Editing Openai/6908ee36-d3b4-8010-a468-28e0753d08ae (section)

=== B. Map the stretch to a tunneling action (WKB/instanton) ===

If you picture a true tunnel between adjacent coherence states with an internal attempt frequency ω0\omega_0ω0, then a standard estimate is

τ∼πω0 eS/ℏ,\tau \sim \frac{\pi}{\omega_0}\,e^{S/\hbar},τ∼ω0πeS/ℏ,
so the stretch factor s≡T⋆/TC=1+εs\equiv T_\star/T_C = 1+\varepsilons≡T⋆/TC=1+ε gives

  Sℏ  ≈  ln⁡ ⁣(s ω0 TCπ)  \boxed{\;\frac{S}{\hbar}\;\approx\;\ln\!\Big(\frac{s\,\omega_0\,T_C}{\pi}\Big)\;}ℏS≈ln(πsω0TC)
Choose ω0\omega_0ω0 according to your ontology:
* If the internal clock is the Compton clock, ω0=ωC=mHc2/ℏ\omega_0=\omega_C=m_Hc^2/\hbarω0=ωC=mHc2/ℏ and ω0TC=h/ℏ=2π\omega_0T_C = h/\hbar=2\piω0TC=h/ℏ=2π. Then

  Sℏ≈ln⁡ ⁣(2s)=ln⁡ ⁣(2(1+ε))  \boxed{\;\frac{S}{\hbar}\approx \ln\!\big(2s\big)=\ln\!\big(2(1+\varepsilon)\big)\;}ℏS≈ln(2s)=ln(2(1+ε))
For your ε=0.033721746\varepsilon=0.033721746ε=0.033721746: S/ℏ≈ln⁡(2.0674)≈0.726S/\hbar\approx \ln(2.0674)\approx 0.726S/ℏ≈ln(2.0674)≈0.726.
That’s a pleasantly modest barrier (action of order ℏ\hbarℏ)—intuitively, just enough to add a few-percent dwell time each step.

==== Practical use in your paper (drop-in) ====
* Definition: Each HU step is delayed by a fractional latency ε\varepsilonε due to an internal spin+tunnel barrier.
* Consequences: T⋆=(1+ε)TC,meff=mH1+ε,Eb=ε1+ε mHc2.T_\star=(1+\varepsilon)T_C,\qquad  m_{\rm eff}=\frac{m_H}{1+\varepsilon},\qquad E_b=\frac{\varepsilon}{1+\varepsilon}\,m_Hc^2.T⋆=(1+ε)TC,meff=1+εmH,Eb=1+εεmHc2.
* If ω0=ωC\omega_0=\omega_Cω0=ωC: the barrier action is S/ℏ=ln⁡ ⁣(2(1+ε))S/\hbar=\ln\!\big(2(1+\varepsilon)\big)S/ℏ=ln(2(1+ε)).

If you later want to separate the “spin” part from the “barrier” part, you can write ε=εspin+εbar\varepsilon=\varepsilon_{\rm spin}+\varepsilon_{\rm bar}ε=εspin+εbar (for small pieces adding linearly) so that

Eb≃(εspin+εbar) mHc2E_b \simeq (\varepsilon_{\rm spin}+\varepsilon_{\rm bar})\,m_Hc^2Eb≃(εspin+εbar)mHc2
to first order, while the exact expression above handles the finite-ε\varepsilonε case.