Metabolic energy pump sustains Fröhlich condensate in microtubules; tubulin tunneling data lack quantum evidence — E8 Intelligence Research

FINDING: Fröhlich condensate in microtubules requires a metabolic energy pump to sustain a non-thermal phonon mode at ~10^11 Hz against thermal decoherence; single-tubulin tunneling-current data provide the only direct experimental handle on tubulin conformational states, but do not yet demonstrate quantum superposition. MATH: - Fröhlich rate equation: dn_k/dt = s_k − γ_k n_k + Σ_{k'} (χ_{kk'} n_k n_{k'} − χ_{k'k} n_{k'} n_k) − Γ_th(T) n_k where s_k = metabolic pump rate, γ_k = linear damping, χ = nonlinear mode coupling, Γ_th = thermal decoherence. - Thermal decoherence rate (Caldeira–Leggett, high-T limit): Γ_th ≈ (2 m k_B T γ) / ħ² → for tubulin dimer m ≈ 110 kDa ≈ 1.8×10⁻²² kg, T = 310 K, γ ≈ 10¹² s⁻¹ (overdamped protein): Γ_th ~ 10¹²–10¹³ s⁻¹. - Fröhlich condensation threshold: pump must exceed s_c ≈ γ_k · (k_B T / ħω_k) per mode. For ω/2π = 10¹¹ Hz, ħω ≈ 6.6×10⁻²³ J, k_B T ≈ 4.3×10⁻²¹ J → ratio ≈ 65. Pump must exceed damping by ~65× to condense. - Coherence lifetime τ_coh ≈ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com