Thermodynamic Computing Membrane

A sparse Ising network — particularly one operating at the thermodynamic critical point (sub-threshold CMOS, where thermal energy kT is comparable to signal energy) — is a physical realisation of the live-boundary. The system sits at the barely-stable state without external tuning because that is the only point where a local rule can keep absorbing perturbation. This is Bak’s sandpile in silicon.

The Membrane Tax in Hardware

The [[membrane-tax]] is the energy cost of crossing from the sparse (natural) state to the dense (artificial) state. In probabilistic hardware, the tax is:

1/c(d) = 2(d−1) / (d · sin(π/d))

where d is the degree (number of fixed couplings per pbit) and c(d) is the spectral efficiency derived from the [[tusk-series]] via the Ramanujan coefficient theory.

At d=16 (the Extropic Z1 chip): 1/c(16) = 9.61 — matching the reported ~10× FLOPs penalty for sparse versus dense networks, with no free parameters.

Connection to sopfr and the Tusk Series

The sum-of-prime-factors function [[sopfr-binding-energy]] is completely additive: sopfr(ab) = sopfr(a) + sopfr(b). The Ising energy E = h·z + z·J·z is extensive: E(A∪B) = E(A) + E(B) for non-interacting subsystems. Both are dominated by their largest term (largest prime factor; strongest coupling). Both have Euler-product generating functions.

The Ramanujan coefficient of sopfr at prime p is a_sopfr(p) = p/(p−1). Combined with the first-difference Fourier factor 2sin(π/p), this gives the spectral efficiency c(p) = p·sin(π/p)/(2(p−1)).

The absolute scale is K₂ = π²/12 = ζ(2)/2, which appears in both the prime spectral theory and the low-temperature specific heat of one-dimensional quantum systems (Sommerfeld expansion).

Optimal Connectivity

For thermodynamic hardware with energy advantage A over dense compute:

Net benefit B(d) = A × c(d)

The optimal connectivity under a minimum-capacity constraint:

d* = min { d : c(d) ≥ c_min }

For the Extropic Z1 (A ≈ 4,680×, language-scale capacity floor at d=16): net benefit ≈ 487×.

Self-organised criticality

The chip operates at sub-threshold CMOS — physically calibrated to the thermodynamic critical point where KAM stability and the Bak sandpile converge. The system does not need to be tuned to criticality; it arrives there because that is where minimum power consumption lands. This is the natural membrane: it stays at the boundary not because anyone holds it there, but because the local rule cannot persist anywhere else.

Published work

• Blog: The Membrane Tax: Calculated (pwt.life/blog/the-membrane-tax-calculated/)
• Paper: Spectral Efficiency of Sparse Ising Networks (DOI: 10.5281/zenodo.23529072)
• Companion: Resolution of Open Question 2 in the Tusk Series (DOI: 10.5281/zenodo.23501055)