E_coh_eq
plain-language theorem explainer
The coherence energy in Recognition Science equals the golden ratio to the negative fifth power. Researchers deriving lepton masses from the phi-ladder cite this equality when assembling structural mass formulas. The proof is a direct unfolding of E_coh, coherence_exponent, octave, and D followed by numerical normalization.
Claim. $E_{coh} = phi^{-5}$, where the exponent is the Fibonacci deficit $2^D - D$ with $D=3$ the unique dimension satisfying the constraint that both $D$ and $2^D$ are Fibonacci numbers.
background
The module establishes that the Fibonacci constraint on dimension D and octave period 2^D forces D=3=F4 and 2^D=8=F6, so the deficit 5=F5 becomes the coherence exponent. Coherence energy is defined as E_coh = phi raised to the negative of this exponent in RS-native units. Upstream definitions include octave as 2^D from MusicalScale and Constants, and coherence_exponent as octave minus D.
proof idea
The term proof unfolds E_coh, coherence_exponent, octave, and D, then applies norm_num to reduce phi ^ (-(2^3 - 3)) directly to phi ^ (-5).
why it matters
This supplies the explicit E_coh value used in downstream ElectronMass theorems such as electron_structural_mass_forced, where rw [E_coh_eq] combines it with yardstick and rung factors to yield m_struct = 2^{-22} phi^{51}. It closes the T7 eight-tick octave and T8 D=3 step in the forcing chain, confirming E_coh is fixed by the Fibonacci-phi framework rather than a free parameter.
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