-matching or collider-prediction theorems that will quote $\kappa_{\lambda_3}^{\mathrm{RS}}=0$ and $\kappa_{\lambda_4}^{\mathrm{RS}}=1/3$ as sharp, parameter-free targets.

Within the forcing chain this is the Higgs-sector readout of J-uniqueness (T5) after the eight-tick and $D=3$ geometry have already fixed the ambient setting.

scope and limits

depends on (3)

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