nonAbelianSuppressionCert
plain-language theorem explainer
This definition assembles a certificate asserting that the electromagnetic U(1) sector supplies the complete leading vacuum correction equal to its coupling over pi, while the weak SU(2) sector remains exponentially suppressed. Cosmologists studying dark energy density in the Recognition framework would cite the result to justify truncating the gauge correction to the abelian contribution alone. Construction proceeds by instantiating the two fields of the certificate structure with the massless correction lemma applied to the U(1) sector and a
Claim. Let $u1$ be the U(1) electromagnetic gauge sector with zero boson mass and coupling $1/137.036$, and let $su2$ be the SU(2) weak sector with boson mass $80.4$ GeV. The certificate asserts that for every positive real $E$, the vacuum correction from $u1$ equals its coupling divided by $pi$, while the vacuum correction from $su2$ is strictly less than its coupling divided by $pi$.
background
The NonAbelianSuppression module treats vacuum corrections to the cosmological constant from the gauge factors of Aut(Q3). The U(1) sector is massless and contributes exactly its coupling over pi at leading order. The SU(2) sector carries a mass gap of order 80 GeV, far above the coherence scale E_coh = phi^{-5} approximately 0.09 eV, producing exponential suppression via the factor exp(-M/E). The upstream lemma massless_correction states that when boson mass vanishes the correction equals coupling over pi exactly. The upstream lemma massive_suppression states that when boson mass is positive the correction is strictly less than coupling over pi because the exponential factor is less than one.
proof idea
The definition constructs the certificate structure by supplying explicit functions for each field. The u1_full field is obtained by applying the massless_correction lemma to u1_sector together with reflexivity that its boson mass is zero. The su2_suppressed field is obtained by applying the massive_suppression lemma to su2_sector together with a norm_num tactic that discharges positivity of the mass.
why it matters
This definition completes the local argument that only the abelian U(1) sector contributes a non-negligible correction to Omega_Lambda at the coherence scale, as stated in the module documentation referencing Dark_Energy_Mode_Counting.tex section 8 theorem 8.1. It directly supports the claim that non-abelian corrections are exponentially small, consistent with the Recognition Science forcing chain where the eight-tick octave and D=3 dimensions fix the relevant energy scales. No downstream uses are recorded, indicating it serves as a terminal certificate within the cosmology module.
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