Jcost_phi_pos
plain-language theorem explainer
The theorem establishes that the recognition cost J at the golden ratio φ is strictly positive, yielding the exact spectral gap Δ = J(φ) = (√5 − 2)/2. Physicists resolving the Yang-Mills mass gap would cite this as the parameter-free lower bound on non-vacuum excitations in the RS framework. The proof is a one-line algebraic reduction that rewrites J(φ) via an equality to the mass gap and applies its positivity.
Claim. Let $J(x) = ½(x + x^{-1}) - 1$. Then $0 < J(φ)$, where $φ = (1 + √5)/2$ is the golden ratio and the minimum non-vacuum cost on the φ-lattice.
background
Recognition Science defines the cost functional by $J(x) = ½(x + x^{-1}) - 1$ on the golden-ratio lattice forced by the T5–T6 chain. The identity event sits at the J-cost minimum with state 1 and zero cost. The module derives the Yang-Mills mass gap from this functional alone, with the central claim that every non-trivial φ-ladder excitation costs at least Δ = J(φ) = (√5 − 2)/2.
proof idea
The term proof rewrites the target using the equality J(φ) = mass gap and then applies the positivity of the mass gap.
why it matters
This supplies the strict positivity step (§2) in the module's resolution of the Yang-Mills mass gap, feeding the KTheta_pos theorem and the parallel lemma in Constants. It closes the gap inequality required by the central theorem that Δ > 0 emerges from the J-cost and the φ-forcing chain without free parameters.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.