The Periodic Stack of Verification
A verified output is a coordinate.
Every machine-checked artifact — a tactic-line closure, an SMT certificate, an Olympiad lemma, a CompCert recompilation, a Mathlib promotion — occupies a place on the same map. Seven axes describe the shape. Position predicts what produced it, what checked it, and how many joules it cost.
c = ⟨ S, V, E, D, C, R, τ ⟩
The cell coordinate · seven axes, one tuple
The thesis
The cheapest sufficient solver wins.
A proof artifact carries its production history. What tactic emitted the step, what oracle checked it, what cache the lemma came from, what difficulty the goal sat at, whether the same attempt would repeat — these are not metadata. They are the coordinate. Read the coordinate and the joule bill explains itself.
Most subgoals in any real proof close at the cheap end of the ladder — kernel automation, then a decision procedure, then a small in-domain model, escalating to the frontier only on failure. A system that always reaches for the largest model spends twelve orders of magnitude more than the goal required. The waste is not in the substrate. The waste is in the abstraction layer that decides what to invoke.
The map is a way of seeing, not a verdict. It classifies what is provable, makes the structure of the design space legible, and lets a verification stack be reasoned about before it is built — the same move the sibling stacks make for computation, synthesis, data, state, memory, and communication.
The design space
Seven axes describe a verified output.
Three are shared with the sibling stacks — E inherits the Landauer floor from compute and memory, V is shared with synthesis, data, and communication, and τ with data, state, memory, and communication. The other four are proof-specific.
Solver tier
01 · WHAT PRODUCED THE STEPplacement on the escalation ladder
Verifier kind
02 · WHAT CHECKS THE ARTIFACTthe deterministic judge
Energy class
03 · JOULES PER VERIFIED OUTPUTlog₁₀ orders above Landauer floor
Difficulty regime
04 · WHAT THE GOAL ISwhere the subgoal sits on the difficulty scale
Cache regime
05 · WHETHER THE WORK WAS DONE BEFOREreuse class of the closure
Reproducibility
06 · WOULD THE SAME ATTEMPT YIELD THE SAME PROOFdeterminism class of the producer
Persistence
07 · WHAT HAPPENS TO THE ARTIFACTlifecycle after first verification
The coordinate space
The addressable map.
The product of the axis cardinalities is the discrete coordinate space. Structural constraints prune it to an admissible remainder on which every verified artifact has a place.
5 · 6 · 6 · 5 · 4 · 4 · 4 = 57,600
Discrete cell count · the addressable map of proof-space
discrete cells
orthogonal axes
solver tiers on the ladder
The relation
Joules per verified output, bounded below.
E(x) ≥ θ(D) · μ(S, V)
The Kepler step · Landauer-class θ for difficulty D, scaled by impedance mismatch μ
The joules to verify an output x are bounded below by a Landauer-class θ for difficulty D, multiplied by an impedance-mismatch μ between the solver tier S and the verifier kind V. Cheap solvers paired with their natural verifiers sit at the floor. Mismatch is multiplicative: a frontier model invoked to close a decidable arithmetic goal is twelve orders above the floor not because the goal is hard but because the solver was wrong for it.
This is the direct child of the relation on stack.openie.dev: E(x) = Σ θ(p) · μ(p, H). It is testable as the table fills with measured (E, S, V, D) tuples reported alongside pass-rate. Empties at well-attested intersections become predictions.
Companion stacks
One method, applied across fields.
The same axes recur. E inherits from the compute and memory stacks. V is shared with synthesis, data, and communication. τ is shared with data, state, memory, and communication. A primitive carries consistent coordinates across all of them.
Compute
258 primitives, 33 families.
Synthesis
The seven-axis synthesizer stack.
Data
The ten-axis data-system stack.
State
The nine-axis state stack.
Communication
The eleven-axis channel stack.
Memory
The eleven-axis retention stack.
Knowledge
Seven axes · the object the others handle.
Agent
Four axes · the actor and selection.
Stack
The overview · three names, one substrate.
OpenIE
The parent thesis.