"Low-dimensional error" in this corpus refers to three different objects. They are\nrelated but not identical, and conflating them produces real mistakes (it produced\nseveral in a 2026-06-02 working session). This page is the canonical disambiguation;\neverything else should link here rather than re-define the terms.
\nThe literature foundation is\narchive/swarm_preprint_review/research/immi_dim01_sloppy_theory.md\n(25+ primary sources). Read it for the full chain; this is the compressed map.
| \n | A. Model manifold (the hyper-ribbon) | \nB. Participation-ratio measure | \nC. Configuration-space error core | \n
|---|---|---|---|
| Lives in | \nmodel prediction / data space | \nobservable space (e.g. C11/C12/C44) | \nconfiguration space Ω ⊂ ℝᵐ | \n
| Is | \nimage y(θ) of the prediction map as parameters vary; a bounded manifold with a geometric hierarchy of widths Wₙ ~ W₀·Δⁿ | \nPR = (Σλ)²/Σλ² of the error covariance across potentials — a scalar effective-dimension measurement | \na low-dim core H with the error boundary a codim-1 tube around it (dim Γ = m−1) | \n
| Role | \nthe theory object: why errors are low-dimensional | \nthe empirical handle: how we measure A from data | \na distinct, more demanding rigor object | \n
| Source | \nTranstrum–Machta–Sethna, PRL 104 060201 (2010); PRE 83 036701 (2011) | \nparticipation-ratio defn (Cell Reports Methods 2022); immi_dim01 §4 | \nrepo-root PDF "A Conditional Universality Theorem for Error Geometry in MLIPs" | \n
| In this repo | \nthe framing; docs/sloppy_models_report.md | \nlupine-distill/src/hypothesis/manifold.rs; HyperRibbonEmpirical.lean | \nnot yet built; docs/science/keystone-reconciliation.md | \n
| Status | \nestablished theory | \nstandard, correct usage (B measures A) | \nconditional theorem; assumptions untested | \n
The model manifold is the surface y(θ) swept out in prediction space as the\nparameters θ vary over all allowed values; the parameters are coordinates on it and the\nmetric is the Fisher information gₐᵦ = JᵀJ. For sloppy models the manifold has a\nhierarchy of widths decreasing geometrically — much longer than it is wide, much wider\nthan it is thick — a hyper-ribbon (Transtrum–Machta–Sethna 2010/2011). The widths track\nthe Fisher eigenvalue spectrum; the hierarchy follows from interpolation/approximation\ntheory and model smoothness (Quinn et al. 2019/2023). Interatomic potentials are in this\nclass (Frederiksen 2004; Wen 2017 [OpenKIM]; Kurniawan 2022). When this corpus says\n"hyper-ribbon," it means object A.
We do not observe y(θ) directly; we observe prediction errors of many potentials on a\nfew properties. Stacking each potential's error vector (e.g. [C11_err, C12_err, C44_err]),\nforming the error covariance, and taking the participation ratio\nPR = (Σλᵢ)²/Σλᵢ² gives the effective dimensionality of that error cloud — a scalar\nbetween 1 and the nominal rank. The IMMI corpus measures PR ≈ 1.05–1.86 of 3 (a near-1D\nribbon with a thin secondary direction). This is the standard, literature-sanctioned way\nto measure the effective dimensionality of A from data (immi_dim01 §4). It is computed\nin lupine-distill/src/hypothesis/manifold.rs\nand recorded in lean-spec/.../Theory/HyperRibbonEmpirical.lean.
\n\nCaveat (do not skip). B is a linear effective-dimension measure and the elastic\nconstants share structural components (the Cauchy relation, mechanical-stability\nconstraints). Jackson–Somers (1991) and Archie (1981) show such shared components produce\na non-zero baseline correlation / PR independent of any shared physics. Any PR or\ncross-model-alignment claim must therefore be tested against a coupling-aware null,\nnot against
\nr = 0. (This is exactly the control the 2026-06-02 A6 test still lacks.)
The repo-root keystone paper studies a different thing: the geometry of the high-error\nregion in configuration space Ω ⊂ ℝᵐ. It proves a conditional universality —\nif there is a shared low-dim core manifold H ⊂ Ω (positive reach), a common monotone\nradial error profile, and small model-specific perturbations, then each model's high-error\nregion is a tube around H and the boundaries are pairwise diffeomorphic. Two cautions the\npaper makes decisively:
m−1, not low; the low-dim object is the\ncore H. A measure-theoretic concentration (like B) must not be called a manifold.C is not the same as A. It is a separate, rigor-first reframing, and the open scientific\nquestion is whether the A→C bridge (A6) holds. See\nkeystone-reconciliation.md.
B is the empirical measure of A; C is a distinct configuration-space object whose link to\nA/B is the untested A6 bridge. A and B are the established program (correct usage). C is the\nfrontier.
\n| Claim / artifact | \nObject | \nNote | \n
|---|---|---|
Hyper-ribbon framing; sloppy_models_report.md | \nA | \nthe theory | \n
HyperRibbonEmpirical PR ≈ 0.40 fractional; manifold.rs | \nB | \nthe measurement of A; coupling caveat applies | \n
hyper-ribbon-universality / mlip-transfer conjectures | \nA via B | \nsurvival of the low-PR structure across potentials/MLIPs | \n
RibbonProjection.lean | \nnone cleanly | \na scalar toy; not A, B, or C — keep as a concentration lemma, not "the ribbon formalized" | \n
keystone paper + keystone-reconciliation.md | \nC | \nconditional; A6 untested | \n
A6 alignment test (a6-alignment-results.md) | \nA→C bridge | \nprovisional; coupling-aware null still owed | \n
If you cannot say which one, the statement is not yet precise enough to publish.
\n"}