\n\nWe are not running LAMMPS. We are formalizing what it means to validate without running it all.
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This page explains why we built a Lean 4 formal specification instead of installing LAMMPS and running simulations.
\nComputational materials science operates on a false dichotomy:
\nThere is a third option:
\nAn interatomic potential claim sits on a spectrum:
\n| Position | \nExample | \nStatus | \n
|---|---|---|
| Formal proof | \nPR ≤ min(P, N) for differentiable potentials | \nProven for synthetic data; real proof needs inverse function theorem | \n
| In between | \nGround-truth NIST errors converge to the synthetic manifold | \nWell-posed conjecture; no ground truth yet | \n
| Physical experiment | \nNIST IPR elastic constants | \nAvailable but not yet computed with our pipeline | \n
The "in between" is where most validation science actually lives. Papers pretend they are at the experiment end. We formalize that they are in the middle.
\nWe embedded 72 FCC + 42 BCC synthetic entries directly into the Lean source. These are not loaded from a file at runtime — they are compile-time constants.
\nWhy this matters: If someone edits the data, the theorems about the data are re-checked at build time. You cannot silently change a value and break a proof without the build failing.
\nTheorems at this layer:
\nsyntheticFccCount = 72)syntheticFccIsSynthetic = true)nistScaffoldAlMissing = true)native_decide)We compute statistics (correlation, participation ratio, Simpson's paradox detection) and prove properties about the computed values.
\nWhy native_decide: Lean compiles the expression to native code, runs it, and checks the result. This is not a symbolic proof — it is a reproducible computation that happens at compile time.
Theorems at this layer:
\npooledRBelowMinus08 — pooled correlation < −0.8fccEamPRBounded — PR ∈ (1.2, 1.3)fccAllSatisfiesHyperRibbon — PR/n < 0.5Some claims require ground-truth data we do not have. Rather than pretend they are proven, we document them as gaps.
\nWhy documented gaps are honest: Every theorem is fully proven. The gaps are not sorry proofs — they are explicit records of what would need to happen to upgrade from synthetic to NIST-backed validation.
Documented gaps:
\nWe could install LAMMPS, download NIST potentials, and compute elastic constants for Al. Here's why we chose not to:
\nScope control. Running one simulation creates pressure to run ten. Formalizing the validation structure is invariant to how many simulations we run.
\nReproducibility. A LAMMPS trace is a reproducible artifact — but only if you save the exact input script, potential file, and LAMMPS version. We formalized what a valid trace looks like (Computation.LammpsTrace.lean) before producing any traces.
Epistemic clarity. Running a simulation gives you a number. Proving a theorem gives you a guarantee that holds for all inputs in a class. We need both, but the formal layer comes first.
\nCost. GPU time is expensive. Compile time is cheap. We can iterate on hypotheses at compile time, then run simulations only for the hypotheses that survive formal scrutiny.
\nOur audit report is not a blog post. It is a computed string generated from proven theorems:
\ndef fullAuditReport : String :=\n "[CLAIM 1] Simpson's Paradox in BCC Elastic Constants\\n" ++\n " " ++ simpsonsParadoxVerdict ++ "\\n\\n" ++\n "[CLAIM 2] Hyper-Ribbon Manifold Dimensionality\\n" ++\n " " ++ hyperRibbonVerdict ++ "\\n\\n" ++\n ...\nThe verdict strings are themselves proven:
\nsimpsonVerdictContainsFabricated — the string contains "FABRICATED"hyperRibbonVerdictContainsConsistent — the string contains "CONSISTENT"If the computed statistics shift, the build fails. The audit cannot drift.
\nThe documented gaps define exactly what needs to happen to close the epistemic boundary:
\nEach step is a well-posed theorem schema. The gap is a TODO list written in logic.
\n| Approach | \nCost | \nGuarantee | \nGap visibility | \n
|---|---|---|---|
| Run everything | \n$$$$ | \nNone (never complete) | \nHidden | \n
| Trust the PDF | \n$ | \nNone | \nHidden | \n
| Formalize the gap | \n$ | \nCompile-time checked | \nExplicit (documented) | \n
We chose the third path. The build passes. The gap is documented. The hypotheses are falsifiable.
\nThat is what it means to be in the in between.
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