Date: 2026-06-16
Scope: paper2/projection-law.tex (PRX master), paper2/immi/projection-law-immi.tex (IMMI companion), paper/immi-paper.tex (classical-potential discovery substrate), the lean-spec formalization, and the library-site public surface.
Status: Working papers, not yet submitted.
Reviewer: Kimi Code CLI (internal; adversarial review requested before further publication).
The projection-law manuscript is an ambitious, cross-layer attempt to turn a\nqualitative worry about model-ensemble agreement into a geometric law with\nmachine-checked theory, pre-registered factorial experiments, and honest failure\nreporting. The empirical signal is genuinely striking: errors cluster by\nconstraint (functional, training functional, pseudopotential table) rather than\nby implementation at three layers of one stack, and the anisotropy is conserved\nacross paradigm replacements. The formal core is now substantially extended\n(convex consensus, PR gauge, decoupling, affine decomposition, smooth non-convex\nlocal law, finite-sample concentration) and build-locked in Lean 4 with zero\nsorry.
Before any journal submission, however, several issues need attention. The most\nimportant are: (1) the logical relationship between the new affine/smooth theorems\nand the global consensus/gauge claims is oversold in places; (2) the empirical\nsample sizes and permutation floors limit the strength of the MLIP and DFT\nfactorial claims; (3) internal counts and cross-references were inconsistent\n(181 vs. 225 theorems; 4 vs. 7 theorems) — already fixed during this review;\nand (4) the public-facing lupine.science marketing page and the GCS-hosted\nworking-paper PDF remain stale.
Recommendation: Address the conceptual-clarity issues below, run the\nadversarial multi-agent review pass recommended in TARGETING.md, upload the\nrebuilt PDF to a stable versioned URL, and only then submit.
replication/error-geometry/ kit makes this auditable.AffineDecomposition, SmoothProjection, and FiniteSampleConcentration\nmodules; Vision.lean #checks all seven core theorems. lake build is\ngreen (2891 jobs, 0 sorry).Theorem 5 (affine decomposition) shows that for a closed affine reachable set
\nK=a+L, the residual splits into a shared bias b∈L⊥ and a\nwithin-family component ξ(p)∈L. The paper then writes:
\n\n\n"Theorem 5 turns the bias-plus-noise spectrum of Theorem 3 from a modeling\nassumption into a derivable consequence for affine families."
\n
This is too strong. The affine decomposition gives an orthogonal split of the\nresidual, but Theorem 3 additionally assumes:
\nNone of these follow from the affine decomposition alone. What Theorem 5\ndoes license is the existence of a shared bias direction and an orthogonal\nwithin-family subspace; the isotropic-noise model remains an assumption that\nmust be justified empirically (which the paper does, but not as a theorem).
\nSuggested fix: Reword to: "Theorem 5 turns the bias-plus-within-family\nsplit of Theorem 3 from a modeling assumption into a derivable consequence for\naffine families; the isotropic-noise gauge remains an empirical regularity\nsupported by the data in §...".
\nTheorem 6 (smooth non-convex local normal cone) is pointwise: at a local\nminimizer of ∥T−f(x)∥, the residual is orthogonal to the tangent space.\nThe text correctly notes that it is "not the global consensus theorem," but\nimmediately thereafter the MLIP analysis treats cross-architecture cosines of\n0.95–0.99 as evidence that the same residual is shared across architectures.\nThat inference requires a global or near-global consensus theorem, which does\nnot hold for general non-convex families.
\nSuggested fix: Add an explicit bridge: the local theorem justifies testing\ntangent-space orthogonality; the empirical clustering is then interpreted as\nlocal minimizers landing near the same fitted point / normal space, not as a\nformal uniqueness result. Avoid language that suggests Theorem 6 "licenses"\nthe global consensus conclusion.
\nTheorem 7 gives an entrywise Hoeffding bound for the empirical second-moment\nmatrix. The paper also notes (correctly) that the participation ratio is\ncontinuous where the denominator is non-zero. But continuity plus entrywise\nconvergence does not give a concrete sample-complexity bound for $|\\widehat{\\rm\nPR} - {\\rm PR}|$ because PR is a ratio of quadratic forms and the denominator\ncan be arbitrarily small. The manuscript does not currently use this theorem to\nquantify uncertainty in the measured PR values.
\nSuggested fix: Either (a) derive a finite-sample PR bound under a\nnon-degeneracy assumption on the population spectrum, or (b) present Theorem 7\nas a proof-of-concept that the empirical second moment converges to the\npopulation object, with PR uncertainty handled empirically by bootstrap (which\nthe classical-layer analysis already does). Do not imply that the entrywise\nbound alone governs PR convergence.
\nThe MatPES 4×2 test has only n=8 cells. The exact permutation
\np-value is 0.029, which equals the lattice resolution floor 1/70. The\nregistered effect-size prediction also failed: observed separation 0.085 vs.\nregistered threshold 0.30. The paper reports this honestly, but the\nresulting claim should be framed as "direction of clustering is significant\nbut the predicted magnitude is not supported" rather than as a clean\nconfirmation of the law at the MLIP layer.
\nSuggested fix: In the abstract and layer summary, qualify the MLIP result\nas "significant directional clustering by training functional, with the\nregistered effect-size component failing." This is already in the body; make\nsure the summary voice matches.
\nFour of seven predictions failed. Because the predictions were not all testing\nthe same hypothesis, simple Bonferroni correction is not obviously required,\nbut the reader needs a clearer account of which predictions were primary vs.\nauxiliary. The paper mentions that round 2 will use a single-primary-endpoint\ndesign; for the present manuscript, a short paragraph specifying the\nhierarchical testing structure would help.
\nThe MLIP layer compares model predictions to experimental references for FCC\nmetals and to Materials Project (DFT) references for BCC metals. The classical\nlayer uses experimental references throughout. This mixing is acknowledged in\npaper/immi-paper.tex but should also be flagged in the projection-law\nmanuscript, because it means the MLIP-layer residual is not measured against a\nuniform reference standard.
The paper is careful to distinguish ensemble error-vector ribbons from\nparameter-space sloppy-model ribbons, but the term still invites confusion\namong readers familiar with Transtrum/Sethna. Consider adding a one-sentence\ngloss at first use in the abstract: "ribbon here means low-dimensional\nerror-vector geometry, not the parameter-manifold hyper-ribbons of sloppy\nmodels."
\nThe Lean artifact is the strongest part of the package. The mapping from paper\ntheorems to Lean declarations is now:
\n| Paper theorem | \nLean declaration | \nFile | \n
|---|---|---|
| Theorem 1 (normal-cone criterion) | \nConvexProjection.IsBestApproxOn.residual_mem_normalCone | \nConvexProjection.lean | \n
| Theorem 2 (consensus) | \nProjectionLaw.IsBestApprox.residual_eq | \nProjectionLaw.lean | \n
| Theorem 3 (gauge) | \nSpectrumBridge.prSpectrumFin_biasNoise etc. | \nSpectrumBridge.lean | \n
| Theorem 4 (decoupling) | \nErrorGeometry.axis_pr_one, ribbon_consensus_decoupled | \nErrorGeometry.lean | \n
| Theorem 5 (affine decomposition) | \nAffineDecomposition.AffineFamily.decomposition | \nAffineDecomposition.lean | \n
| Theorem 6 (smooth local law) | \nSmoothProjection.SmoothFamily.residual_orthogonal_to_tangent | \nSmoothProjection.lean | \n
| Theorem 7 (finite-sample concentration) | \nFiniteSampleConcentration.empiricalSecondMoment_entrywise_concentration | \nFiniteSampleConcentration.lean | \n
All seven are #checked in Vision.lean and the full lake build is green.\nThe computationallyProvenCount was bumped to 77.
One gap: the paper's reproducibility sections previously claimed "181\ntheorem and lemma declarations" and "four theorems of this paper"; this was\ninconsistent with the updated formal core. It has been corrected to "225"\nand "seven" in both .tex sources and the PDFs rebuilt. Make sure any other\nmanuscripts (e.g., paper/immi-paper.tex, paper/review-ready/*.tex) are\nsimilarly audited.
paper2/quality_gate.py passes locally: 42 citations, 42 bibliography entries,\n4 figures, no placeholder hits.replication/error-geometry/ kit is commit-versioned and publicly served....v2026-06-11b.pdf) is stale and should be replaced or\nremoved to avoid confusion.library.lupine.science: the working-papers.html page was updated with the\nnew theorem count and PDF link and will auto-deploy via\n.github/workflows/deploy-library-site.yml.lupine.science: no marketing-page source is present in this repo. If the\nmain site currently links to the old GCS PDF, that link is now stale.atlas/atlas-view agent guides (llms.txt, llms-full.txt) still describe\nthe general research program accurately and do not need urgent change.FINAL_DRAFT_REPORT.md says "14 pp." Update to 15.projection-law-immi.tex still has\nORCID: 0009--0000--0000--0000 (to be completed).FINAL_DRAFT_REPORT.md lists this as a known remaining\nmanual step; placeholder DOIs in the manuscripts need to be replaced before\nsubmission.\\citep{...} consistently\nexcept in a few Related Work sentences where the author-year intent would be\nclearer with \\citet{...}; this is a polish item, not a blocker.paper/immi-paper.tex still says "180-theorem Lean 4 corpus" in the\nworking-papers HTML; the HTML was updated but the source .tex was not.library-site/src/reports/working-papers.html to point there instead of the\nGitHub raw file (which is fine as a stopgap but not a permanent submission\nartifact).TARGETING.md\nbefore submitting anywhere.The projection-law package is submission-ready in substance, but it is not yet\npolished enough for a flagship journal. The formalization is sound, the\nempirical design is strong, and the failure reporting is admirable. The main\nrisk is overclaiming how far the new affine/smooth/finite-sample theorems\nextend the core law. Fix those conceptual framing issues, clean up the public\nsurfaces and placeholders, and run the planned adversarial review before\npressing submit.
\nThe first pass of fixes was applied in the same cycle:
\npaper2/projection-law.tex and immi/projection-law-immi.tex already state that Theorem 5 gives the bias/within-family split, not the isotropic-noise gauge; that Theorem 6 is pointwise and does not imply global consensus; and that the finite-sample bound is entrywise, with PR continuity only preventing discontinuous jumps. Table 1 and the MLIP abstract now note the permutation-floor nuance.library-site/src/assets/papers/projection-law-v2026-06-16.pdf and projection-law-immi-v2026-06-16.pdf; working-papers.html and the catalog article point to those URLs.library-site/scripts/catalog.js) and linked it from the Working Papers page.Remaining open gates: fill ORCID and Zenodo DOI placeholders; run the adversarial multi-agent review pass in TARGETING.md; update the external lupine.science marketing page (source not in this repo).