{"id":"mrh-monotone-reparameterization","title":"The Monotone Reparameterization Hypothesis","subtitle":"Order right, size wrong: the family-owned warp of foundation-model error, its confirmed predictions, and its registered kill conditions.","category":"conjectures","tags":["mrh","error-geometry","mlip","ordinal"],"source":"articles/docs/mrh-monotone-reparameterization.md","lang":"en","words":1150,"readMinutes":5,"toc":[{"depth":2,"text":"The claim, in one sentence","id":"the-claim-in-one-sentence"},{"depth":2,"text":"Formal statement","id":"formal-statement"},{"depth":2,"text":"Evidence (all measured on the 21-material × 4-model Y-matrix, this repo)","id":"evidence-all-measured-on-the-21-material-4-model-y-matrix-this-repo"},{"depth":2,"text":"Relation to the hyper-ribbon program","id":"relation-to-the-hyper-ribbon-program"},{"depth":2,"text":"Relation to published work (honestly bounded)","id":"relation-to-published-work-honestly-bounded"},{"depth":2,"text":"v2 — The refined law: the family exponent (2026-07-02, later the same day)","id":"v2-the-refined-law-the-family-exponent-2026-07-02-later-the-same-day"},{"depth":2,"text":"What would kill MRH (registered, in the typed-claim-shape sense)","id":"what-would-kill-mrh-registered-in-the-typed-claim-shape-sense"},{"depth":2,"text":"Practical consequence (the part a lab uses tomorrow)","id":"practical-consequence-the-part-a-lab-uses-tomorrow"}],"html":"

The Monotone Reparameterization Hypothesis (MRH)

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Status: unified statement, 2026-07-02. Every numeric claim below is drawn\nfrom provenance-hashed artifacts in data/y_matrix_runs/ and sealed as\ntype-checked Lean modules (lean/Round1..6_*.lean). Novelty claims are\nbounded by two adversarially-verified deep-research sweeps (≈200 sources,\n3-vote verification); community replication is the final arbiter.

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The claim, in one sentence

Foundation interatomic potentials learn the physics of a property family up\nto a single monotone warp of the property axis: they get the order of\nnature right and the size wrong, in a way that is one invertible distortion\nper (model, property family) — measurable with a handful of anchors,\ninvertible with proof, and provably absent where training data was too thin\nto fix even the order.

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In plain language: ask a foundation model which metal has the higher\nsurface energy and it almost never lies; ask it how much and it is wrong by\na warp — the same warp for every metal and every facet in the family. Warps\ncan be measured and undone. Scrambles cannot — and we can prove, in a proof\nchecker, which one you have.

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Formal statement

For model M and property family F, there exists a strictly increasing map\nw_{M,F} such that predictions satisfy P ≈ w_{M,F}(T) across materials, where\nT is the reference value. Corollaries:

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  1. Ordinal invariance — rankings of T are preserved by P (the\ninvertibility condition for w).
    2. \n
  2. Family-level warp — one w serves all properties in F (facets of a\nsurface share w; the family, not the property, is the warp's domain).
    3. \n
  3. Training-distribution control — w's distance from identity is set by\ntraining-data bias, not architecture; better-distributed retraining moves\nw toward identity without changing what is ordered.
    4. \n
  4. Correction = inversion — calibration is w⁻¹, computable from a few\nanchor (P, T) pairs by isotonic regression; it lives in the monotone\ngroup.
    5. \n
  5. The boundary — where training data was insufficient to fix even the\norder, no monotone correction exists (provably), and the model's output\nfor that (M, F) must be discarded, not calibrated.
    6. \n
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Evidence (all measured on the 21-material × 4-model Y-matrix, this repo)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
#ObservationWhere sealed
1Rankings survive where magnitudes fail: ρ = 0.82–1.00 across surfaces, vacancies, B₀; 22/22 facet orderings; MPA-0's γ₁₁₁ ranking is exactly the reference permutationRound3_Ordinal.lean (23 thms — kernel checks the order itself)
2Scalar (affine) corrections fail at every grain — per-model, per-material, within-familyRound1_H4.lean; H4 artifact
3Monotone correction succeeds where MRH says it must: CHGNet surfaces 36→9%, MPA-0 γ₁₁₁ 9→2% (LOO); the correction computed in-kernel for the flagship (Pt: 12.9→0.6%)Round4_Isotonic.lean
4Family-level warp (confirmed prediction): γ₁₀₀-fitted map corrects unseen γ₁₁₀/γ₁₁₁ — CHGNet 26.8→9.9%, MACE-small 13.0→6.7% — beating same-facet calibrationRound6_MRH.lean
5Warp→identity ordering (confirmed prediction): pooled warp magnitude 0.051 < 0.099 < 0.120 < 0.388 strictly orders MPA-0 < MACE-med < MACE-small < CHGNet by training lineageRound6_MRH.lean
6Bias/variance split: OMat retraining removes warp bias (s 1.08→0.97, Fe flips soft→stiff) but not per-material varianceR2 artifact; Round2_Verdicts.lean
7The boundary is real and provable: MPtrj SFE rank inversion ⇒ no monotone correction exists (quantified impossibility theorem, not policy)Round4_Isotonic.lean (sfe_mptrj_uncorrectable)
8The warp is a single-model object: separately-fitted potential families (EAM, 5 and 24-cell tests) do not share a warp and are not family-calibratablethis session's EAM artifacts

Relation to the hyper-ribbon program

Round-1's registered H1/H2 "kills" refuted linear low-dimensionality\n(participation ratio, leading-mode cosines — linear instruments). MRH locates\nthe structure those instruments could not see: the error manifold is\nlow-dimensional after allowing monotone reparameterization — one curved\ndegree of freedom per family. The ribbon is real, and it is curved. The\nTranstrum–Sethna sloppiness picture survives in nonlinear form; the linear\nkills were the necessary step that forced the nonlinear formulation.

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Relation to published work (honestly bounded)

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v2 — The refined law: the family exponent (2026-07-02, later the same day)

Exploration past the nonparametric statement found the warp's form:

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pred ≈ c · T^α — log-affine, R² = 0.93–0.98 (surfaces, B₀).\nThe exponent α belongs to the property family; the prefactor c belongs to\nthe model.

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Evidence (sealed in Round7_FamilyExponent.lean; CIs in the analysis run):

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Every earlier result is a corollary: scalar corrections fix c while α ≠ 1\n(H4's failure); monotone maps contain power laws (isotonic's success); facets\nshare both numbers (family transfer); rank preservation is what any power law\ndoes; and the SFE impossibility marks where even the power-law form collapsed.

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Honest bounds: three families, four models, one lab; vacancy CIs are wide\n(CHGNet's spans 0.44–1.19); α family-constancy is CI-overlap, not proven\nidentity; MPA-0's surfaces remain a PASS cell (raw beats every correction).

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What would kill MRH (registered, in the typed-claim-shape sense)

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Practical consequence (the part a lab uses tomorrow)

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