{"id":"z1-union-anchor-economics","title":"Z1 Sparse-DFT Pilot — Union-Anchor Economics","subtitle":"Cross-model cost analysis from recorded campaign artifacts: 558 naive vs 154 union anchors (72.4% fewer evaluations, 3.62×); union cost scales 139.5→154 from one to four models.","category":"validation","tags":["savings-stack","z1","union-anchor","sparse-dft","economics"],"source":"articles/docs/analysis/z1-union-anchor-economics.md","lang":"en","words":773,"readMinutes":4,"toc":[{"depth":2,"text":"1. Method","id":"1-method"},{"depth":2,"text":"2. Headline numbers (recomputed)","id":"2-headline-numbers-recomputed"},{"depth":2,"text":"3. The 29-vs-30 reconciliation","id":"3-the-29-vs-30-reconciliation"},{"depth":2,"text":"4. Discrepancy vs the prior chat-session numbers","id":"4-discrepancy-vs-the-prior-chat-session-numbers"},{"depth":2,"text":"5. Scaling note — union grows sub-linearly in model count","id":"5-scaling-note-union-grows-sub-linearly-in-model-count"},{"depth":2,"text":"Reproduce","id":"reproduce"}],"html":"

Z1 Sparse-DFT Pilot — Union-Anchor Economics

Schema: lupine.z1.union_anchor_economics.v1 · Data: data/candidates/z1-union-anchor-economics.json · Script: tools/analysis/union_anchor_economics.py · Recorded: 2026-07-21

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Cross-model cost analysis of the frozen sparse-anchor protocol\n(docs/plans/2026-07-20-sparse-dft-pilot-preregistration.md) on the locked\n30-path Z1 panel, computed from the recorded Round-4 float64 campaign\nartifacts only — no DFT was run. This note is the durable record of numbers\npreviously computed ad hoc in a chat session; the recomputed values below are\nauthoritative where they differ (see §4).

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1. Method

Per model m and path p, the model's recorded predicted_image_energies_ev\nprofile (completed predictions only, mirrored from\ngcp/sparse-dft-pilot/run_pilot.py:guided_paths) selects extrema, and the\nfrozen rule (gcp/mlip-cell-runner/z1_sparse_dft.py, byte-equivalent copy in\nthe analysis script) builds the anchor set:

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A(m,p) = {0, n−1, argmin} ∪ {argmax−w … argmax+w}, clamped and de-duplicated;\nw = 2 when n ≤ 6 images, else 1.

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Sources (sha256 recorded in the JSON): panel\ngs://shed-489901-atlas-inputs/z1/data/candidates/z1_nebdft2k_barriers.lock.json;\nartifacts gs://shed-489901-atlas-outputs/z1/campaign-float64/<model>/cell_result.json\nwith model dirs chgnet, mace-mp-small, mace-mp-medium, mace-mpa-0-medium.\nAll 30 panel paths are analyzed, including the seven large-cell paths deferred\nfrom DFT execution (data/candidates/z1-sparse-dft-deferred.json lists\nindices 2, 8, 10, 18, 20, 23, 28 — its prose says "six", the list holds seven;\nthey are excluded from GPAW runs, not from this prediction-side analysis).

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2. Headline numbers (recomputed)

\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
BasisPathsModel-path pairsNaiveUnionSavedRatio
All analyzable paths (primary)29/30111/12055815472.4%3.6×
Fully-covered paths only26/30104/12052013673.8%3.8×

Cross-model agreement over the 29 analyzable paths:

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\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
StatisticAvailable-models conventionAll-four-required convention
Saddle (argmax) exact23/2920/29
Saddle within ±126/2923/29
Both extrema exact10/299/29
Both extrema within ±112/2911/29

3. The 29-vs-30 reconciliation

The panel has 30 paths (indices 0–29). The true analyzable denominator is\n29: path index 14 (mp-756912_1_1_1_0_0) failed CI-NEB convergence under\nthe frozen protocol in all four model artifacts, so no model profile exists\nto guide anchors — it is the one path with incomplete artifacts across\nmodels, and it is excluded with its per-model failure records in the JSON.

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Three more paths have partial coverage (agreement there is computed over the\nmodels present, which is why two conventions are reported): index 8 lacks\nmace-mp-small; index 13 lacks mace-mp-small and mace-mpa-0-medium; index 18\nlacks chgnet and mace-mp-small (all recorded campaign failures, never imputed).\nOnly 26 paths carry all four models.

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4. Discrepancy vs the prior chat-session numbers

The prior session reported: 624 naive vs 132 union (79% fewer, 4.7×), saddle\nagreement 20/29, both-extrema agreement 12/29.

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5. Scaling note — union grows sub-linearly in model count

Union cost saturates quickly: models largely nominate the same images, so each\nadded model costs its full naive share but adds few new unique anchors.

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\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
ModelsNaive (mean over subsets)Union (mean over subsets)Union, cumulative fixed order (chgnet → +mace-mp-small → +mace-mp-medium → +mace-mpa-0-medium)
1139.5139.5142
2279.0147.8144
3418.5152.0152
4558.0154.0154

Going from 1 → 4 models multiplies naive cost ~4× but union cost only ~1.10×\n(139.5 → 154 mean; 142 → 154 cumulative). Sharing anchors across an ensemble\nof guides is therefore close to free: four models' worth of guidance for ~10%\nmore DFT than one, and the marginal union growth per added model is already\nflat at k=3.

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Reproduce

python3 tools/analysis/union_anchor_economics.py \\\n    --out data/candidates/z1-union-anchor-economics.json           # pulls from GCS\npython3 tools/analysis/union_anchor_economics.py --local <dir> \\\n    --out data/candidates/z1-union-anchor-economics.json           # offline artifacts\n
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