Treecode-AI · research program and company

The cost of reasoning is a measurable property of the verifier.

Every system that reasons today — a chain of thought, a program synthesizer, an agent planning its next action — proposes a few candidate steps, scores them, and keeps the promising ones. That makes the space of answers a tree, and a tree with scored steps is a known object of statistical physics. Treecode uses that physics to predict, with no adjustable parameters, when a search finds the correct chain and how wide it has to be — from numbers measured on the verifier in an afternoon — and then tests the prediction against real learned verifiers, pre-registered, with the results published as they come in.

The object. b candidates per step, d steps, b^d chains, one of them correct (red). A beam keeps B prefixes alive per level. The whole question is whether the red chain survives the pruning.

The line

A verifier either clears a threshold or it does not — and depth has nothing to do with it.

Give each step an energy (minus the verifier's score) and each chain the sum of its steps. Wrong chains then form Derrida's random energy model, the simplest spin glass; the one correct chain is a planted low-energy state. Among b^d wrong chains the lowest reaches −σ d√(2 log b), and the correct chain sits at −δ d. It is the ground state exactly when

\frac{δ}{σ} > \sqrt{2 \log b}

δ: the verifier's mean margin per step; σ: its spread on wrong steps; b: candidates per step.

Depth cancels. Whether the correct chain can be found by energy at all is set by the per-step margin and the branching factor alone. Below the line, no width of search succeeds at depth; above it, search can succeed at any depth. It is the same algebra as the Bryngelson–Wolynes condition for a protein sequence to fold rather than misfold — the founder's field — and it is the book's freezing criterion.

b = 102.15b = 26 (our library)2.55b = 1003.03

Derivation 1 in the primer (sign in) →

Two energy pictures: the band of wrong-chain energies in gray; in (a) the correct chain sits inside the band, in (b) it lies below the band, separated by a gap. — The REM picture and the gap. The *b^d* wrong chains form a band; its lowest member reaches −σ√(2 log b) per step. (a) A margin smaller than that leaves the correct chain inside the crowd, unfindable by energy however wide the search. (b) A larger margin puts it below every wrong chain, separated by a gap of δ − σ√(2 log b) per step.

What it predicts

How wide the beam must be, and how that grows with depth.

Above the line, the number of wrong continuations that undercut the correct prefix has a power-law tail whose exponent is set by how far above the line the verifier sits:

ρ^{*} = \frac{{(δ / σ)}^{2}}{2 \log b} - 1, B \propto d^{1 / ρ^{*}}

ρ*: the Pareto exponent of competitors; B: beam width; d: depth.

At the threshold ρ* = 0 and every moment diverges. At ρ* = 1 the mean number of competitors diverges — this is exactly Gallager's computational cutoff rate of sequential decoding, because the tree is a tree code of orthogonal signals on a Gaussian channel; the formulas coincide. At twice the threshold ρ* = 3 and a sixteen-fold increase in depth costs only 2.5× the width. The per-level loss bound K B^−ρ is proved; the rest is simulated to depth 160 and width 10,000 in the monograph and reproduced to the printed digits by the code.

Derivation 2 in the primer (sign in) →

Beam width needed to reach ninety percent of the ceiling, against depth, for three verifier margins; the lines are the width law anchored at depth ten. — The width law on simulated trees (b = 10). Markers: the smallest width that reaches 90% of the exact-recovery ceiling at depths 10, 40, 160. Lines: B ∝ d^1/ρ*. At δ/σ = 2.5 the width explodes with depth; at 4.0 it barely moves.

Why it matters for cost

Levinthal's count: a step verifier is the whole game.

A protein cannot find its fold by trying conformations one by one, yet it folds in milliseconds, because the landscape rewards partial progress. Reasoning systems obey the same arithmetic. A verifier that can only judge complete answers cannot prune, so an uninformed proposer must produce of order b^d chains. A verifier that judges steps, with a margin above the line, lets a beam examine

b^{d} against d b B

complete chains for an outcome verifier, against nodes expanded by a beam with a step verifier above threshold.

For b = d = 10 that is 10¹⁰ against 10⁴: six orders of magnitude, and the difference between a system that needs a data center and one that runs on the processors already in the building. The target, therefore, is a learned step verifier whose measured margin clears the line — and a step verifier below the line is no better than none.

Verifier evaluations needed with an outcome verifier (b^d) and with a step verifier above threshold (d·b·B). Left pair: the monograph's reference case. Right pair: the first search in this repository, where the scorer turned out to be an outcome verifier, and what a step verifier would cost on the same tree.

Measurements

A theory with no free parameters, tested against real verifiers.

The code reproduces the monograph's tables to the printed digits, the predictor reproduces the model when fed the model's own samples, and the harness is closed on an oracle before any real system is scored. The measurements themselves — the pre-registered Milestone 1 test of learned step verifiers, the ARC-AGI-1 calibrations and every run behind them — live on the Treecode platform, where each experiment can be re-run with its parameters and its history is kept.

Pre-registered, with a kill criterion

The protocol, the informativeness rule and the refutation rule are fixed in a dated file before any result exists; every departure from the registered parameters is recorded with the run.

The harness is checked on the model first

Oracle step scores drawn from the model itself go through the same real program search and must reproduce the model's own numbers before a real system is scored.

Refutations are kept

A model with no free parameters is useful because it can lose. When it does, the measurement names the input that was taken too coarsely, and the next registration carries the correction.

The research program · CPU-only · twelve weeks

Each milestone is a measurement.

M0Apparatus2026-10-01done
M1Learned step verifier2026-10-02first pass
M2Proposer, coverage, depth 3–6weeks 3–6planned
M3Width law on a real systemweeks 6–9planned
M4Preprint and releaseweeks 9–12planned

The funded program beyond M4 is the monograph's WP2–WP6: a small recurrent core with fixed-point halting, family supervision, attention thermodynamics inside the core, externalised knowledge, and the CPU systems path — each with the same pre-registered discipline as WP1.

What it is for

Three products fall out of one measurement.

CPU-native reasoning inference

Program-synthesis and structured-reasoning engines — data transformation, spreadsheet logic, code repair, step-verified formal reasoning — whose search runs on commodity cores, priced per solved task. Levinthal's count is why: a step verifier above the line turns b^d into d·b·B, and the width law keeps B small when the verifier sits at twice the threshold.

For on-premise and air-gapped enterprises, edge devices, and anyone whose constraint is per-task cost at volume.

The search-cost oracle

A service that takes a team's verifier calibration data — the score histograms of correct and wrong steps — and returns the beam width, the samples and the dollars needed for a target success at a target depth, and whether the next dollar is better spent on verifier margin or on compute. Every test-time-compute and agent team makes this decision blind today.

For teams building reasoning models, agents and RL-with-verifiers pipelines; for providers who price inference.

Design as search

Molecules, proteins and materials are searched the same way — propose a modification, score it, keep the promising ones. The same harness measures whether a docking score or a stability predictor is a step verifier above threshold for the design tree it is used on, and how wide that search must be.

The longer-horizon application, and the one where the physics is native.

Open-core. The model, the harness and the benchmark results are open; trained proposers and verifiers, the CPU inference stack and the oracle service are licensed. The ARC Prize is the public proving ground: a measured, reproducible result there, with cost per task on CPUs, is the demonstration that the method is real.