cancellation_of_RS_preservation
plain-language theorem explainer
If the RS Higgs EFT bridge preserves longitudinal unitarity by setting its scalar-exchange residue to the negative of the gauge residue, the high-energy cancellation condition follows at once. Researchers checking consistency of Recognition Science models with electroweak unitarity bounds would cite this result. The argument is a direct one-line application of the SM cancellation theorem to the RS hypothesis.
Claim. Let $a_{gauge}, a_{scalar,RS} ∈ ℝ$. If $a_{scalar,RS} = -a_{gauge}$, then $a_{gauge} + a_{scalar,RS} = 0$.
background
The module studies high-energy longitudinal vector-boson scattering $W_L^+ W_L^- → W_L^+ W_L^-$, where gauge-exchange diagrams produce an amplitude growing as $s^2/v^4$ that violates perturbative unitarity above roughly 1 TeV unless cancelled by scalar exchange. CancellationCondition is the requirement that the gauge and scalar residues sum to zero at leading order. RSPreservesLongitudinalUnitarity states that the RS theory, once the canonical-normalisation map of HiggsEFTBridge fixes Λ(v) and the J-cost Taylor expansion fixes the scalar coupling, produces a scalar residue exactly opposite the gauge residue.
proof idea
The proof is a one-line wrapper that applies cancellation_of_SM_hypothesis to the hypothesis h. The RS preservation condition matches the algebraic form required by the SM cancellation implication, so the result follows immediately by substitution and ring simplification.
why it matters
This theorem supplies the conditional step that RS satisfies the structural cancellation needed for longitudinal unitarity preservation, completing the master bridge certificate in the module. It shows that the scalar coupling extracted from the J-cost geometry enforces the same cancellation identity as the SM, consistent with the phi fixed point and the eight-tick octave. The module notes that a full kinematic four-point amplitude with Mandelstam variables remains open.
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