phi_plus_inv
plain-language theorem explainer
The golden ratio satisfies φ + φ^{-1} = √5. Researchers deriving the Yang-Mills mass gap in Recognition Science cite this fact to fix the numerical value of the gap Δ = (√5 - 2)/2. The proof is a short algebraic reduction that substitutes the reciprocal relation φ^{-1} = φ - 1 and normalizes the resulting polynomial.
Claim. Let φ denote the golden ratio. Then φ + φ^{-1} = √5.
background
Recognition Science places the golden ratio φ as the self-similar fixed point forced by the J-cost functional on the discrete lattice {φ^n | n ∈ ℤ}. The module derives the Yang-Mills mass gap from this lattice by showing that every non-vacuum excitation carries cost at least J(φ) = (√5 - 2)/2. The J-cost itself is J(x) = ½(x + x^{-1}) - 1, and the gap emerges once the minimal rung n = ±1 is evaluated exactly.
proof idea
The proof substitutes the upstream reciprocal identity φ^{-1} = φ - 1, simplifies the explicit definition of φ, and applies ring normalization to obtain √5 on the right-hand side.
why it matters
This identity is invoked directly by the exact formula J(φ) = (√5 - 2)/2 that supplies the mass gap Δ in the central theorem of the module. It closes the algebraic step between the φ-forcing chain (T5-T6) and the explicit positive gap required for the RS resolution of the Millennium problem. The downstream J-cost theorem and the full spectral-gap statements both depend on it.
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