ADCD — Anomaly-Driven Correction Discovery

Physics-constrained symbolic regression that discovers correction terms — not equations from scratch. The same logic that led from Newton to Einstein, from Rayleigh–Jeans to Planck.

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Science rarely discovers from a blank slate — it corrects. ADCD automates the step between anomaly and theory correction: given a classical law and data that disagrees with it, it searches for the minimal physically-valid correction term $\Delta$ — passing every candidate through dimensional, asymptotic, and complexity gates before a single parameter is ever fit.

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⚡ Key Features

• Correction-First Paradigm — Starts from a known classical law, not a blank slate. Focuses the search space on the discrepancy $\Delta$ between theory and experiment.
• Cascaded Physics Gates — AST complexity, dimensional homogeneity, transcendental guardrails, and asymptotic consistency (ARC) gates screen out unphysical candidates before running parameter-fitting.
• JAX-Traced L-BFGS-B Optimizer — Highly optimized parameter-scaled differentiable fitting with multi-restart log-uniform initialization.
• BIC Model Selection — Employs the Bayesian Information Criterion (BIC) to rank models, favoring simpler physical theories over overly complex numerical fits.
• Residual Feature Intelligence — Extracts mathematical features (monotonicity, curvature, oscillation, decay) from residuals to bias proposal templates.
• Phase 2: Multivariable Discovery — Buckingham Π group decomposition + per-variable Sequential ARC + variance-factorization separability detection for multi-input physical laws.
• Real-World Validated — Successfully identifies correct structural classes on Mercury's perihelion (GR), Lamb Shift (QED), Muon g-2 (Schwinger), and Blackbody (Planck).

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📦 Installation

Install the stable package from PyPI:

pip install adcd

Or install from source:

git clone https://github.com/apiprdt/PhysicsPaper.git
cd PhysicsPaper
pip install -e ".[dev]"

Verify your installation:

pytest tests/

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💻 Quick Start

1. High-Level Scientific API

Running ADCD on predefined physics benchmarks is extremely simple:

import adcd

# 1. Load a pre-defined benchmark scenario (e.g. Relativistic Kinetic Energy)
scenarios = adcd.get_all_scenarios()
scenario = scenarios[0]

# 2. Run discovery in a single line!
result = adcd.discover_correction(scenario, max_iterations=5, proposer="mock")

# 3. View the best fit
print(f"Discovered correction: {result.best_expr}")       # θ₀ * (v/c)**2
print(f"LaTeX representation:  {result.export_latex()}")   # \theta_0 \left(\frac{v}{c}\right)^2
print(f"Parameters:            {result.best_theta}")
print(f"BIC Score:             {result.best_bic:.2f}")

# 4. Plot residuals
result.plot_residuals()

2. Custom Experimental Datasets

For custom datasets, use the adcd.fit function:

import numpy as np
import adcd

# Your custom data
x = np.linspace(1.0, 5.0, 100)
X = {"x": x}
y_classical = 2.0 * x
y_observed  = 2.0 * x + 0.5 * x**2   # True correction is 0.5 * x^2

# Run ADCD
result = adcd.fit(
    X=X,
    y_obs=y_observed,
    y_classical=y_classical,
    limit_variable="x",
    limit_direction="0",
    correction_mode="additive"
)

result.summary()

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📊 Benchmark Results

Headline (primary claim): a mean structural recovery of 80.4% (±7.4%) across sixteen independent seeds (95% bootstrap CI [76.7%, 84.0%]). The reference seed=42 below is disclosed explicitly as the highest-performing seed (94.4%) — the mean, not the peak, is the claim. Full per-seed × per-noise breakdown ships in results/seed_distribution.json.

1. Multi-Seed Distribution (primary result, Mock Proposer)

Noise level	ADCD mean (16 seeds)	ADCD worst seed	ADCD best (seed=42)
0%	86.8% (±9.8%)	66.7% (6/9)	100% (9/9)
1%	81.2% (±14.6%)	44.4% (4/9)	100% (9/9)
5%	77.1% (±10.0%)	66.7% (6/9)	88.9% (8/9)
10%	76.4% (±12.3%)	55.6% (5/9)	88.9% (8/9)
Overall	80.4% (±7.4%)	69.4% (25/36)	94.4% (34/36)

2. Reference-Seed Detail (seed=42, Mock Proposer)

Scenario	Tier	0% Noise	1% Noise	5% Noise	10% Noise
Relativistic KE	Textbook	✓	✓	✓	✓
Yukawa Gravity	Textbook	✓	✓	✓	✓
Anharmonic Spring	Textbook	✓	✓	✓	✓
Screened Coulomb	Cross-Domain	✓	✓	✗	✗
Net Radiation	Cross-Domain	✓	✓	✓	✓
Nonlinear Drag	Cross-Domain	✓	✓	✓	✓
Mystery-A (tanh²)	Synthetic	✓	✓	✓	✓
Mystery-B (sinc)	Synthetic	✓	✓	✓	✓
Mystery-C (log-quotient)	Synthetic	✓	✓	✓	✓
Overall		100%	100%	88.9%	88.9%

3. PySR Comparison (same residual, 5% noise — the structural-selection test)

The gap is seed-independent: even ADCD's worst of 16 seeds beats PySR fair, and PySR with doubled budget cannot reach ADCD's worst seed.

Method (5% noise)	0%	1%	5%	10%
ADCD (ours, seed=42)	9/9 (100%)	9/9 (100%)	8/9 (88.9%)	8/9 (88.9%)
ADCD multi-seed mean	86.8%	81.2%	77.1%	76.4%
ADCD worst of 16 seeds	66.7%	44.4%	66.7%	55.6%
PySR fair (100 iter, 60s)	4/9 (44.4%)	5/9 (55.6%)	1/9 (11.1%)	5/9 (55.6%)
PySR generous (2× budget)	4/9 (44.4%)	4/9 (44.4%)	5/9 (55.6%)	2/9 (22.2%)

At 5% noise the gap is +66.0 points (ADCD multi-seed mean 77.1% vs PySR fair 11.1%). Doubling PySR's budget (generous, 55.6%) does not close it — and that doubled-budget figure still sits below ADCD's worst of 16 seeds (66.7%). PySR was run once per (scenario, noise); ADCD across 16 seeds. PySR non-monotonic under noise; ADCD stable.

3. Phase 2: Multivariable Benchmark

Scenario	Variables	ADCD Solved	Notes
Yukawa Mass-Ratio	m, M, r, r₀	✓	Π groups: m/M, r/r₀
Turbulent Drag	v, ρ, A, C_D	✓	Separable multiplicative
Coupled Oscillator	k, m, Ω, ω₀	✗	Mixed functional form
Van der Waals MV	a, b, P, V, T	✗	Requires 3rd Π group
Overall		2/4 (50%)	Baseline: 0/4

4. Real-World Physical Constants

Validation on historical anomalies using physical constants from JPL DE440, NIST, and CODATA:

Physical Scenario	Discovered Correction	Converged	Class Match	NMSE
Mercury Perihelion (GR)	θ₀·vc²	—	✓ polynomial	1.11e-05
Hydrogen Lamb Shift (QED)	θ₀(n/θ₁)^(-θ₂)	✓	✓ power_law	1.82e-18
Muon g-2 (Schwinger)	θ₀(α/π)^θ₁	✓	✓ polynomial	7.94e-07
Blackbody (Planck)	-1 + e^(-f/θ₁)	—	✓ exponential	2.59e-02

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📁 Project Structure

adcd-v3.0.0/
├── src/adcd/                       # Installable package
│   ├── __init__.py                 # Public API (fit, discover_correction)
│   ├── anomaly_scenarios.py        # 9 standard + 3 blind + 4 multivariable scenarios
│   ├── arc_scorer.py               # Asymptotic consistency gate (ARC)
│   ├── buckingham_pi.py            # [Phase 2] Buckingham Π group engine
│   ├── coarse_evaluator.py         # Coarse numerical pre-filter
│   ├── correction_orchestrator.py  # Main multi-iteration discovery loop
│   ├── dimensional_checker.py      # Dimensional homogeneity + transcendental gate
│   ├── jax_optimizer.py            # JAX L-BFGS-B optimizer
│   ├── llm_proposer.py             # Mock + Gemini + OpenAI proposers
│   ├── metrics.py                  # NMSE, BIC, structural classification
│   ├── multivar_orchestrator.py    # [Phase 2] Multivariable correction pipeline
│   ├── pipeline.py                 # Stage 1 filter cascade
│   ├── real_data_loader.py         # Real-world data loading (JPL, NIST, CODATA)
│   ├── residual_factorizer_v2.py   # [Phase 2] Variance-decomposition separability
│   ├── result.py                   # CorrectionResult object
│   └── sequential_arc.py           # [Phase 2] Per-variable Sequential ARC checker
├── tests/                          # Unit + integration tests
├── paper/                          # LaTeX source (main.tex) + figures
├── data/                           # Input datasets (SPARC, cosmic chronometers, growth rate)
├── scripts/                         # Table generation and verification scripts
├── run_correction_discovery.py     # Benchmark runner
└── README.md                       # This file

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📖 Citing This Work

If you use ADCD in your research, please cite:

@software{erdita2026adcd,
  author    = {Erdita, Muhammad Afif},
  title     = {{Anomaly-Driven Correction Discovery (ADCD): Physics-Constrained
                Symbolic Regression for Evolutionary Scientific Discovery}},
  year      = {2026},
  publisher = {Zenodo},
  version   = {3.0.0},
  doi       = {10.5281/zenodo.20534940},
  url       = {https://doi.org/10.5281/zenodo.20534940}
}

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🔬 Reproducibility

Every quantitative claim in this project is reproducible from committed scripts. No number is hand-typed.

# Regenerate the 9-scenario benchmark (seed=42)
python run_correction_discovery.py

# Multi-seed study (16 seeds × 9 scenarios × 4 noise levels)
python run_reproducibility.py

# Build the per-seed × per-noise anti-cherry-pick artifact
python scripts/generate_seed_distribution.py    # → results/seed_distribution.json

# Guard: fails loudly if any headline number drifts
python scripts/verify_paper_claims.py

# SPARC MOND robustness study
python -m adcd.experiments.sparc_robustness

The full test suite must pass before any release:

pytest tests/ -q

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📄 License

This project is licensed under the MIT License.