{"id":"savings-techniques-synthesis-2026-07-21","title":"The Savings Stack and the Theorem Commons","subtitle":"Synthesis of the nine digests: seven layers of compute savings, each with the same hole — the missing correctness certificate.","category":"references","tags":["literature-review","savings-stack","synthesis","correctness-certificate"],"source":"articles/lit-review/savings-techniques-synthesis-2026-07-21.md","lang":"en","words":2097,"readMinutes":10,"toc":[{"depth":2,"text":"1. The argument in one paragraph","id":"1-the-argument-in-one-paragraph"},{"depth":2,"text":"2. The cost wall, briefly","id":"2-the-cost-wall-briefly"},{"depth":2,"text":"3. The seven savings layers, and the hole in each","id":"3-the-seven-savings-layers-and-the-hole-in-each"},{"depth":2,"text":"4. What we have measured ourselves","id":"4-what-we-have-measured-ourselves"},{"depth":2,"text":"5. The theorem commons","id":"5-the-theorem-commons"},{"depth":2,"text":"6. What we do not claim","id":"6-what-we-do-not-claim"},{"depth":2,"text":"7. Next measurements","id":"7-next-measurements"},{"depth":2,"text":"References","id":"references"}],"html":"
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Provenance: director synthesis, 2026-07-21. Draws on nine deep-research digests initially materialized verbatim in this directory (savings-surrogate-neb.md, savings-active-learning.md, savings-delta-multifidelity.md, savings-abstention-economics.md, savings-dft-systems.md, savings-electronic-surrogates.md, savings-path-sampling-algorithms.md, supercomputing-atomistic-history.md, research-compute-resourcing.md); six digests subsequently received narrowly scoped editorial corrections replacing the retracted 624/132 union-anchor figures. It also draws on two primary records: data/candidates/z1-union-anchor-economics.json (+ docs/analysis/z1-union-anchor-economics.md) and docs/plans/2026-07-20-sparse-dft-pilot-preregistration.md. The identifier-level citation audit in citation-verification-2026-07-21.md verified every extracted arXiv ID and DOI; literature numbers remain qualified by each digest's stated access level and [UNVERIFIED] claim flags. Every number marked [derived] is our arithmetic on stated inputs, not a published value.

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The savings stack and the theorem commons

Scope: compute-savings techniques in atomistic and molecular simulation, 2015–2026, organized as one argument: the field has independently discovered seven layers of savings, each layer has a known hole, and the holes are all the same shape — the absence of a correctness certificate. That shape is where Lupine lives, and it is why a shared, formally verified theorem library is not a philosophy but the next savings technology.

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1. The argument in one paragraph

Atomistic simulation spent forty years making FLOPs cheaper (capex per peak PFLOP fell ~30× in a decade: Tianhe-2 ≈ <!--MATH0-->0.22M/PF — see supercomputing-atomistic-history.md) and the last ten years discovering that the real enemy is not the FLOP price but the evaluation count: how many times you must call the expensive oracle at all. Every successful savings technology of the ML era — active learning, surrogate NEB, Δ-learning, abstention gates — is an evaluation-count reduction with an unpriced correctness risk attached. Our measured contribution is that correctness-gated evaluation, shared across models, is itself a savings technology with a remarkable property: four independent models of guidance cost only ~10% more DFT than one (measured, §4). If that property generalizes — and the scaling curve says it should — then a commons in which teams contribute theorems and anchors back is a machine where we all get faster together, at no marginal cost to anyone.

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2. The cost wall, briefly

The backdrop (full version in supercomputing-atomistic-history.md, research-compute-resourcing.md):

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So the interesting question is never "what does a FLOP cost" but "who lets you make fewer oracle calls, and when are they wrong."

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3. The seven savings layers, and the hole in each

Each layer below is covered in depth by its own digest; one strongest measured number is quoted per layer, with its hole.

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  1. Surrogate-accelerated saddle search (savings-surrogate-neb.md). Local GP surrogates cut true evaluations by ~an order of magnitude, replicated for a decade: Koistinen/Jónsson 2017 (JCP 147, 152720); Garrido Torres 2019 (PRL 122, 156001), 5–25× fewer calls, cost decoupled from image count; Goswami 2025 (JCTC 21, 7935), ~10× on 500 molecular reactions. Hole: GP variance is a sampling-density signal, not an accuracy bound (stated plainly in the 2026 tutorial review); each search re-trains and then discards its surrogate.
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  2. Active learning (savings-active-learning.md). Uncertainty/committee-triggered labeling removes 2–4 orders of magnitude of DFT: VASP on-the-fly MLFF skips >99% of first-principles steps (Jinnouchi/Kresse 2019); DP-GEN labeled 0.0044% of ~650M explored configurations (Zhang et al., PRM 3, 023804); FLARE trains reactive force fields in ~100–250 calls. Hole: every mainstream trigger is sufficient-but-not-necessary — DP-GEN's own authors concede the committee can agree and be wrong; FLARE's GP variance underestimates error under strong extrapolation; rare events are the documented miss.
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  3. Δ-learning, multi-fidelity, transfer (savings-delta-multifidelity.md). MatterSim fine-tuned to revPBE0-D3 water with 30 high-fidelity configurations vs 900 from scratch (30×; arXiv:2405.04967); DPA-2 reports 1–2 orders less downstream data; classic Δ-ML reaches DFT enthalpies from 1–10% of the labels (Ramakrishnan 2015). Hole: fine-tuning reintroduces PES softening (OMat24 analysis), fails out-of-distribution, and costs a training run per model per chemistry — the wrong shape for breadth.
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  4. Abstention economics (savings-abstention-economics.md). The cleanest production measurement: AdsorbML finds the DFT-level adsorption minimum in 87.4% of ~1000 systems while paying full DFT only on the ~13% tail (Lan et al., npj Comput. Mater. 2023). Hole: no published budget prices abstention across models — every number is per-model/per-potential. The cross-model sharing economy is unmeasured territory.
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  5. Systems-level DFT acceleration (savings-dft-systems.md). GPU ports give 3–20× per node above a size threshold; reduced-rank exact exchange (ACE/ACE-ISDF) cuts hybrid-DFT cost ~2 orders (1,000-atom Si HSE in 10 min on 2,000 cores); Periodic Pulay gives 3× fewer SCF iterations on hard metals; ML density guesses cut ~20–33% of SCF iterations (ELECTRAFI: −20% total cost). Hole: these cut the cost per evaluation, not the count — and the aggressive settings (mixed precision, loose mixing) carry exactly the silent-failure risk that gates exist to catch.
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  6. Electronic-structure surrogates (savings-electronic-surrogates.md). ML Hamiltonians eliminate 100% of SCF at inference (DeepH: 10³× on a MoS₂ supercell; HamGNN: 4,284-atom Si Hamiltonian in 36 s); M-OFDFT reaches chemical accuracy at 27.4× on a 738-atom protein. Hole: predicted Hamiltonians do not give variationally reliable energies/forces for MD or NEB — and ML functionals fail silently (DM21's transition-metal convergence failures).
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  7. Path and sampling algorithms (savings-path-sampling-algorithms.md). Freezing/growing-string: a TS guess in ~20–90 gradient calls (Marks et al.); ART nouveau converges a DFT saddle in 50 force evaluations vs 463 unbiased; REST2 folds trpcage with 10 replicas where T-REMD needs 48; OPES explores ~10× faster than WTMetaD. Hole: cheap exploration distorts or skips the TS region (OneOPES says so explicitly), and string methods converge to second-order saddles at measured rates (2/16, 7/24 in Marks et al.) — a wrong-curvature failure that is, note, a violated-theorem condition.
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Seven layers, one repeated hole: no certificate. The field's triggers, variances, and committees all estimate where the model is probably wrong. None can say where physics is definitely violated.

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4. What we have measured ourselves

All numbers in this section trace to committed records; nothing here is from chat memory.

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Sparse anchors work on real metal. Path mp-760344, sparse barrier 0.5567 eV vs reference 0.5244 eV → 32.2 meV error, WIN against the frozen ≤40 meV gate. Anchor cost on a 4-vCPU local box: 2,483–3,668 s each (~48 min mean, ~3.2 vCPU-hours, ~<!--MATH1-->0.06/vCPU-h [derived]). Known line item: the GPAW-vs-VASP convention offset drifts ~122 meV along the profile (tracked as theorem-line T1).

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The sparse protocol was validated before any DFT ran. On recorded campaign artifacts, the preregistration's simulation showed sparse-protocol barrier MAE of 1.2–9.4 meV at ~7 anchors/path versus ~50–100 dense evaluations, with the saddle image located exactly on 82–86% of paths and within ±1 on 89–93% (docs/plans/2026-07-20-sparse-dft-pilot-preregistration.md).

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Union anchors: the measured sharing economy (data/candidates/z1-union-anchor-economics.json, recomputed 2026-07-21; supersedes earlier informal figures of 624/132/79%, which were arithmetic drift and are retracted):

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The multiplicative stack [derived estimate, undemonstrated as a product]. Union-sparse evaluation (154 anchors vs a 4-model dense accounting of 4 × 50–100 = 200–400 per path) is a ~38–76× reduction in evaluation count; per-anchor cost levers from layer 5 (mixed precision ~2×, ML density guess ~1.25×) compose orthogonally to ~95–190×. Each factor is individually cited; the product is ours to demonstrate — that demonstration is the pilot program now running.

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5. The theorem commons

Here is the network effect, stated as an engineering property rather than a hope.

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Theorems are non-rival goods with zero marginal sharing cost. A Lean-checked physical-law theorem — curvature signature at a first-order saddle, energy–force consistency, symmetry constraints, boundary conditions — is contributed once and then gates every simulation for everyone, forever. Unlike training data, a theorem leaks nothing about the contributor's chemistry or commercial targets. It is the shareable residue of work a serious team does anyway: every failed simulation is a candidate theorem, and formalizing your own failure converts your most expensive knowledge into a permanent asset for the commons that pays you back in everyone else's theorems.

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The demand side already exists — the literature is asking for it. Seven digests, working independently, came back with the same hole: learned triggers are sufficient-not-necessary (DP-GEN), GP variance saturates (FLARE), ensemble σ decorrelates from true error (Annevelink–Viswanathan line), fast exploration skips the TS region (OneOPES), string methods land on second-order saddles (Marks), ML functionals fail silently (DM21). Every one of those failure modes is a physical-law violation detectable without learning anything. A commons of formal theorems is precisely the supply for a demand the field has been circling for five years.

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Union anchors are the evaluation-side commons, and we have already measured its economics. Within one lab, four models share anchors at 72–74% savings, sub-linearly with model count (§4). Across labs the same arithmetic applies to anchor libraries: a content-addressed store mapping structure-hash → DFT energy/forces lets one team's anchor validate every other team's model on the same path. Where theorems are zero-leak by construction, anchor sharing is opt-in per project — the commons has two tiers and members choose their exposure.

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The contribution loop. Run your campaign on the commons → your model's failures are caught by existing theorems (you save evaluations immediately) → your novel failures get formalized as new theorems (you contribute) → every member's gates get sharper (false-accept and false-reject rates both fall) → evaluations saved compound across the network. The gate quality is a monotone function of membership. That is what "we all get faster together" means mechanically: the commons converts each team's worst day into everyone's permanent speedup.

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6. What we do not claim

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7. Next measurements

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  1. Chgnet × 23 active paths (local pilot, running): verdict against the ≤40 meV gate plus measured wall-hours — replaces the derived unit-cost figures with measured ones. Seven ≥159-atom paths deferred (data/candidates/z1-sparse-dft-deferred.json), verdicts PENDING, not excluded.
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  2. Union-anchor driver variant: evaluate the 154 unique anchors once and assemble per-model barriers from the shared pool — the direct experimental test that shared evaluation loses nothing vs per-model evaluation.
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  3. Gate-versus-gate pricing: theorem-gate false-accept/false-reject rates against a GP-variance gate on the same panel, in units of barrier error per DFT call — the abstention-economics baseline the literature lacks.
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  4. Convergence-loosening revalidation: Gamma-point vs (2,2,2) and h=0.20 vs 0.18, one path each, adopt-if-≤5 meV — a further 4–8× and ~40% per-anchor lever pending a preregistration amendment.
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References

Primary records: data/candidates/z1-union-anchor-economics.json; docs/analysis/z1-union-anchor-economics.md; tools/analysis/union_anchor_economics.py; docs/plans/2026-07-20-sparse-dft-pilot-preregistration.md; data/candidates/z1-sparse-dft-deferred.json.\nDigests (this directory): the nine files listed in the provenance header. Literature citations live inside the digests with per-item access level (full text vs abstract) and [UNVERIFIED] flags.

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