Physics Intermediate Representation (PIR): A Structured Framework for Automated Physical Law Discovery from Data — Version 3.2

PIR is a research framework for automated physical law discovery combining physics-aware symbolic search, soft dimensional consistency filtering, residual-driven refinement, sparse model selection, and a hybrid loss (MSE + Optimal Transport, alpha=0. 7, beta=0. 3). It produces typed scientific output objects (Equation, PIRDifferentialEquation, PIRHamiltonian, PIRLagrangian, PIRSystem) decoupling data-level discovery from knowledge-level reasoning. Version 3. 2 (May 2026) additions over v3. 1: 1. Section 9. 3 — Head-to-head comparison with gplearn 0. 4. 3 on the full PIR-Bench-20 under an identical evaluation protocol (same datasets, same seeds, same success criterion, same DCS formula). Result: PIR 100% mean DR / DCS 0. 838 vs gplearn 50% / 0. 672. PIR wins on 14/20 tasks, tied on 6/20, gplearn wins on 0/20. Eight catastrophic gplearn failures (0% DR) on structurally demanding tasks (Lagrangians, FK, Jacobian, pendulum, inverse-square). Six perfect ties confirm the benchmark contains genuinely solvable tasks. This retires the standard SR baseline gap identified in v3. 1 critique. 2. Section 11 Conclusion — Stale "noise ≤ 0. 05" parenthetical removed; now consistent with Abstract, Table 1, Section 10. 4, and Appendix A (all of which correctly reported sigma in 0. 01, 0. 05, 0. 10, 0. 20). 3. PySR head-to-head and component ablation are documented as deferred to v3. 3 (Python 3. 13 / PythonCall. jl C-API incompatibility for PySR; benchmark-runner cache requires invalidation logic for component ablation). The Langevin-step ablation already published in v3. 1 is unaffected. All Table 1 numbers, feature comparisons, and design constants are unchanged from v3. 1. Bundle contents: - PIRArchitectureᵥ3₂. pdf — main paper (13 pages) - PIRArchitectureᵥ3₂. tex — LaTeX source- comparisonₜable. md,. csv,. json — gplearn comparison artifacts- rungplearnₕeadtoheadᵥ2. py — reproducibility script- aggregateₚysrcomparison. py — aggregation script Reproducibility: the gplearn comparison reruns end-to-end via the two included Python scripts on the public PIR codebase. Per-run JSON artifacts and provenance trail are preserved.