phi_fifth_bounds
plain-language theorem explainer
The lemma pins the fifth power of the golden ratio between 10.7 and 11.3. Workers on RS constants, alpha-band devices, or zero-parameter gravity cite this interval for numerical predictions. The proof reduces via the Fibonacci recurrence identity and closes the bounds with linear arithmetic on the tighter phi estimates.
Claim. $10.7 < phi^5 < 11.3$ where $phi = (1 + sqrt(5))/2$ satisfies the self-similar fixed-point equation and $phi^5 = 5 phi + 3$.
background
Recognition Science derives constants from the forcing chain in which phi emerges as the unique self-similar fixed point (T6). The Constants module supplies numerical bounds on powers of phi for downstream applications in photobiomodulation, chemistry, and gravity. The module sets the fundamental RS time quantum tau_0 = 1 tick. Upstream, the identity lemma states: Key identity: phi^5 = 5 phi + 3 (Fibonacci recurrence). phi^5 = phi times phi^4 = ... = 5 phi + 3.
proof idea
The term proof first rewrites the target using the Fibonacci identity to replace phi^5 with 5 phi + 3. It then invokes the lemmas establishing 1.61 < phi < 1.62. Linear arithmetic on these inequalities yields the desired bounds on the linear expression.
why it matters
This result supplies the numerical anchor for phi^5 approx 11.09 that appears in hbar = phi^{-5} and G = phi^5 / pi. Downstream theorems apply it directly: phi_fifth_in_alpha_band for EEG coherence, born_exponent_in_range for ionic bonds, kappa_bounds for gravitational coupling, and cosmological predictions for BAO scales. It closes a step from the T6 phi fixed-point toward T8 D = 3 by providing concrete bounds.
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