Verifier-guided discovery of exact high-order mimetic operators with large language models

Introduction

• Operator design must satisfy accuracy, energy, spectral, and PDE-error requirements simultaneously, since meeting one criterion does not ensure the others.The design problem is framed as a search over representations, constraints, and objectives followed by independent verification.Source: Introduction · p. 2

Problem and fixed-closure obstruction

• The fixed MOLE/Corbino–Castillo closure retains a discrete Gauss identity but its order-six and order-eight Dirichlet blocks have four non-real modes, ruling out an SPD self-adjoint similarity.The proposition establishes real non-positive spectrum when the positive norms satisfy the endpoint-support condition; the reference boundary term extends over closure rows instead.Source: Problem and fixed-closure obstruction · pp. 2–3, Figure 1 · p. 3

Verifier-guided search

• The LLM emits a typed construction program, while a deterministic compiler generates coefficients and an independent verifier checks identities, positivity, physical modes, conditioning, and PDE accuracy.Selected results receive exact rational certification; a proposal is rejected for even one spurious physical mode or failed affine identity.Source: Verifier-guided search · pp. 3–4
• The proposed architecture makes LLMs responsible for structural hypotheses, while deterministic mathematics produces and validates operators.The program grammar spans norm class, boundary closures, row-specific supports, and optimization objectives.Source: Verifier-guided search · p. 4

Verifier feedback increases reliable yield

• 55.0% of full-metric-feedback programs and 53.3% of illumination-archive programs passed, versus 13.3% for uniform random search.The expert-informed prior reached 50.0%; at the common 12-call budget, it was the only arm whose best-so-far area under the curve remained significant after correction.Source: Verifier feedback increases reliable… · p. 4, Figure 3 · p. 5
• Full metrics yielded 85.1 externally verified programs per million tokens, versus 80.2 for illumination, 45.7 for structure only, and 37.9 for no archive.Of 638 LP-infeasible programs with an identified binding group, 637 (99.8%) failed endpoint-supported extended-Gauss compatibility equations; one-step textual refinement did not explain the feedback effect.Source: Verifier feedback increases reliable… · p. 5

Exact constructions recover weighted spectral structure

• The strongest candidate had no extra low modes or non-real eigenvalues and PDE RMS error 8.91 × 10^-8; it reduced the prior positive-diagonal 6–3 candidate’s spectral-radius constant by 25.4% and PDE error by 62.7-fold.The MOLE/Corbino–Castillo reference remained about 69 times more accurate on this manufactured problem, so the candidate is a distinct trade-off rather than a uniform improvement.Source: Exact constructions recover weighted… · pp. 5–6, Table 2 · p. 6
• All four exact candidates remained real and free of extra low modes through m = 640, while the leading candidate’s apparent refinement rate fell from 6.34 to 4.67.At γ = 0.08, mean worst relative eigenvalue drift was 0.01149 for the leading candidate versus 0.01294 for the order-six reference; this was a targeted Monte Carlo diagnostic, not a general robustness proof.Source: Exact constructions recover weighted… · p. 6
• The candidate’s certified heat-equation energy is contractive, while the order-six reference reaches 1.066 transient energy amplification; both have the same RK4 limit, ∆tmax/h2 = 0.4517.The comparison is norm-specific; the recovered structure supplies an energy certificate and SPD formulation, not a larger explicit step.Source: Exact constructions recover weighted… · pp. 6–7
• At m = 200, all seven exact constructions make −QILI symmetric positive definite, enabling CG; none of the four references meets both weighted-symmetry and positivity conditions under native quadratures.Deliberately applying CG to the symmetrized order-six reference system gave solution error 0.278 despite residual 5.0 × 10^-11; this diagnoses symmetrization, not failure of the original discretization.Source: Exact constructions recover weighted… · pp. 7–8, Table 3 · p. 7

• LLMs choose representation-level features, while deterministic compilation and verification decide whether those choices define a valid operator.The observed increase from 13.3% random yield to 55.0% under full feedback indicates verifier information can guide models toward denser feasible regions in this grammar.Source: Discussion · p. 8
• The strongest individual operator came from the structure-only arm, whose aggregate rate did not significantly exceed random search, separating best-object quality from search-policy yield.Source: Discussion · p. 8
• The authors report exact LLM-originated candidates, not previously unknown operator families, with novelty screening scoped to a partial topology library.The comparison found no screened order-six candidate matching the loaded MOLE/Corbino–Castillo reference; several other operator families remain outstanding.Source: Discussion · p. 8
• The study is limited to one-dimensional uniform grids, and its block-positive cone is a sufficient inner approximation rather than the full banded semidefinite cone.Source: Discussion · p. 8
• Changing closure and norm architecture can recover exact weighted structure and a real physical spectrum at high interior order, with deterministic verification grounding the mathematical claim.The propose–compile–verify–exactify pattern may apply where models explore representational choices and deterministic or symbolic tests enforce validity.Source: Discussion · p. 9

Program representation and deterministic compilation

• Programs specify operator orders, norm and closure architecture, support, coefficient bounds, positivity margins, and objectives; a deterministic LP compiler instantiates the operators on design and holdout meshes.Polynomial exactness is imposed through shared boundary-template moment equations, with local block norms represented by a sufficient inner approximation of the positive-definite cone.Source: Program representation and… · p. 9

Independent verifier and exactification

• The verifier checks algebraic identities, norm positivity, Dirichlet spectra, conditioning, and manufactured-problem errors against proposer-independent thresholds.It matches the first five eigenvalues to exact values and separately counts extra low-frequency modes.Source: Independent verifier and exactification · pp. 9–10
• Selected floating-point solutions are reconstructed in a coupled rational affine system, then certified by exact norm minors and verification on unseen meshes.Only free coordinates are placed on a common dyadic grid; pivot coordinates are solved exactly and the original equations are rechecked.Source: Independent verifier and exactification · p. 10
• The novelty screen covers same-order staggered operators, while parameterized higher-order families remain incomplete and cross-topology comparison needs a degree-of-freedom map.The screen canonicalizes scale, sign, reflection, and unequal closure widths.Source: Independent verifier and exactification · p. 10
• Downstream audits measure heat-equation energy and stability limits and test linear-solver applicability without using these diagnostics as search objectives.Solver tests distinguish the original matrix from a symmetrized diagnostic and do not compare iteration counts across solver families.Source: Independent verifier and exactification · p. 10

Search protocol and statistics

• Across 1,200 solver calls, the protocol compares random, surrogate, expert-prior, and LLM search conditions using per-call verifier acceptance and seed-level inference.Paired sign-flip randomization retains difference magnitudes; Holm correction is applied, and a secondary endpoint integrates best-so-far utility over the first 12 calls.Source: Search protocol and statistics · pp. 10–11

Code availability

• The repository contains the self-contained notebook, postprocessing and release-verification scripts, and code to recompute the reported audits.The audits include provider transport, program diversity, prediction coverage, exact candidates, energy, and solver consequences.Source: Code availability · p. 11