aqei-numerical-validation

Numerical validation pipeline for AQEI (Averaged Quantum Energy Inequality) causal stability. This repository contains the discrete causal poset simulation tools, homology stability testing, Mathematica symbolic search, and computational physics manuscript — separated from the formal Lean 4 verification work in the companion aqei-bridge repository.

Relationship to companion repos

Repo	Role
aqei-bridge	Lean 4 formal proofs of causal stability; bridge conjecture; formal verification track
energy-tensor-cone	AQEI cone convexity proofs, extreme ray verification (submitted to Physical Review D)
this repo	Discrete causal poset simulation, H₁ homology stability testing, Mathematica AQEI search, computational physics manuscript

The Mathematica search in this repo outputs JSON candidates that feed into the Lean formalization pipeline in aqei-bridge. The orchestrator (aqei-bridge/python/orchestrator.py) coordinates both sides.

What is in this repo

Papers

• papers/aqei-numerical-validation.tex — Manuscript for computational physics journal (CPC/JCP target). Covers the hybrid symbolic-numeric AQEI search pipeline, FFT-based perturbation testing, homology stability, and multi-ray overlap analysis.

Python pipeline (python/)

Script	Purpose
poset_homology_proxy.py	Core tool: H₁ computation, FFT perturbation, stability sweeps
minkowski_poset.py	Generate discrete Minkowski causal grids
causal_graph_tools.py	Graph causality analysis utilities
poset_interval_tools.py	Order interval utilities
ctc_scan.py	Closed timelike curve detection (Tarjan's SCC)
multi_ray_analysis.py	Multi-ray Jaccard overlap (path-connectedness proxy)
sweep_parameters.py	Parameter space sweeps
sweep_analysis.py	Aggregate sweep statistics

Mathematica (mathematica/)

Script	Purpose
search.wl	Batch symbolic AQEI constraint search
search.nb	Notebook interface for interactive exploration
visualize_results.wl	Post-processing and plotting
results/	JSON output from search runs

Docs (docs/)

• h1_stability_results.md — Empirical H₁ stability test results (100% invariance over 100 trials)
• matlab_comsol_integration.md — MATLAB/COMSOL analog gravity integration guide
• phase4_searches.md — Parameter sweep workflow documentation

Run artifacts (runs/)

• h1_stability_sweep/ — Stability sweep results (Minkowski 10×10 baseline, FFT perturbation tests)

Key results

• H₁ invariance: 100% stability over 100 trials (50 mild ε=0.05 + 50 strong ε=0.3)
• Baseline: Minkowski 10×10 grid, 121 nodes, 310 edges, dim Z₁ = 190
• No CTCs found in any baseline Minkowski runs
• AQEI constraint candidates: 20+ top candidates from Mathematica search feeding into Lean formalization

Quick start

Prerequisites

• Python 3.8+ with NetworkX, NumPy, SciPy
• Wolfram Engine / WolframScript on PATH (optional, for Mathematica)
• MATLAB R2025b with PDE/Symbolic toolboxes (optional, for analog gravity work)
• COMSOL 6.4 with Acoustics Module (optional, for acoustic horizon model)

Run the Python pipeline

cd python

# Generate discrete Minkowski poset
python poset_homology_proxy.py compute-z1 --tmax 10 --xmax 10

# Run H₁ stability sweep
python poset_homology_proxy.py perturb-fft minkowski.json --trials 50 --epsilon 0.05

# Scan parameter space
python poset_homology_proxy.py scan-minkowski-perturb --tmax-range 5 10 15 --epsilon 0.1

Run all tests

./run_tests.sh

Run Mathematica AQEI search

wolframscript -file mathematica/search.wl

Repo layout

aqei-numerical-validation/
├── python/
│   ├── poset_homology_proxy.py     # Core: H₁ computation and perturbation tests
│   ├── minkowski_poset.py          # Discrete Minkowski grid generation
│   ├── causal_graph_tools.py       # Graph causality analysis
│   ├── poset_interval_tools.py     # Order interval utilities
│   ├── ctc_scan.py                 # CTC detection (Tarjan)
│   ├── multi_ray_analysis.py       # Multi-ray Jaccard overlap proxy
│   ├── sweep_parameters.py         # Parameter space sweeps
│   └── sweep_analysis.py           # Sweep statistics
├── mathematica/
│   ├── search.wl                   # Batch AQEI symbolic search
│   ├── search.nb                   # Interactive notebook
│   ├── visualize_results.wl        # Post-processing
│   └── results/                    # JSON outputs + plots
├── papers/
│   └── aqei-numerical-validation.tex  # CPC/JCP manuscript (computational physics)
├── docs/
│   ├── TODO.md                     # Active tasks
│   ├── h1_stability_results.md     # Empirical stability results
│   ├── matlab_comsol_integration.md # MATLAB/COMSOL guide
│   └── phase4_searches.md          # Sweep workflow docs
├── runs/
│   └── h1_stability_sweep/         # Stability sweep artifacts
├── tests/
│   ├── python_tests.sh             # Python unit tests
│   ├── mathematica_tests.sh        # Mathematica tests
│   └── test_pipeline.py            # Integration pipeline tests
└── run_tests.sh                    # Master test script

Mathematical framework

H₁ proxy for causal stability

The discrete H₁ homology proxy tracks 1-cycles in the causal graph. Under AQEI-admissible perturbations:

$$\dim H_1(P') = \dim H_1(P) \quad \text{for } |\delta| < \epsilon$$

This is the discrete analogue of the bridge conjecture (proven formally in aqei-bridge).

FFT perturbation testing

The cmd_perturb_fft subcommand applies low-pass FFT perturbations to edge weights, testing whether H₁ is invariant. The empirical results (100% invariance) support the conjecture and guide Lean proof strategies.

Publication

Target venue: Computational Physics Communications (CPC) or Journal of Computational Physics (JCP)

See papers/aqei-numerical-validation.tex for the full manuscript (12 pages, currently in draft).

License

MIT (see LICENSE).