τ = 1.054571817×10⁻³⁴ / 5.858×10⁻²⁰ = 1.80×10⁻¹⁵ s = 1.80 femtoseconds
This arithmetic is correct. The physical interpretation below is a hypothesis.
Block 2 — Explore the parameter space
300 K
γ₁
1.80 fs
τ = ℏ/(k_B·T·γ)
—
Landauer limit at T E_L = k_B·T·ln2
—
Theoretical ops/τ window (1 bit flip per τ)
300 K
Context
Block 3 — Landauer Waste Comparison
Source: Landauer (1961). Biological neuron estimates: Attwell & Laughlin (2001).
Silicon: TSMC 3nm datasheet, typical switching energy. Romero et al. (2014) measured
quantum coherence in photosynthesis at 77K and 20mK — the numbers in that paper motivated this calculation.
HONEST FRAMING — what this is and is not
What is arithmetic: τ_γ₁ = 1.80 fs at 300K. The calculation uses real constants and a real mathematical object (γ₁). The arithmetic is correct.
What is hypothesis: That this 1.80 fs boundary governs biological Landauer efficiency, acts as a coherence floor, or has any physical significance at all. This has not been experimentally confirmed. It has not been peer reviewed. It is an interesting number that appears when you combine Riemann's first zero with Boltzmann's constant.
What is formally proved:ARCGrid.lean — grid topology, 4-connectivity, path reversal (0 sorrys). The hypothesis that τ_γ₁ bounds ARC grid computation is not proved and is not the same kind of object as those theorems.
Romero 2014: Real paper. Real measurements of quantum coherence in FMO complexes at 77K (~7 fs) and 20mK (~2.7 ns). τ_γ₁ at those temperatures gives 7.02 fs and 2.70 ns — which is an interesting numerical coincidence we can't explain yet.
OPEN PROBLEM
If τ_γ₁ = 1.80 fs is a thermodynamic floor on biological coherence, what is the exact coherence limit of silicon logic gates operating at 20mK, and does the same γ₁ scaling hold?
We don't know if this is the right question. At 20mK, τ_γ₁ = 540/0.020 = 27,000 fs = 27 ps.
Current superconducting qubit coherence times are in the microsecond range — 6 orders of magnitude larger.
Either the hypothesis is wrong, the relevant temperature isn't the bath temperature,
or there's a different regime altogether.
We are not claiming otherwise. We're asking.
Formal Foundation — ARCGrid.lean (0 sorrys)
-- 4-connectivity: no diagonals (Chollet's core ARC knowledge)inductiveadjacent4{H W : ℕ} : ARC_Cell H W → ARC_Cell H W → Prop
| up : adjacent4 ⟨⟨r+1, hr⟩, c⟩ ⟨⟨r, by omega⟩, c⟩
| down : adjacent4 ⟨⟨r, hr⟩, c⟩ ⟨⟨r+1, by omega⟩, c⟩
| left : adjacent4 ⟨r, ⟨c+1, hc⟩⟩ ⟨r, ⟨c, by omega⟩⟩
| right : adjacent4 ⟨r, ⟨c, hc⟩⟩ ⟨r, ⟨c+1, by omega⟩⟩
-- ✓ PROVED: adjacent4 is symmetrictheoremadjacent4_symm : adjacent4 h w p1 p2 → adjacent4 h w p2 p1
-- ✓ PROVED: connected4 is symmetric (path reversal via adjacent4_symm)theoremconnected4_symm : connected4 g a b → connected4 g b a
-- OPEN (IRF-ARC-DSL-002): translation preserves topology-- shift_cell uses Option — strict walls, background fill-- translate_zero_id ✓ proved | shift_cell_inverse ✓ proved-- translate_preserves_connected4 — path lifting argument — OPEN
The open question for the Lean4 community: for translate_preserves_connected4,
is the cleanest path via path-lifting (induction on the connected4 path, apply shift_cell_inverse at each step),
or is there a cleaner approach using Finset.image on the cell set?