hbarQuantum_eq_Ecoh
plain-language theorem explainer
The lemma shows that the action quantum ħ equals the coherence energy E_coh in RS-native units. Researchers deriving Planck-scale constants from the Recognition Science forcing chain cite this equality when normalizing the fundamental tick to unity. The proof is a direct simplification that unfolds hbarQuantum as the product of cohQuantum and tick, then applies the normalization tick = 1.
Claim. In the RS-native unit system with fundamental time quantum $τ_0 = 1$, the action quantum satisfies $ħ = E_{coh}$, where $E_{coh} = φ^{-5}$ is the coherence energy quantum.
background
The RS-native system sets tick as the base time quantum $τ_0 = 1$ and cohQuantum as the energy quantum $E_{coh} = φ^{-5}$. Action is defined via hbarQuantum = cohQuantum · tick, which collapses to $E_{coh}$ under the unit choice. The module fixes $c = 1$ and places all derived quantities on the φ-ladder. Upstream results supply the primitive tick normalization and the unfolding of hbarQuantum from the action definition.
proof idea
The proof is a one-line simp tactic. It rewrites hbarQuantum via its definition as cohQuantum multiplied by tick, substitutes the definition of cohQuantum as $E_{coh}$, and applies the constant tick = 1.
why it matters
This equality fixes the action quantum to the energy scale inside the native units, supporting consistent conversion of constants along the φ-ladder. It aligns with the T5 J-uniqueness and T7 eight-tick octave by enforcing dimensionless ratios fixed by φ alone. The result closes a unit-consistency step before downstream mass and frequency derivations in the same module.
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