Jcost_phi_eq_massGap
plain-language theorem explainer
The theorem identifies the recognition cost J evaluated at the golden ratio φ with the mass gap constant Δ. Researchers resolving the Yang-Mills mass gap via Recognition Science cite this step when linking the J-cost functional to the spectral gap on the φ-lattice. The proof is a one-line wrapper that applies the exact closed-form computation of J(φ).
Claim. J(φ) = Δ where Δ = (√5 − 2)/2 is the mass gap constant on the φ-lattice.
background
Recognition Science defines the J-cost functional by J(x) = ½(x + x⁻¹) − 1 on the golden-ratio lattice {φⁿ | n ∈ ℤ}. The mass gap Δ is the minimum positive cost of any non-vacuum excitation on this lattice. The module derives the Yang-Mills mass gap directly from the J-cost functional forced by the unified forcing chain (T5 J-uniqueness and T6 phi fixed point). Upstream results supply the exact formula J(φ) = (√5 − 2)/2 together with the φ-ladder structure and the ledger factorization that calibrates J.
proof idea
The proof is a one-line wrapper that applies the sibling theorem Jcost_phi_exact, which unfolds the definition of Jcost and rewrites via phi_plus_inv to reach the closed form (√5 − 2)/2.
why it matters
This equality supplies the exact identification Δ = J(φ) required by the complete Yang-Mills mass gap theorem yang_mills_mass_gap_complete and by the downstream mass_gap_from_phi result. It fills the central step in the module's resolution of the Millennium problem: the gap emerges from the RCL identity together with the eight-tick octave and D = 3 forcing. The result touches the open question of whether the same Δ bounds excitations in all three gauge sectors simultaneously.
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papers checked against this theorem (showing 1 of 1)
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A lattice with gain and loss mimics a black hole horizon
"Γ ∝ exp(−8πM|γ|) = exp(ΔS_{B-H}), with ΔS_{B-H} = −8πM|γ|."