YMGapCertificate
plain-language theorem explainer
The YMGapCertificate encodes the Recognition Science resolution of the Yang-Mills mass gap by requiring an exact positive gap value J(φ) = (√5 - 2)/2, a strict lower bound on all nonzero φ-ladder excitations, rigidity against sequences approaching zero cost, nonabelian sectors gapped with the abelian sector gapless, and gauge ranks matching the Standard Model. Researchers addressing the Millennium problem would cite it for deriving the gap from the J-cost functional and the φ-forcing chain with zero free parameters. The declaration is a sorry_s
Claim. A certificate asserting that the recognition cost satisfies $J(φ) = (√5 - 2)/2$, the mass gap Δ is positive, every nonzero φ-ladder element obeys Δ ≤ J(φ^n), no sequence of nonzero ladder elements has J-cost tending to zero at infinity, the color and weak gauge sectors are gapped while the hypercharge sector is gapless, the cube-derived gauge ranks equal the Standard Model ranks (3,2,1), and the gap prediction remains unfalsified.
background
Recognition Science defines the J-cost by J(x) = ½(x + x⁻¹) - 1 on positive reals; it obeys the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The φ-ladder is the discrete set {φ^n | n ∈ ℤ}, with the minimal positive cost at the first rung n = ±1. The module derives the Yang-Mills gap from the φ-forcing chain (T5 J-uniqueness through T8 D = 3) together with gauge ranks extracted from the cube geometry.
proof idea
The declaration is a sorry stub. Its fields are populated by direct application of the lemmas Jcost_phi_exact for the exact gap value, massGap_pos for positivity, spectral_gap for the universal lower bound, gap_rigidity for the non-convergence property, and SU2_SU3_gapped for the nonabelian gaps.
why it matters
This structure supplies the central certificate for QG-005, the Recognition Science resolution of the Yang-Mills mass gap. It is consumed by the theorem yang_mills_gap_cert, which constructs an explicit inhabitant, and by yang_mills_gap_cert_nonempty, which proves the certificate is inhabited. The construction closes the derivation chain RCL → T5 → T6 → T2 → T8 → GaugeFromCube, confirming that the gap emerges from J-cost on the φ-lattice with no free parameters.
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