phi_self_similar
plain-language theorem explainer
The golden ratio satisfies the quadratic φ² = φ + 1, which encodes the self-similar Fibonacci property required for the information ledger. Researchers deriving recognition-event scaling from the Recognition Composition Law cite this identity when building the phi-ladder. The proof is a direct one-line application of the defining equation for φ.
Claim. The golden ratio satisfies the equation $φ^2 = φ + 1$.
background
The InformationIsLedger module identifies information with the physical ledger: every recognition ratio x > 0 carries a J-cost J(x) ≥ 0, with J(x) = 0 precisely when x = 1. The golden ratio φ is introduced as the unique positive real satisfying the self-similarity relation that encodes “the next level contains the previous two.” This theorem sits inside the IC-001 sequence that unifies J-cost with Shannon entropy.
proof idea
The proof is a one-line wrapper that applies the upstream lemma phi_sq_eq, which states the key identity φ² = φ + 1 derived from the quadratic x² - x - 1 = 0.
why it matters
The result supplies the self-similarity step inside the IC-001 certificate, which aggregates J-cost properties to certify that information is the ledger. It aligns with the T6 forcing step in which φ is the self-similar fixed point of the Recognition Composition Law. The downstream ic001_certificate string explicitly lists this theorem among the derived facts confirming the ledger interpretation.
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