J_bit_explicit
plain-language theorem explainer
Researchers deriving the recognition wavelength from ledger bit costs cite this equality to express J_bit in closed algebraic form. It substitutes the golden ratio directly into the cost functional to obtain an explicit expression without the J operator. The proof is a one-line reflexivity that follows immediately from the definition of J_bit_val as J phi.
Claim. The bit cost satisfies $J_{bit} = (φ + φ^{-1})/2 - 1$.
background
In the PlanckScaleMatching module the bit cost is introduced as the evaluation of the unique cost functional at the self-similar fixed point: J_bit_val := J phi. The module derives λ_rec ≈ 0.564 ℓ_P by setting this bit cost equal to the curvature cost 2λ² over the eight faces of the Q3 hypercube and restoring SI units via c³λ²/(ℏG) with a 1/π averaging factor. The functional J itself is the one forced by the T5 uniqueness condition, J(x) = (x + x^{-1})/2 - 1, equivalently written cosh(log x) - 1.
proof idea
The proof is a term-mode reflexivity. It unfolds the definition J_bit_val := J phi and applies the explicit algebraic form of the J functional at x = phi.
why it matters
The result supplies the explicit algebraic handle on the bit cost that enters the extremum condition of Conjecture C8. It therefore supports the subsequent steps that equate J_bit to J_curv(λ_rec) and obtain λ_rec = ℓ_P / √π. The equality sits inside the T5–T6 segment of the unified forcing chain where J-uniqueness and the phi fixed point are already established.
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