Physics Intermediate Representation (PIR): A Structured Framework for Automated Physical Law Discovery from Data — Version 3.3

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 3.3 (May 2026) — External-benchmark evaluation upgrade. KEY NEW RESULT (Section 9.4):PIR evaluated on a 44-equation Tier A subset of the Feynman Symbolic Regression Database (Udrescu & Tegmark 2020) under a SRBench-compatible held-out numerical-equivalence protocol (median relative error < 5% on a disjoint 500-point test set). Verified per-task DR: 27.3% (11/44 fully recovered, 3 partially, 30 not recovered). Fully recovered equations include kinetic energy 1/2*m*(v^2+u^2+w^2), gravitational PE m*g*z, Coulomb-like inverse-square components q1*r/(4*pi*epsilon*r^3), friction mu*Nn, photon energy hbar*omega, and angular momentum quantization n*hbar. For context (from published SR literature, different protocols, orientation only): gplearn ~6%, BSR ~6%, DSR ~37%, PIR v3.3 27.3%, AI Feynman 2.0 ~74%. PIR is positioned between naive genetic programming and neural-augmented systems. FAILURE-MODE ANALYSIS (30 unrecovered tasks):- F1: Single-variable nonlinear (5 tasks like 1/2*k*x^2, q/C) — PIR's basis-construction does not auto-generate x^n for single variables. Cheapest v3.4 fix.- F2: Transcendental unary functions (12 tasks needing exp/log/sqrt/arcsin) — PIR's runner currently restricts unary set to {sin, cos}. Grammar extension targeted for v3.4.- F3: Complex multi-variable interactions (13 tasks) — structure detector misses dominant variable pairings in 5+ variable equations. Targeted for v3.5 architectural work. ALSO NEW:- Section 3.4: Formal success-criterion split between PIR-Bench-20 legacy protocol (substring tokens + error threshold) and v3.3 held-out numerical-equivalence protocol (SRBench-compatible).- Section 9.5: PIR-Bench-20 re-verification under the stricter held-out protocol is in progress; result will be published in v3.3.1 or v3.4 regardless of outcome. UNCHANGED FROM v3.2:- PIR-Bench-20 results (Table 1, mean DCS 0.840)- gplearn head-to-head (Section 9.3, PIR 100% / gplearn 50% mean DR; 14W/6T/0L)- All design constants (alpha=0.7, beta=0.3, gamma=0.2, Langevin T=500) DEFERRED TO v3.4 OR LATER:- Tier B and Tier C Feynman evaluations- Formal SRBench submission (sklearn-compatible PIR algorithm)- PySR head-to-head (Python 3.13 / PythonCall.jl upstream issue)- Full PIR component ablation Bundle contents:- PIR_Architecture_v3_3.pdf — main paper (14 pages)- PIR_Architecture_v3_3.tex — LaTeX source- feynman_loader.py — manifest-driven Feynman dataset generator- verify_feynman_tierA.py — held-out structural verifier (SRBench-style)- aggregate_feynman_tierA.py — per-task results aggregator- tier_a_verified.md / .csv / .json — verified per-task results Reproducibility: the Feynman Tier A evaluation reruns end-to-end from the PIR repository using the included scripts. The Feynman manifest (FeynmanEquations.csv) is sourced from Udrescu & Tegmark's MIT page and not redistributed.