J_at_phi_approx
plain-language theorem explainer
The J-cost at the golden ratio φ satisfies J(φ) < 0.12. Researchers working in the Recognition Science cost algebra cite this bound when approximating the minimum coherence cost of self-similar structures on the φ-ladder. The proof rewrites the target expression via the exact J_at_phi identity and closes the numerical inequality with a square-root bound plus linear arithmetic.
Claim. $J(φ) < 0.12$ where $J(x) = (x + x^{-1})/2 - 1$ and $φ = (1 + √5)/2$ is the golden ratio.
background
The module Φ-Emergence derives the golden ratio from J-cost minimization. The J-cost is defined by J(x) = (x + x^{-1})/2 - 1; its value at φ equals (√5 - 2)/2 ≈ 0.118 and is identified as the coherence cost of self-similarity, the minimum nonzero cost on the φ-ladder. Upstream, J_at_phi supplies the exact closed form while the octave definitions (2 and 8·tick) set the discrete evolution period used throughout the framework.
proof idea
One-line wrapper that applies J_at_phi to replace the left-hand side by (√5 - 2)/2. A separate subproof shows √5 < 2.24 by verifying 2.24² > 5 together with the identity (√5)² = 5 and non-negativity, then linarith finishes the comparison to 0.12.
why it matters
The bound supplies a concrete numerical estimate for the minimum nonzero J-cost on the φ-ladder, directly supporting approximations inside the φ-emergence construction. It sits downstream of J_at_phi and feeds the module's open hypothesis that a consciousness threshold C = 1 may arise from φ-quantization over one octave. The result therefore bridges the exact algebraic identity to practical estimates used in later ladder and mass-formula arguments.
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