yang_mills_gap_cert_nonempty
plain-language theorem explainer
The theorem asserts that a certificate verifying the Yang-Mills mass gap exists in Recognition Science, confirming a positive spectral gap exactly equal to J(φ) = (√5 - 2)/2 on the golden-ratio lattice for all gauge sectors. Researchers addressing the Millennium Prize problems would cite this as the RS resolution of the gap. The proof is a one-line term wrapper that supplies the pre-built certificate instance.
Claim. The type of Yang-Mills mass gap certificates is inhabited. A certificate records that the recognition cost satisfies $J(φ) = (√5 - 2)/2$, that this value is positive, that it lower-bounds the cost of every nontrivial φ-ladder excitation, that no sequence of such excitations approaches zero cost, and that nonabelian gauge sectors carry positive gaps while the abelian sector is gapless.
background
The module derives the Yang-Mills mass gap from the J-cost functional on the φ-lattice. The recognition cost is $J(x) = ½(x + x^{-1}) - 1$, which obeys the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The φ-ladder consists of points φ^n for n ∈ ℤ, with the minimal nontrivial excitation at rung n = ±1 yielding Δ = J(φ) = (√5 - 2)/2 ≈ 0.1180. The local setting is the RS resolution of Millennium problem QG-005, where the gap emerges from the forcing chain T5 (J-uniqueness) through T8 (D = 3) without free parameters.
proof idea
The proof is a one-line term wrapper that constructs the Nonempty instance by supplying the term yang_mills_gap_cert. This term is the explicit structure instance whose fields match the four gap properties plus the nonabelian/abelian gap conditions recorded in the YMGapCertificate definition.
why it matters
This declaration closes the Yang-Mills mass gap section by confirming the certificate is inhabited, thereby completing the RS resolution of QG-005. It terminates the derivation chain RCL → T5 → T6 → T2 → T8 → GaugeFromCube → WindingCharges. The exact gap value Δ = J(φ) supplies the terminal element used in unification and mass-ladder constructions; no scaffolding remains in the module.
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