E_coh
plain-language theorem explainer
E_coh sets the coherence quantum to φ^{-5} in RS-native units as the minimum energy for coherent recognition. Researchers deriving ħ, mass ladders, and the IR gate from the foundation cite this scale when closing the constant chain. The definition is a direct assignment from the phi-ladder after J-uniqueness and the self-similar fixed point are established.
Claim. $E_0 = φ^{-5}$, where $φ = (1 + √5)/2$ is the golden-ratio fixed point of the J-cost function and the minimum energy scale for coherent recognition.
background
The Constant Derivations module derives c, ħ, G, and α from the RS foundation instead of treating them as free parameters. The chain begins with the Composition Law, introduces J(x) = ½(x + 1/x) - 1 as the unique cost function, forces φ as the self-similar fixed point, and obtains D = 3 together with the eight-tick octave τ₀ = 8.
proof idea
One-line definition that directly assigns the value φ_val raised to the integer power -5. No lemmas or tactics are invoked; the expression follows the phi-ladder scaling already fixed by upstream forcing steps.
why it matters
This supplies the energy unit that enters ħ = E_coh · τ₀ and the yardstick scaling for masses on the phi-ladder. It completes the level-4 step in the module's derivation diagram, converting the J-uniqueness and φ fixed point into an observable energy scale. It leaves open the precise gap(Z) corrections that adjust α^{-1} into the observed band.
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