pith. sign in
lemma

hbar_eq_phi_inv_fifth

proved
show as:
module
IndisputableMonolith.Constants
domain
Constants
line
313 · github
papers citing
4 papers (below)

plain-language theorem explainer

The lemma establishes that the reduced Planck constant equals the inverse fifth power of the golden ratio in RS-native units. Researchers deriving quantum gravity relations from Recognition Science would reference this identity when normalizing action to the fundamental tick. The proof unfolds the definitions of hbar as the product of cLagLock and tau0, then simplifies directly to phi to the negative fifth.

Claim. In RS-native units with fundamental tick duration equal to 1, the reduced Planck constant satisfies $ħ = φ^{-5}$.

background

Recognition Science fixes the fundamental time quantum τ₀ as one tick, with the tick definition set to 1. The coherence energy is identified with the locked constant cLagLock, which equals φ^{-5}. The reduced Planck constant is defined as the product ħ := E_coh · τ₀, where E_coh stands for cLagLock. Upstream results supply tau0 as the tick and cLagLock as φ^{-5}, with the module establishing RS-native units where c = 1.

proof idea

The proof is a one-line wrapper that unfolds the definitions of hbar, cLagLock, tau0, and tick, then applies simp to reduce the product directly to φ^{-5}.

why it matters

This lemma supplies the explicit value of ħ required by bounding theorems such as hbar_bounds and hbar_lt_one. It underpins the quantum-gravity identities in QuantumGravityOctaveDuality, including G = φ^5 / π and G/ħ = φ^{10}/π. The result closes the derivation of the action quantum from the phi-ladder and feeds the eight-tick octave relations.

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papers checked against this theorem (showing 4 of 4)