phi_eleventh_eq
plain-language theorem explainer
The identity φ¹¹ = 89φ + 55 follows from the golden ratio recurrence and supplies the next Fibonacci coefficient after the tenth power. Researchers bounding mass hierarchies or scaling ladders in Recognition Science cite it to obtain concrete lower bounds on φ-powers. The proof is a five-step calc chain that multiplies the tenth-power identity by φ, reduces the quadratic term, and collects coefficients with ring.
Claim. $φ^{11} = 89φ + 55$
background
The Constants module derives RS-native constants from the golden ratio φ, whose powers appear in the phi-ladder for mass formulas. The local setting treats τ₀ = 1 tick as the fundamental RS time quantum. Upstream phi_sq_eq records the defining relation: “Key identity: φ² = φ + 1 (from the defining equation x² - x - 1 = 0).” The immediately preceding phi_tenth_eq states φ¹⁰ = 55φ + 34, which is the direct input to the present reduction.
proof idea
A calc block begins with φ¹¹ = φ · φ¹⁰. The tenth-power identity is substituted, the expression is expanded by ring, the quadratic relation φ² = φ + 1 is applied, and a final ring step collects the coefficients 89φ + 55.
why it matters
The lemma is invoked by phi_11_hierarchy_lower to prove φ¹¹ > 180, a conservative bound used for large hierarchies. It advances the phi-ladder construction that underlies the mass formula yardstick · φ^(rung - 8 + gap(Z)). The identity is consistent with T6, where φ is forced as the self-similar fixed point, and contributes to the eight-tick octave structure.
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