AutoMOOSE: An Agentic AI for Autonomous Phase-Field Simulation

1 Introduction

• MOOSE input construction couples thermodynamics, kinetics, discretization, solvers, and platform syntax, while errors can yield failures or completed runs that violate physics.Undocumented settings and differences in resolution, tolerances, or initial conditions can alter quantitative results without visible error.Source: §1 · pp. 1–2
• Mesoscale phase-field modeling has received comparatively little attention from agentic systems, and input generation alone cannot establish physical validity.The paper frames the challenge as converting stochastic model output into reproducible simulations whose physical validity is quantitatively tested.Source: §1 · p. 2
• AutoMOOSE uses six specialized agents to orchestrate MOOSE simulations from natural-language prompts through execution and adversarial physics testing.Its Skeptic tests results against conservation laws, asymptotic limits, and scaling relations.Source: §1 · p. 2
• Across 25 grain-growth tasks, AutoMOOSE produced valid inputs for all 25, measurable coarsening in 19, and 15 fully credible results.The unsuccessful cases traced to two identifiable generator defects.Source: §1 · p. 2
• A 1000-simulation ensemble recovered the prescribed activation energy to within 1% after finite-size extrapolation.The conserved Fe–Cr simulation showed 6.3×10^-6 relative mass drift, monotonic free-energy dissipation, and evolution toward the equilibrium tie-line.Source: §1 · p. 2

2.1 Phase-Field Model for Grain Growth

• The grain-growth demonstration uses the multiphase Allen–Cahn model, where continuous grain order parameters evolve by free-energy minimization without a conservation constraint.Double wells stabilize bulk grain regions, overlap penalties separate neighboring grains, and mesh spacing must satisfy h ≤ wGB/4 to resolve interfaces.Source: §2.1 · p. 3

2.2 Overview of Agentic Workflow

• Six agents pass a shared simulation plan through cognitive input construction, execution and analysis, then independent epistemic verification.The plan carries model, domain, mesh, boundary conditions, solver and runtime settings, and requested outputs; each agent reads and enriches this structured state.Source: §2.2 · p. 5

2.4 Input Writer Agent

• The Architect converts natural-language intent into a structured plan, and the compound Input Writer assembles a syntax-validated MOOSE input through six dependency-ordered sub-agents.Its mesh and materials sub-agents enforce interface resolution and compute physically grounded coefficients; benchmark inputs matched 9/10 expert-reference blocks with 0.30% relative kinetic-rate error.Source: §2.3 · p. 5, §2.4 · pp. 6–7

2.5 Runner Agent

• The Runner alone interacts with MOOSE, dispatches temperature-sweep cases concurrently, and routes completed output or failures to the next agent.Timestamped, self-contained run directories preserve inputs, logs, outputs, provenance, and parsed metrics for exact reproduction; live log streaming supports monitoring.Source: §2.5 · p. 7, §2.5.1 · p. 7, §2.5.2 · p. 7

2.6 Reviewer Agent

• The Reviewer admits runs that reach target integration time while distinguishing recoverable adaptive-timestep cutbacks from genuine failures; bounded recovery makes at most three logged correction attempts.Recovery can cut back timesteps, extend integration, or refine the mesh, but corrected runs must independently complete and survive Skeptic falsification.Source: §2.6 · p. 8, §2.6.1 · p. 8, §2.6.2 · p. 8

2.7 Visualization Agent

• The Visualization agent extracts GrainTracker counts, fits inverse grain count over time to obtain coarsening rates, and performs Arrhenius regression across temperatures.It returns numerical results and a natural-language interpretation, including notable features such as low-temperature suppression of coarsening kinetics.Source: §2.7 · p. 8, §2.7.1 · p. 9, §2.7.3 · p. 9

2.7.2 Arrhenius regression and consistency check

• Qfit = 0.296 eV versus specified Q = 0.23 eV, with R2 = 0.90–0.95 at T ≥600 K; coarsening is suppressed at 300 K.The passage attributes the low-temperature suppression to the Arrhenius reduction of L(T), consistent with the Burke–Turnbull framework.Source: §2.7.2 · p. 9

2.8 Skeptic Agent

• The Skeptic independently falsifies completed simulations against equation-specific physical invariants, reporting survivors as credible and diagnosing failures for remediation.This addresses physically incorrect runs that can complete and produce well-formed output, which the Reviewer’s operational screen cannot detect.Source: §2.8 · p. 9

2.8.3 Verdict, diagnosis, and recovery

• The Skeptic requires applicable physical invariants to pass, flags failures with diagnoses, and supports bounded timestep corrections only for divergence failures; corrected runs must pass independent rechecking.Grain-growth checks cover completion, non-increasing grain count, seed-count consistency, parabolic scaling (R2 ≥0.90), and Arrhenius consistency; conserved-dynamics checks require relative mass drift ≤10^-5, net free-energy decrease, and approach to the equilibrium tie-line.Source: §2.8.1 · p. 9, §2.8.3 · p. 10, §2.8.2 · p. 10

2.9 Plugin Architecture

• A two-function plugin contract keeps the six-agent orchestration physics-agnostic: plugins generate MOOSE inputs and parse simulation time-series into structured observables.A PLUGIN metadata dictionary supplies UI labels, executable and parameter configuration, presets, sweepable parameters, result keys, and the Architect’s physics-specific system prompt.Source: §2.9 · p. 10, §2.9.1 · p. 10, §2.9.2 · p. 10

2.9.4 Plugin registry and extensibility

• The grain-growth plugin supports two Allen–Cahn formulations, two initial-condition types, 2D/3D geometries, and seven presets; a dynamically loaded Cahn–Hilliard Fe–Cr plugin extends coverage to conserved dynamics.The Fe–Cr plugin uses the split composition–chemical-potential form with CALPHAD free energy and mobility; ferroelectric-switching and solidification plugins are registered as stubs.Source: §2.9.3 · p. 10, §2.9.4 · p. 11, §2.10 · pp. 11–12

2.11.1 Server architecture

• The MCP layer decouples client access from simulation execution, exposing identical tool surfaces over stdio and SSE transports and using JSON schemas and permissions to govern tool discovery and access.The server and FastAPI simulation backend run independently; backend endpoints provide plugin lookup, logs, run status, and health checks.Source: §2.11.1 · p. 12, §2.11.2 · p. 12, §2.11.3 · pp. 12–13

2.11.4 Verification arc

• The MCP verification arc routes completed runs from Runner to Reviewer and then Skeptic, reporting only runs that both execute and survive falsification; divergence failures may enter bounded recovery.MCP also supports headless sweeps and tool composition with external services, without a browser session or human in the loop.Source: §2.11.4 · p. 13, §2.11.5 · p. 13

2.12 Parameter Sweep Automation

• Parameter sweeps expand a simulation plan into concurrent cases, propagate each temperature through mobility and rate extraction, fit Arrhenius behavior, and retain metadata provenance for reproducibility.The demonstrated temperature sweep uses 300, 450, 600, and 750 K; its total wall-clock time is bounded by concurrent dispatch rather than the serial sum.Source: §2.12 · pp. 13–14

2.13.1 Benchmark-driven generator corrections

• Across 25 end-to-end grain-growth tasks, the Skeptic exposed two generator defects—placeholder LinearizedInterface constants and aggressive 3D time-stepping—that were corrected and re-executed.The benchmark scored tasks against five pre-registered gates: parseable input, completion, coarsening, Burke–Turnbull kinetics, and a positive rate constant; reported benchmark outcomes reflect the original configuration before fixes.Source: §2.13.1 · p. 14, §2.13 · p. 14

2.14 User Interface and Demonstration Workflow

• At T ≥600 K, the four-temperature demonstration recovered Burke–Turnbull kinetics with R2 = 0.90–0.95 and completed prompt-to-analysis in under two hours.Users specified the sweep, model, geometry, and analysis goal in natural language; the interface populated a simulation plan and displayed concurrent runs, solver logs, input files, and fitted results.Source: §2.14 · pp. 14–15

3.1 End-to-End Workflow Demonstration

• Concurrent dispatch reduced four-case sweep wall time 1.8× versus serial execution (22,820 s versus 41,261 s), a scheduling effect rather than solver acceleration.The full prompt-to-kinetics campaign took approximately 380 minutes; active human effort was limited to composing the prompt.Source: §3.1 · p. 15

3.2 Benchmark Results

• 25/25 tasks produced parseable inputs, 19/25 ran and coarsened, and 15/25 passed all five gates and independent Skeptic falsification.The parabolic-kinetics gate was recovered for 18/25 tasks at production integration length.Source: §3.2 · p. 16, Table 6 · p. 17

3.3 Second Physics Domain: Spinodal Decomposition

• 6.28 × 10^-6 relative Cr-mass drift met the 10^-5 tolerance, while free energy decreased 0.4% to a plateau and composition approached the equilibrium tie-line.The final extreme compositions were 0.826 Cr-rich and 0.239 Fe-rich, against targets of 0.823 and 0.236 mol fraction Cr.Source: §3.3 · p. 17, Table 7 · p. 17, Figure 4 · p. 18

3.4 Input File Fidelity

• 6 of 12 structural blocks matched the expert reference [9] exactly, 4 were functionally equivalent, and 2 differed through valid mesh or executioner choices.The exact matches included all physics-critical blocks; AutoMOOSE also avoided duplicating GrainTracker in Postprocessors.Source: §3.4 · p. 19, Table 9 · p. 21

3.4.1 Fidelity at scale

• Across 1,000 inputs all 12 blocks were complete; across 100 matched pairs every physics-determining block matched exactly, with divergences confined to Executioner and Outputs.Those operational blocks allow valid autonomous choices, while matched-cell kinetics showed slope 1.023, Pearson r = 0.9887, and median absolute deviation 3.4%.Source: §3.4.1 · pp. 19–20, Figure 5 · p. 19

3.5 Grain Growth Kinetics

• Linear regression of N−1(t) against t showed AutoMOOSE rate constants within 0.5% of the human reference at 450 K, but 22–26% divergent at higher temperatures, reflecting different Voronoi tessellations.The passage attributes this to stochastic variability intrinsic to small-system statistics, not agent error.Source: §3.5 · p. 22

3.6 Arrhenius Analysis and Consistency Check

• AutoMOOSE yielded Qfit = 0.296 eV (R2 = 0.994), versus 0.267 eV (R2 = 0.996) for the human reference, with the offset attributed to finite-size effects.The two fits were highly linear; their Qfit difference was 0.029 eV, with seed-driven differences in higher-temperature coarsening rates contributing to the spread.Source: §3.6 · pp. 22–23, Figure 8 · p. 23

3.6.1 Ensemble convergence of the activation energy

• Q∞ = 0.228 eV versus the input Q = 0.230 eV, a 0.9% agreement after finite-size correction, validates the ensemble Arrhenius analysis.The 1000-run campaign found converged ensembles at N0 = 100 and 200; N0 = 500 was excluded because it was resolution-limited at 6.5 cells per grain.Source: Figure 10 · p. 24, Figure 9 · p. 24, §3.6.1 · p. 25

3.7.1 Execution-screen catches during plugin development

• The execution screen caught three input-defect classes during plugin development before physically meaningless results could propagate.A duplicate GrainTracker declaration made all four sweep runs fail on the first attempt; removing the erroneous [Postprocessors] entry restored runnable inputs on the second attempt.Source: §3.7.1 · p. 25, §3.7.2 · p. 25, Table 12 · p. 26

3.7.3 Class II: Duplicate solver parameter

• A duplicated solve_type = PJFNK key and an unsupported Exodus interval parameter were corrected by deduplication and parameter removal, respectively.Removing the unused parameter restored Exodus output without changing physics or solver settings.Source: §3.7.3 · p. 25, §3.7.4 · p. 25, Table 12 · p. 26

3.7.5 Skeptic falsification of benchmark failures

• The Skeptic falsified the same LinearizedInterface and three-dimensional benchmark cases that failed the gates, identifying two generator defects that were corrected and re-executed.The fixes replaced placeholder copper constants and reduced the initial time step while strengthening the 3D preconditioner.Source: §3.7.5 · p. 26

3.7.6 Closed-loop recovery module

• The recovery module applies bounded time-step corrections to divergence-class failures and accepts a regenerated run only after independent completion and Skeptic checks.Component validation confirmed that completed runs are not corrected based on stale status records and that inapplicable corrections are suppressed; formulation failures are surfaced rather than forced.Source: §3.7.6 · p. 26

3.8 The Value of Task Decomposition: Pipeline versus a Single Non-Decomposed Agent

• AutoMOOSE produced valid inputs for 8/8 core tasks on every invocation, while Claude Opus 4.8 averaged 5.0/8 (range 3–7) and GPT-4o produced 0/8.The comparison tests G1 input validity, the stage a bare single agent can attempt; functional and physical correctness are evaluated separately, with later pipeline stages screening, falsifying, and recovering results.Source: §3.8 · pp. 26–27, Table 13 · p. 27

3.9 Skeptic Falsification as Methodology

• The Skeptic admitted exactly 15/25 benchmark runs that satisfied physics invariants and falsified the remainder, including completed runs that violated parabolic scaling or failed in three dimensions.The same falsification machinery applies across domains, using conserved-dynamics invariants for the Fe–Cr spinodal simulation.Source: §3.9 · p. 27

3.10 Agent-Narrated Interpretation

• Across four temperatures, the Visualization agent identified coarsening at T ≥450 K, attributed zero coarsening at T = 300 K to Arrhenius suppression, and reported Qfit = 0.296 eV.For the 450 K example, grain count fell from 15 to 13 (13.3%) over 4000 ns; the fit had R2 = 0.749 and rate constant ˜k = 2.863×10−6 ns−1.Source: §3.10 · pp. 27–28

4.1 Agentic Decomposition Versus Single-Turn Prompting; 4.4 Comparison to Related Approaches

• Specialized agents avoid an informationally overloaded single prompt, while AutoMOOSE extends prior input-generation tools with execution, diagnosis, analysis, and physics-grounded falsification.Modularity also permits users to intercept the workflow at arbitrary stages.Source: §4.1 · p. 28, §4.4 · p. 29

4.2 Physical Validity and Pipeline Consistency

• 15 of 25 benchmark tasks produced fully credible runs; the ensemble recovered Q∞ = 0.228 eV against 0.230 eV after finite-size correction.Failures were confined to two diagnosable generator defects.Source: §4.2 · p. 28
• The conserved Fe–Cr spinodal run satisfies mass-conservation and free-energy-dissipation laws to a stated tolerance and approaches the equilibrium tie-line.Source: §4.2 · p. 28

4.3 Role of Pre-trained Knowledge Versus Retrieval Augmentation

• Across 100 matched GBEvolution pairs, physics-determining blocks match human references exactly, but pre-trained knowledge alone is likely insufficient for less common physics.RAG over MOOSE documentation and official examples is identified as a near-term extension.Source: §4.3 · p. 29

4.5 Reproducibility as a First-Class Outcome

• Each run’s provenance record captures the executable path, parameters, host, MPI ranks, runtime, and random seed to support reproduction without external systems.Source: §4.5 · p. 29

4.6.1 RAG over MOOSE documentation

• Recently added MOOSE objects and non-standard formulations can prompt plausible but incorrect parameter names, motivating a planned documentation-backed retrieval layer.Source: §4.6.1 · pp. 29–30

4.6.2 Extended recovery policy

• Recovery currently handles divergence-class failures through bounded timestep correction; broader recovery and large-scale validation on genuine breakdowns remain future work.Source: §4.6.2 · p. 30

4.6.3 Extended physics coverage; 4.6.4 Active learning and autonomous discovery

• Ferroelectric switching and solidification remain plugin stubs, while autonomous parameter-space exploration through Bayesian optimization is planned rather than demonstrated.Source: §4.6.3 · p. 30, §4.6.4 · p. 30

• The six-agent pipeline closes the loop from intent to quantitative result, with a Skeptic that tests completed runs against physical laws and an MCP server exposing structured tools.Source: §5 · p. 30
• The authors present AutoMOOSE as an open-source framework for connecting natural-language specifications to reproducible, physics-validated phase-field campaigns.Source: §5 · pp. 30–31
• On the pre-registered 25-task grain-growth benchmark, AutoMOOSE generated valid inputs for all tasks and produced 15 fully credible runs, with failures localized to two generator defects.Source: §5 · p. 30
• The ensemble recovered activation energy within 1% after finite-size correction, and the Fe–Cr simulation satisfied conservation and dissipation laws while approaching the equilibrium tie-line.Controlled ablations also found valid input generation on all 8 tasks for the deterministic pipeline, versus 3–7 of 8 for the strongest bare model.Source: §5 · p. 30
• Planned extensions include documentation retrieval, activation of ferroelectric and solidification plugins, active learning, and integration with HPC job schedulers.Source: §5 · p. 31

• The supplementary information is titled “AutoMOOSE: Agentic AI for Autonomous Phase-Field Simulation.”Source: Supplementary Information · p. 34

S1 Agent System Prompts

• The six deployed prompts require structured JSON communication; the supplement records their approximate token counts and preserves original prompt formatting.Source: S1 Agent System Prompts · p. 34

S1.1 Architect Agent (f1)

• The Architect converts natural-language intent into a plan covering formulation, geometry, boundary conditions, sweep parameters, and solver strategy.Unsupported physics is surfaced as an error with available plugins listed.Source: S1.1 Architect Agent (f1) · p. 34

S1.2 Input Writer Agent (f2)

• The Input Writer coordinates six sub-agents to render MOOSE files in dependency order, applying phase-field parameter bridges and validating cross-references.Its prompt places GrainTracker in UserObjects and prohibits its placement in Postprocessors.Source: S1.2 Input Writer Agent (f2) · pp. 34–35

S1.3 Runner Agent (f3)

• The Runner constructs and monitors execution commands, writes run provenance to record.json, and does not modify inputs or rerun without Reviewer instruction.Source: S1.3 Runner Agent (f3) · p. 35

S1.4 Reviewer Agent (f4)

• The Reviewer screens execution completion; separate recovery logic diagnoses failures, applies specified corrections, and requests confirmation for low-confidence or unknown cases.Source: S1.4 Reviewer Agent (f4) · pp. 35–36

S1.5 Visualization Agent (f5)

• The Visualization agent fits grain-coarsening and Arrhenius relations, reports fit statistics, and flags kinetics anomalies when R2 is below 0.90.Source: S1.5 Visualization Agent (f5) · p. 36

S1.6 Skeptic Agent (f6)

• The Skeptic selects invariants by governing dynamics: grain-growth checks include monotonicity and scaling, while conserved-dynamics checks include mass drift, energy dissipation, and phase separation.It returns a credible verdict only when all applicable invariants pass; its diagnosis can feed recovery.Source: S1.6 Skeptic Agent (f6) · p. 36

S2.1 Code Availability

• AutoMOOSE is released under the MIT License with source code, the GrainGrowth plugin, example runs with provenance records, documentation, and agent prompts.Source: S2.1 Code Availability · p. 37

S2.2 Codebase Structure

• Table S14 separates user-facing files requiring configuration from infrastructure modules that need no modification for standard deployment.Source: Table S14 · p. 37, S2.2 Codebase Structure · p. 37

S2.3 Installation Requirements

• Deployment requires Python ≥3.11, Node.js ≥18, a compiled MOOSE phase_field-opt binary, and configured language-model credentials; the backend is model-agnostic.Reported results use Claude Opus 4.8 and Sonnet 4.6, while Qwen2.5-32B-Instruct is also supported.Source: S2.3 Installation Requirements · p. 37

S2.4 MCP Server Tool Index

• The MCP server exposes ten tools over stdio and SSE transports for Claude Desktop, Claude Code, and programmatic clients.Source: S2.4 MCP Server Tool Index · p. 37, Table S15 · p. 38

• The equation index maps main-text expressions to the pipeline component that uses each expression and its role.Source: S3 Equation Index · p. 38, Table S16 · p. 38
• Its entries are organized by equation, expression, component and system-prompt reference, and role.Source: S3 Equation Index · p. 39
• Equation 18 assigns floc(c) to the spinodal plugin as the local free energy for Fe–Cr.Source: S3 Equation Index · p. 39

S4.0.1 Campaign design

• The AutoMOOSE ensemble contains 1000 runs across five initial grain counts, five temperatures, and up to 40 seeds per cell, versus 100 human-written runs.Both campaigns use a fixed 1000 × 1000 box and 1282 mesh; the AutoMOOSE campaign is ten times larger.Source: S4.0.1 Campaign design · p. 39

S4.0.2 Admissibility criteria

• A trajectory qualifies with coarsening to ≤0.60 N0, at least five distinct grain-count levels, at least six fit points, and R2 ≥0.90; cells also require ≥0.60 seed yield and five valid seeds.For N0 = 100 and 200, all five temperatures clear the cell gate with R2 > 0.999.Source: S4.0.2 Admissibility criteria · p. 39

S4.0.3 Aggregation and uncertainty

• Q∞ = 0.228 eV is the finite-size-extrapolated activation energy; N0 = 500 was excluded as resolution-limited at 6.5 cells per grain.Source: S4.0.3 Aggregation and uncertainty · p. 39
• Rate constants are ensemble means with standard errors; activation energy uses standard-error-weighted regression, bootstrap confidence intervals, and inverse-variance-weighted finite-size fitting.Source: S4.0.3 Aggregation and uncertainty · p. 39

S4.0.4 Per-cell seed yield and resolution

• Table S17 reports each populated cell’s seed yield, resolution, rate constant, and gate outcome, making the yield and resolution exclusions auditable.Cells enter fits only with yield ≥0.60 and at least five valid seeds; fewer than 8 cells per grain is marked resolution-limited.Source: S4.0.4 Per-cell seed yield and… · pp. 39–40, Table S17 · p. 39

• All 1000 AutoMOOSE inputs and 100 human reference inputs contain all twelve well-formed top-level MOOSE blocks, and all physics-determining blocks match exactly across 100 pairs.Systematic differences occur only in the operational [Executioner] and [Outputs] blocks, for which no canonical form exists.Source: S5 Input-File Fidelity: Full Per-Block… · p. 40

S6 Baseline Comparison: Per-Run Detail

• AutoMOOSE scored 8/8 on every run across the eight core grain-growth tasks, with zero variance.The pipeline is deterministic and produces identical output on every invocation.Source: S6 Baseline Comparison: Per-Run Detail · p. 40
• Across eight core grain-growth tasks, bare Claude Opus 4.8 scored 5.0 (range 3–7)/8 strictly and 5.3 (range 3–8)/8 leniently; Sonnet 4.6 scored 0.3 strictly, with lenient scoring allowing unused items.Source: S6 Baseline Comparison: Per-Run Detail · p. 40
• GPT-4o scored 0/8 in every run under both scoring rules; GPT-5.x was unavailable, making GPT-4o the strongest GPT model evaluated.Each bare-model condition was repeated three times; the single-agent baseline attempted only input generation and was scored on G1.Source: S6 Baseline Comparison: Per-Run Detail · pp. 40–41

S7 Evaluation Prompt Set

• Each fixed-set task is a single natural-language prompt run through the full pipeline without human intervention, then scored against five pre-registered gates and independently tested by the Skeptic.Source: S7 Evaluation Prompt Set · p. 41

S7.0.1 Gates

• The gates progress from parseable input and completion without solver divergence to coarsening, Burke–Turnbull R2 ≥0.90, and a physically valid fitted rate constant.A dash indicates a later gate was not reached because an earlier gate failed.Source: S7.0.1 Gates · p. 41

S7.1 Prompts

• The prompt set varies temperature, grain count, mesh resolution, and model formulation, including GBEvolution and LinearizedInterface cases.Examples include temperature sweeps and refined or coarse-mesh conditions.Source: S7.1 Prompts · pp. 41–42
• LinearizedInterface prompts specify copper simulations at 500–525 K with 24 grains, while other prompts vary temperature and copper material parameters.Source: S7.1 Prompts · pp. 41–42
• Other cases vary random seeds and grain density, and include three-dimensional simulations on HEX8 meshes alongside high-temperature and finely refined runs.Source: S7.1 Prompts · pp. 42–43

S7.2 Gate outcomes

• The task table reports initial-to-final grain counts, Burke–Turnbull fit R2, gate outcomes, and verdicts of credible, near-miss, or falsified.The listed outcomes include credible high-temperature and density cases, alongside falsified 3D cases.Source: Table S18 · p. 43, S7.2 Gate outcomes · p. 44
• 15 of 25 tasks pass all five gates and the Skeptic’s falsification battery, while 19 run to a correctly coarsening microstructure and all 25 produce parseable inputs.The shortfalls are localized to the linearized-interface material defect and three-dimensional time-stepping divergence.Source: Table S18 · p. 43, S7.2 Gate outcomes · p. 44

S8.0.1 CALPHAD free energy

• The CALPHAD-fitted Fe–Cr free-energy expansion uses coefficients A–G and unit-conversion factors; c0 = 0.4677 lies within the spinodal range c ∈ [0.364, 0.712].The reported range is linearly unstable, so the alloy decomposes spontaneously.Source: S8.0.1 CALPHAD free energy · p. 44

S8.0.2 Numerical setup

• The split Cahn–Hilliard model runs on a 25 × 25 nm2 periodic domain with 100 × 100 QUAD4 elements and noisy initial composition c0 = 0.4677.Uniform random noise has amplitude 0.02, and CSV postprocessors record mass, free energy, composition extrema, and feature count at every step.Source: S8.0.2 Numerical setup · p. 44

S8.0.3 Skeptic invariant scores

• Cr mass drift is 6.28 × 10−6, free energy decreases 0.4% to a stable plateau, and composition separates toward the equilibrium tie-line.At final time, cmin = 0.239 and cmax = 0.826, with one Cr-rich precipitate; the tie-line is 23.6/82.3 mol% Cr at 500 °C.Source: S8.0.3 Skeptic invariant scores · p. 44
• Because Cr-rich domains reach the box scale before a clean coarsening window develops, the validation makes no Lifshitz–Slyozov L(t) ∼ t1/3 scaling claim.Instead, it rests on composition snapshots and the S1–S3 invariants.Source: S8.0.3 Skeptic invariant scores · p. 44

S9 Complete Input File Comparison

• The comparison covers the T = 450 K, 15-grain GBEvolution case on a 12 × 12 mesh with uniform refinement level 3 and copper parameters.Source: S9 Complete Input File Comparison · p. 45
• Six of 12 blocks match exactly, four differ only in ways reported to have no effect on output, and two use different but physically valid solver or mesh choices.Source: S9 Complete Input File Comparison · p. 45
• Grey inline comments trace parameter values to the natural-language prompt through P (Eq. (9)).Source: S9 Complete Input File Comparison · p. 45