phi_sixth_eq
plain-language theorem explainer
The sixth power of the golden ratio equals eight times phi plus five, extending the Fibonacci recurrence pattern. Researchers computing quark or lepton mass ratios in the Recognition Science framework cite this when establishing bounds such as phi to the sixth between 17 and 18. The proof is a short algebraic calculation that chains the fifth-power identity with the square identity through successive ring expansions and rewrites.
Claim. The golden ratio satisfies $phi^6 = 8 phi + 5$.
background
In the Constants module the golden ratio phi is the self-similar fixed point whose powers obey Fibonacci recurrences. The square identity states phi squared equals phi plus one, the defining relation from the quadratic x squared minus x minus one equals zero. The fifth-power identity states phi to the fifth equals five phi plus three, obtained by the same recurrence chain. These identities supply the algebraic steps for higher powers used in mass-ladder predictions. The module treats the RS time quantum tau zero as one tick, the fundamental unit in native units.
proof idea
The proof is a calc block that begins with phi to the sixth equals phi times phi to the fifth. It rewrites the second factor via the fifth-power identity, expands the product, substitutes the square identity for phi squared, and finishes with ring simplification to eight phi plus five.
why it matters
This identity is invoked by the seventh-power lemma and by the mass-ratio bounds for phi to the sixth and phi to the seventh in NumericalPredictions. It supplies the Fibonacci step that populates the phi-ladder used for generation mass ratios and neutrino squared-mass ratios. The relation instantiates the self-similar fixed point of the framework and supports the chain from T6 toward explicit mass formulas on the ladder.
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