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Research note

An Order of Effort: Where Correction Laws Hold, and Where the Boundary Begins

The measured ladder from proven corrections to proven impossibility — and the order in which our next rounds climb it

A row of four increasingly specialized test bays — bulk compression, defect scanning, reaction vessel, interface microscopy — each material environment routed to the equipment its complexity requires.
Brand film: An Order of Effort: Where Correction Laws Hold, and Where the Boundary Begins

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The ladder: seven steps from calm to storm

We spent two campaigns measuring where foundation models fail hardest — barriers and interfaces, the observables where errors run to electron-volts and every cheap fix dies. People reasonably ask why we started at the deep end. The answer: a boundary is only visible from the far side. Now that we've walked it, the whole ladder is in view — where our correction layer already delivers, where only boundary theorems can speak, and the order of effort for the rounds ahead. This is that ladder, with the receipts.

Why map the hard end first

Barriers and interfaces are where MLIPs drift farthest: 135–243 meV of barrier error against a 40 meV gate across four foundation models, adsorption underbinding up to +26 eV. We deliberately ran there first — not because it was kind, but because a trust layer that works there works anywhere, and one that fails there was never going to be one. The result, honestly recorded: no low-dimensional correction survives on that error field, and we can prove it in Lean 4. But the same measurement produced something more useful than a win: the complete map of difficulty, from observables where corrections already pass to domains where the only honest answer is abstention.

The ladder, as measured

The boundary theorem as referee

A branching sample rail at a quiet allocation station: simple coupons enter a rapid screen while one complex interface coupon is diverted to a reference chamber — The physical switch routes work according to evidence difficulty

Tier 1 — statics (lattice constants, volumes, relaxed bulk). Our correction layer already wins here: direction-gated corrections cut lattice-constant error from 1.60% to 0.33% and 1.75% to 0.74% in a preregistered round. The error structure is small, one-sided, and tame — and the runtime cost is trivial.

Tier 2 — elastic response (bulk modulus, elastic constants). Harder, and we mark it honestly: our own kill condition fired on B0. The theorem work here is boundary theorems — classification of which cells may correct and which may not — not correction laws.

Tier 3 — surface energies. A smooth, low-dimensional error field: our environment-resolved correction cut one facet error from 9.7% to 1.5% at ~15% runtime overhead. Correction laws exist because the field is smooth enough to have an inverse.

Tier 4 — defect formation energies. Cross-model spreads of 0.5–1.8 eV with sign instability. Corrections need per-defect anchoring; the theorem work is scope restriction.

Tier 5 — transition-state barriers. Where we just spent two campaigns. The verdict: energy levels fail beyond rescue by any cheap correction, but energy geometry — saddle locations — is right 82–93% of the time. The boundary theorems price the only honest route: sparse DFT anchors at model-chosen extrema.

Tier 6 — interface adsorption. Structured underbinding fields to +26 eV; frozen corrections made the holdout worse. Only on-policy corrections (fitted during the search itself) have a chance here.

Tier 7 — magnetic and excited-state properties. No measurement path exists in current models. Not a correction domain — an abstention domain, by construction.

The rule that keeps the ladder from fighting itself

Non-competition is structural, and it’s the part worth stealing: each correction family is licensed only where the family below it provably cannot act. Level shifts are dead everywhere — a theorem kills them (barrier_shift_invariant), so they never compete. Slope corrections are licensed only by sign stability. Environment-field corrections apply only where a smooth field exists. Anchors are priced exactly: two extrema, exactness, and the model demoted to geometry guide. And a single wobble bound referees all of it: no information-free correction can beat the model’s profile wobble, ever. The ladder is not a pile of tricks; it’s an ordered stack of licenses with a machine-checked referee.

A high-pressure material cell enclosed by three nested mechanical containment rings, with the innermost ring carrying a restrained indigo boundary — The nested rings prevent a correction from extending beyond its tested material regime

What the next rounds look like

Round easy-first (now): Tier 1, where corrections are already proven. The goal is a model-independent runtime win: our system improving any model’s simulation loop on relaxed-structure observables, with overhead measured per cell. The correction layer is in the runner; the instrumentation is next.

Sparse-DFT pilot (executing): Tier 5’s priced route — models guide, DFT measures only at model-chosen extrema. In exact simulation on the locked panel, that reproduces barriers at 1–9 meV mean error at roughly 7 evaluations per path, against a gate the raw models miss by 3–6×. The memorandum of theory is frozen before results (PDF); the campaign fills its slots.

The boundary at each step: when a round hits the tier where its correction family provably can’t act, the boundary is already written down — that’s what the Lean layer is for.

The upside, stated plainly

What does this process buy? Three things nobody else in this space is selling. Measured trust: every accuracy claim hash-locked and preregistered, so “the model says” becomes “here is what the model is worth, per observable.” Machine-checked limits: the correction boundary means you never spend on a correction that can’t work — the impossibility is proved, not discovered. Priced accuracy: where corrections pass, runtime wins at laptop cost; where they can’t, sparse anchors deliver DFT-grade accuracy at ~10× less DFT compute — with the cost declared before the measurement, not after. That’s the offer, and the receipts are the pitch.

A phased research hall with a simple coupon rack nearest the viewer and progressively enclosed thermal, reaction, and interface test bays farther away — Samples advance only when each increasingly specialized test bay supplies evidence

Receipts