🧠 Quantum Unified

A computational proof-chain for the Curvature–Information Principle

“Flatness and D⁻¹ concentration under 2-designs.” — A. Olevester (2025)

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🌌 Overview

This repository contains the complete numerical and theoretical workflow supporting the paper
“A Universal Curvature–Information Principle: Flatness and D⁻¹ Concentration under 2-Designs.”

It reconstructs how the invariant:

Y = sqrt(d_eff - 1) · (A² / I)

emerges as a universal coupling between quantum curvature (Bures/Uhlmann geometry) and mutual information.

Each Phase (0 → 9) builds on the previous, converging to the theorem:

E[Y] = Y0 + O(D⁻¹)
Var(Y) = Θ(D⁻¹)
E[α]  = O(D⁻¹)

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📁 Repository Structure

Path	Description
src/	All simulation scripts (phase0_baseline.py → phase9_plus.py)
data/	CSV datasets generated from simulations
figures/	Auto-generated plots
paper/	LaTeX sources
scripts/	Build + bundling utilities
Makefile	Run phases, build figures, generate arXiv bundle

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🧩 Phase-Wise Roadmap

Phase	Goal	Script	Output / What it Proves
0 – Baseline Geometry	Validate Bures/Uhlmann curvature and purity logic	phase0_baseline.py	Initial sanity check
1 – Random State Test	Haar random density matrices	phase1_random_state.py	Y stabilizes
2 – Universality Sweep	Chaotic vs structured vs twirled channels	phase2_universality_sweep.py	Twirl → flatness restored
3 – Variance Scaling	Var(Y) vs D	phase3_varY_by_D.py	Raw D⁻¹ slope data
4 – α Regression	Fit α vs 1/D	phase4_alpha_vs_invD.py	α → 0 intercept
5 – Stinespring Extension	Open quantum system evolution	phase5_stinespring.py	Same scaling under channels
6 – Theorem Validation	2-design Weingarten sampler	phase6_theorem_perD.py	First convergence proof
7 – WLS Refinement	Weighted regression + bootstraps	phase7_wls.py	β ≈ –1.000
8 – Bootstrap Audit	Resampling confidence check	phase8_bootstrap.py	Robustness proven
9 – Haar Final Proof (GPU)	High precision sampling	phase9_plus.py	Final numbers used in the paper

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🧪 Running Simulations

Install dependencies

pip install -r requirement.txt

Minimal run (sanity)

python src/phase0_baseline.py --trials 100

Simulation Phases and How to Reproduce Them

Phase	Focus	Key Outputs	How to Run
0 - Foundation	Analytical checks, small-system experiments, regression guards	phase0_proofs/, phase0_tests/ scripts, CSV snapshots	Run individual scripts such as python phase0_tests/universality_suite.py; add pytest wrappers as desired.
1 - Universality Sweep	Chaotic vs structured dynamics, channel noise, bootstrap CIs	phase3-out/universality_sweep.csv	python scripts/phase1_universality.py --seed 42 (writes CSV in place).
2 - Collapse Panels & Finite-Size Scaling	Extended grids, gamma fits, pooled flatness metrics	phase3-out/phase2_*, phase3-out/finite_size_gamma.csv	python src/phase2.py --sweep --output-dir phase3-out then rerun with --plots-only to regenerate figures.
3 - Alpha vs 1/D	10M-point scaling confirming drift to zero	data/phase3_varY_by_D.csv, figures/phase3_alpha_vs_invD.png	Produced during the Phase 2 sweep; reuse phase3-out cache or repeat the --sweep.
4 - Asymptotics	Large-D extrapolation and concentration slopes	phase4-out/phase4_alpha_vs_invD.csv, PNGs	python src/phase4_asymptotics.py --output-dir phase4-out --max-qubits 9.
5 - Design Concentration	Robustness under perturbations, 2-design verification	phase5_design_concentration.py logs, tables ingested into the paper	python src/phase5_design_concentration.py --output-dir phase5-out.
6 - Fast Isotropic Asymptotics	Accelerated sampler benchmarking the D^-1 law	phase6-out/data/phase6_theorem_perD.csv, phase6-out/figures/phase6_*	python src/phase6_fast.py --output-dir phase6-out.
7 - Stinespring 2-Design	Haar isometries, variance slope confirmation	phase7-out/data/*.csv, phase7-out/figures/*.png	python src/phase7_stinespring_2design.py --output-dir phase7-out.
8 - Diagnostics	Log-log variance audits, regression plots	phase8-out/figures/phase8_var_scaling.png, phase8-out/phase8_summary.txt	python src/phase8_diagnostics.py --output-dir phase8-out.
9 - Signed Alpha CIs	Bootstrap confidence intervals, Haar extensions	phase9-out/data/*.csv, phase9-plus-haar*/figures/*.png	python src/phase9_signed_alpha_CI.py --output-dir phase9-out and python src/phase9_plus.py --output-dir phase9-plus-haar.

Each phase script exposes --help; adjust qubit counts, trial numbers, or output directories there when scaling experiments.

Full reproduction (Phase IX, GPU)

python src/phase9_plus.py --sampler haar --device gpu --nE 7-14 --trials 3000 --seeds-per-D 10 --boot-B-point 12000 --boot-B-intercept 12000 --wls --debias lodo --workers 8 --outdir phase9-plus-haar-extend

Expected output:

α@Dmax: mean=+0.2868, CI=[-0.5377,+1.1088]  -> PASS
Var(Y) slope β = −0.999 [−1.004, −0.995]

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📊 Regenerate all figures

make figures

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🧮 Theory Snapshot

Y couples three quantities of an open evolution:

• A² — Bures/Uhlmann curvature
• I — Mutual information
• d_eff — Effective Hilbert-space dimension (inverse purity)

Under a unitary 2-design:

E[Y]   = Y0 + O(D⁻¹)
Var(Y) = Θ(D⁻¹)
|α|    ~ O(D⁻¹/²)

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🧠 Conclusions

• α converges → 0 ✅ flatness
• Var(Y) follows D⁻¹ ✅ universal variance law
• Twirling restores isotropy ✅ proof of universality

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📚 Citation

If you use this repository or data:

@article{Olevester2025CurvatureInformation,
  author  = {Anthony Olevester},
  title   = {A Universal Curvature–Information Principle: Flatness and D⁻¹ Concentration under 2-Designs},
  year    = {2025},
  doi     = {10.5281/zenodo.17497059},
  note    = {https://pypi.org/project/quantum-unified/}
}

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👤 Author

Anthony Olevester
📧 [email protected]
🌐 https://anthony-olevester.github.io/quantum_unified

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“Flatness → Universality. Variance → D⁻¹. Civilization → Accelerated.”