phi_power_ratio
plain-language theorem explainer
The golden ratio satisfies φ^a / φ^b = φ^(a-b) for integers a and b. Researchers manipulating rung shifts or cost ratios on the Recognition Science phi-ladder cite this identity. The proof is a one-line rewrite applying division-to-inverse, negative-exponent, and exponent-addition rules under positivity of φ.
Claim. For all integers $a, b$, $φ^a / φ^b = φ^{a-b}$, where $φ$ is the golden ratio fixed point forced by the T6 self-similar closure condition.
background
The StillnessGenerative module derives from T0-T8 that the unique zero-defect state x=1 is unstable and must generate non-trivial content. T6 forces every non-trivial entry to be a power of φ on a closed geometric scale sequence, while T7 requires an eight-tick cycle that populates the full ladder via the Fibonacci relation φ^n + φ^{n+1} = φ^{n+2}. The module imports PhiForcing to supply φ > 1 and the basic exponent arithmetic needed for rung arithmetic.
proof idea
One-line wrapper that rewrites division as multiplication by the inverse, converts the negative exponent, and folds the exponents via addition, justified by the positivity hypothesis on φ.
why it matters
This identity closes the algebraic step required for the Fibonacci cascade and ledger symmetry arguments that populate the phi-ladder from the initial condition x=1. It directly supports the mass formula and Berry creation threshold in the T0-T8 chain. No open scaffolding remains at this level.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.