phi_lt_onePointSixTwo
plain-language theorem explainer
The golden ratio satisfies φ < 1.62. Researchers in acoustics, aesthetics, and applied physics cite the bound when locating derived quantities inside narrow empirical intervals such as T60 reverberation or photobiomodulation energies. The proof simplifies the definition of phi, establishes sqrt(5) < 2.24 by comparing squares, and closes with linear arithmetic.
Claim. Let φ denote the golden ratio. Then φ < 1.62.
background
Phi arises as the self-similar fixed point forced at step T6 of the UnifiedForcingChain. The Constants module records its basic algebraic and numerical properties in RS-native units where the fundamental time quantum equals one tick.
proof idea
The tactic proof first simplifies the definition of phi. It then proves sqrt(5) < 2.24 by showing 5 < (2.24)^2 via norm_num and applying the monotonicity of the square root. Linear arithmetic finishes the comparison.
why it matters
Paired with the matching lower bound, this result places phi inside (1.61, 1.62). It feeds downstream results including optimalT60_band for concert-hall reverberation and Jphi_numerical_band for visual beauty scores. The interval supports phi-ladder constructions and the eight-tick octave throughout the Recognition framework.
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