phi_pow_fibonacci_sum_le
plain-language theorem explainer
The golden ratio satisfies the recurrence φ^n + φ^{n+1} = φ^{n+2} for every natural number n. Researchers building the phi-ladder mass spectrum in Recognition Science cite this to justify the Fibonacci addition rule for successive rungs. The argument is a one-line linear-arithmetic reduction that applies the defining recursion of phi.
Claim. For every natural number $n$, $φ^n + φ^{n+1} = φ^{n+2}$, where $φ$ is the golden ratio obeying $φ^2 = φ + 1$.
background
The QuantumGravityOctaveDuality module proves that gravitational and quantum constants are locked by the octave factor 8, with all results derived from the J-cost functional. Section §5 states that the mass ladder is Fibonacci because powers of phi obey the same addition rule as the sequence itself. The upstream lemma phi_fibonacci_recursion supplies the base recurrence used here; Constants.octave defines one octave as 8 ticks, the fundamental RS evolution period.
proof idea
This is a one-line wrapper that applies the upstream lemma phi_fibonacci_recursion at the given n and closes the equality via the linarith tactic.
why it matters
The result fills QG-004 in the module documentation: the fermion mass spectrum satisfies the Fibonacci recurrence. It contributes to the complete QG Octave Duality Certificate that unifies quantum mechanics and gravity through the 8-tick recognition cycle starting from J-cost and phi. The lemma touches the T6 forcing of phi as self-similar fixed point and the T7 eight-tick octave; the zero used_by count indicates it remains an intermediate step toward the certificate.
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