Jcost_phi_exact
plain-language theorem explainer
The recognition cost at the golden ratio equals (√5 − 2)/2. Researchers resolving the Yang-Mills mass gap in the Recognition Science framework cite this identity for the explicit spectral gap value Δ. The proof is a direct algebraic reduction that substitutes the sum identity φ + φ⁻¹ = √5 into the definition of J and normalizes the expression.
Claim. $J(φ) = (√5 - 2)/2$
background
In Recognition Science the cost functional is J(x) = ½(x + x⁻¹) − 1 evaluated on the golden-ratio lattice {φⁿ | n ∈ ℤ}. The module derives the Yang-Mills mass gap directly from this functional, showing that the minimum non-vacuum excitation cost is exactly J(φ). The upstream result φ + φ⁻¹ = √5 is obtained by unfolding the definition of φ and applying algebraic simplification.
proof idea
The proof unfolds the definition of Jcost, rewrites the expression using the phi sum identity, and applies ring normalization to reach the closed form.
why it matters
This supplies the exact gap value required by the Yang-Mills gap certificate theorem, which asserts positivity, rigidity, and applicability to non-abelian sectors. It occupies the first position in the module structure for QG-005 and follows from J-uniqueness together with the phi fixed-point forcing in the T5–T6 chain.
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