phi_has_positive_info_cost
plain-language theorem explainer
The J-cost of the golden ratio φ is strictly positive, confirming that φ encodes non-trivial information in the Recognition Science ledger. Researchers deriving information costs from recognition ratios cite this when showing balanced states carry zero cost while deviations like φ incur positive cost. The short proof rewrites J(φ) in squared form and verifies positivity via division of positive terms.
Claim. The information cost of the golden ratio satisfies $J(φ) > 0$, where $J(x) = (x-1)^2/(2x)$ for $x > 0$ and $φ > 1$.
background
In the InformationIsLedger module every recognition ratio x > 0 carries a J-cost J(x) that quantifies its information content in the ledger. The cost is zero precisely when x = 1 (perfect balance, no information) and positive otherwise, with the algebraic identity J(x) = (x-1)^2/(2x) supplied by the upstream lemma Jcost_eq_sq. The module setting treats information as the physical ledger itself, so that any deviation from unity represents concrete physical content rather than an abstract label.
proof idea
The proof is a one-line wrapper that first rewrites J(φ) via Jcost_eq_sq using the fact that φ ≠ 0. It then applies div_pos to the resulting quotient, supplies pow_pos on the squared numerator (invoking one_lt_phi to guarantee the base exceeds 1), and closes with linarith on the positive denominator.
why it matters
This result supplies IC-001.21 inside the Information IS the Ledger claim and feeds directly into the ic001_certificate that records the derived status of the full set of information-cost theorems. It instantiates the general principle that any x ≠ 1 carries positive cost, with φ serving as the self-similar fixed point forced by the T5–T6 steps of the unified forcing chain. No open scaffolding remains for this specific positivity statement.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.