IR_gate_identity
plain-language theorem explainer
The IR gate identity equates the reduced Planck constant to the product of coherence energy and base time scale under the Recognition Science derivation. Researchers calibrating ħ from φ-based quantities cite this when tracing the constant chain. The proof is a one-line simplification that unfolds the definition of the derived ħ and substitutes the supplied equality.
Claim. If $ħ = E_{coh} · τ_0$, then $ħ$ equals the derived expression $ħ_{derived}(E_{coh}, τ_0)$, where $E_{coh} = φ^{-5}$ in RS units.
background
The RRF Foundation module derives all physical constants from φ through gate identities. The explicit chain is φ → E_coh → τ₀ → c → ħ → G → α^{-1}. The IR gate is the first link: ħ = E_coh · τ₀. E_coh is defined as φ^{-5} (≈ 0.09 eV). Upstream results supply the octave as 8 ticks and the φ-tier structure for nuclear densities.
proof idea
One-line wrapper that applies the definition of hbar_derived together with the hypothesis hbar = e_coh * tau_0 via simplification.
why it matters
This theorem supplies the IR gate identity that anchors the constant derivation chain in the module. It feeds later steps toward α^{-1} = 128 ln(π/2) + 45 ln φ + curvature_correction. The result closes the direct link between coherence energy and ħ in the φ-based framework.
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