E_coh_pos
plain-language theorem explainer
The declaration proves that the coherence quantum, defined as the golden ratio raised to the negative fifth power, is strictly positive. Researchers deriving vacuum energy bounds or mass ladders in Recognition Science cite this positivity when establishing derived quantities. The proof reduces directly to the positivity of integer powers of a positive real number after unfolding the definition.
Claim. Let $E = (1 + 5^{1/2})/2$ denote the golden ratio. The coherence quantum defined by $E := E^{-5}$ satisfies $E > 0$.
background
The Constant Derivations module derives physical constants as ratios of RS-native quantities built from the J-cost function and the golden ratio. The coherence quantum is defined as the golden ratio to the negative fifth power and represents the minimum energy scale for coherent recognition events. This positivity is presupposed in the phi forcing principle, which asserts that the golden ratio satisfies its quadratic equation together with the positivity of the minimum cost bit and the coherence quantum.
proof idea
The term proof unfolds the definition of the coherence quantum to the golden ratio raised to the negative fifth power, then applies the zpow_pos lemma to the known positivity of the golden ratio and the negative integer exponent.
why it matters
This result supplies the positivity hypothesis required by the phi forcing principle and by vacuum energy positivity in the Cosmology module. It closes a step in the constant derivation chain from the Composition Law through J-uniqueness to the eight-tick octave and derived constants in RS-native units. The algebraic fact holds once the golden ratio is established as the unique positive fixed point.
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