One law, measured
Compute is nearly free in the physics. The real cost is always at the boundary. This is the measured defense of that sentence — across seven substrates and two computing paradigms, on open tooling.
The claim
Computation is universal — the same finite set of mathematical primitives underlies every workload — but its energy is not. Run a primitive on the right physics and it is nearly free; the cost is at the boundary. Everything below is the measured defense.
The method — implement everything, then measure it
- Implement all of it. The Periodic Stack — 265 mathematical primitives — each emits a buildable, verified artifact: 265/265 REAL, 0 stubs, guarded against drift.
- Collapse it. 265 primitives → ~135 reconfigurable cores → 8 chiplets → one SiP, benchmarked by density, not TOPS.
- Measure it, don't model it. Every number is real sky130 / CORNERSTONE silicon or a first-principles simulation, tool named, every budget labeled.
The substrate axis — four orders of magnitude to the Landauer floor
The same elementary switch, seven ways (energy per switch):
| substrate | aJ / switch | grounding |
|---|---|---|
| digital CMOS 130 nm | 1000 | yosys + OpenSTA |
| PIM crossbar | 660 | ngspice |
| analog translinear | 165 | ngspice + sky130 devices |
| reversible adiabatic | 35 | ngspice + sky130 devices |
| SFQ superconducting | 0.247 | first-principles RCSJ |
+ photonic (layout-measured on CORNERSTONE, energy-budgeted) and acoustic (wave-simulated FDTD). Crossed with a representation axis (ternary / int8 / LNS / FHE), it is a 2-D energy map — every cell a real number. See the measured evidence →
The compute is nearly free; a boundary sets the true cost.
Measured, with a shape on each substrate: digital pays for data movement; analog for the ~5–6 bits mismatch lets it deliver; reversible for the clock ramp; photonic for thermal hold + O/E; PIM for the ADC per column; acoustic for IDT transduction. SFQ is the rung where the law explains a win — its 0.25 aJ switch isn't the point; even with the ~1000× cryocooler it wins ~1000× at the system level, because it nearly erases the data-movement boundary that dominates CMOS.
Acted on, then generalized
Acted on: match the conversion to what the physics delivers, and stay native as long as possible. Optimizing conversion energy blind drives a crossbar to 128×128 — where it delivers 0.1 effective bits and digitizes noise. The compute-aware optimum (4-bit, 16×16 tile) runs at 10.7 fJ/MAC, ~396× under the measured digital MAC — honestly bounded. (2025 SOTA independently derives the same rule: "Compute SNR-Optimal ADCs.")
Generalized — it survives a change of paradigm. An Ising machine doesn't compute a number, it relaxes into an answer (the oscillator machine found the true max-cut on 7/7 graphs). There the boundary isn't conversion in time — it's connectivity in space (all-to-all O(N²) coupling vs O(N) planar wiring). Same law, two faces: conversion for arithmetic, connectivity for optimization.
Two proofs it's real, not rhetoric
- RTL → tapeable open silicon. The universal FHE/ZK datapath (NTT butterfly) routed to a DRC-clean GDS on sky130 — 0 violations / 145 rules — the reference layout the field only ever publishes as closed-node projections.
- The law settles a live question. Optical NTT: measured, the twiddle-multiply optics can remove is only 17% of the butterfly; the nonlinear mod-q reduction (83%) can't be optical. The real photonic-FHE win is interconnect — data movement — exactly the boundary the law names.
What it means
The design problem is not the compute substrate — the multiply is solved everywhere. It is the boundary: convert as little as possible, to only the bits the physics delivers, staying in the native domain and native connectivity across as many operations as you can. That is where every substrate's real energy is decided — and it is the same on all of them.
And it is all open and live: 265 primitives downloadable, the composition computable, the measured evidence surfaced, on a free PDK. The open stack can take any primitive from a math definition to a tapeable layout — and then tell you, honestly, where its energy actually goes.