J_bit_val
plain-language theorem explainer
J_bit_val defines the bit cost as the value of the cost functional J evaluated at the golden ratio φ. Researchers deriving the recognition wavelength λ_rec from ledger curvature would cite this as the fundamental cost of a single ledger bit transition in RS-native units. The definition is a direct assignment of J(phi) with no additional computation.
Claim. Let $J(x) = (x + x^{-1})/2 - 1$. Then $J_{bit} := J(φ)$ where $φ$ is the golden ratio satisfying $φ^2 = φ + 1$.
background
The module derives λ_rec ≈ 0.564 ℓ_P by equating bit cost to curvature cost in the ledger-curvature extremum argument for Conjecture C8. The cost functional J(x) = ½(x + x^{-1}) - 1 is the canonical J-cost, equivalent to cosh(ln x) - 1, drawn from the Recognition Composition Law. Phi is the self-similar fixed point forced as the unique positive solution greater than 1 in the UnifiedForcingChain.
proof idea
One-line definition that directly assigns J(phi) to J_bit_val.
why it matters
This supplies the constant J_bit_val used by extremum_condition to equate J_curv(lambda_rec_from_Jbit) with J_bit_val and by extremum_unique to establish uniqueness. It fills the first step of the derivation chain in the module documentation for Conjecture C8, which yields λ_rec = ℓ_P / √π after face-averaging introduces the factor 1/π. It rests on T6 (phi as self-similar fixed point) and T5 (J-uniqueness) from the forcing chain.
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