AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as Its Third Constraint Block

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This repository accompanies the manuscript “AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as Its Third Constraint Block.” It provides exact, replayable evidence for two fixed convex quadratic programs on which the unmodified direct three-block ADMM has a bounded non-KKT periodic sequence, despite a unique KKT point.

Start here

If you want to...	Start with
Read the mathematical argument	arXiv:2608.14396 · Repository PDF
Verify both counterexamples	Five-minute verification
Inspect the exact machine certificates	certificates/
Understand what each checker proves	Reproducibility contract
Follow the Codex and Kimi discovery routes	Research-stage index
Read the privacy-redacted Codex and Kimi interactions	Consolidated transcript · Raw text
Review time, token, and agent accounting	Computational provenance
Run the independent MATLAB check	MATLAB instructions

The root package is the acceptance layer. The research-process/ archive is historical evidence about how the results were found; it is not a second acceptance layer.

Certified results

Certificate ID	Fixed instance	Exact conclusion	Main evidence
identity_slack_p66_short_v1	m = 2, A = B = I_2, beta = 1	A specified initialization generates a bounded non-KKT orbit of minimal period 66	two independent Python representations and a MATLAB implementation
identity_slack_p23_rational_v1	m = 3, rational QP data, beta = 1	An open invariant set of reduced initializations converges phasewise to a non-KKT orbit of minimal period 23	exact rational replay and Lyapunov certificate

Period 66

• Projection word: (00)^2(01)^64.
• Exact closure and minimal period: 66.
• Strict projection checks: 132/132.
• Minimum signed margin: 0.0037105246944352910173... > 1/1000.
• The orbit is bounded and non-KKT; the QP has a unique KKT point.
• This certificate concerns a deliberately specified initialization. It does not claim attraction, unbounded divergence, or failure of corrected ADMM variants.

Period 23

• All primitive QP coefficients are reduced fractions with numerator magnitude and denominator at most 100.
• The frozen certificate contains the complete exact phase-zero state (x^0,y^0,z^0,lambda^0).
• Exact closure and minimal period: 23.
• Strict projection checks: 69/69, with minimum margin greater than 1/250.
• A rational matrix P satisfies P - M_per^T P M_per > 0 exactly.
• The ellipsoid e^T P e < 1/4000 is return-invariant in the canonical reduced (y,t) state, and every initialization in it converges phasewise to the period-23 sequence.
• This is a neighborhood of initializations for one fixed QP, not a perturbation result for the QP data or a global attraction theorem.

The Kimi Code K3 route originally reached an exact dyadic period-23 replay and an exact Jury local-attraction certificate. The denominator-100 instance and explicit invariant ellipsoid above are later release strengthening; see the route attestation and recompute its public ledger with python python/verify_kimi_provenance.py --check.

Multiplier relaxation for the period-66 QP

The exact relaxation certificate proves three separate statements:

1. one rational Lyapunov matrix works on the strict KKT branch for every tau in [49/100, 51/100];
2. the former period-66 initialization follows a certified strict prefix for 232 steps and enters the invariant ellipsoid for every tau in [1/2 - 10^-10, 1/2 + 10^-10];
3. the strict KKT branch has a unique Schur boundary satisfying 0.9366061114 < tau_c < 0.9366061115.

These are local or fixed-initialization results. They do not establish global convergence from arbitrary initial points.

Five-minute verification

The frozen Python environment is 3.13.5 with SymPy 1.13.3, NumPy 2.1.3, and pytest 8.3.4.

git clone https://github.com/ConanXu-math/identity-slack-admm-cycle-certificate.git
cd identity-slack-admm-cycle-certificate
python3.13 -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt

python python/verify_all.py
python python/verify_research_process_archive.py

The certificate command should end with:

{"checks": [{"name": "period66", "returncode": 0, "status": "passed"}, {"name": "period23", "returncode": 0, "status": "passed"}], "valid": true}

For the full release check used by GitHub Actions:

python python/verify_all.py
python python/verify_research_process_archive.py
python python/verify_kimi_provenance.py --check
python python/export_orbit_66.py
python python/certify_relaxed_multiplier_interval_theory.py
python python/verify_universal_step_obstruction.py --check
python -m pytest -q python/tests/test_relaxed_multiplier_interval_theory.py
python -m unittest discover -s tests -p "test_*.py"
python python/verify_matlab_certificate.py
git diff --exit-code -- certificates/

The last command is part of acceptance: regeneration must leave every tracked certificate byte-for-byte unchanged.

The universal-step check keeps the JSON structure and every exact symbolic field byte-strict. Its six NumPy/LAPACK spectral-radius values are explicitly numerical sanity checks and use absolute tolerance 1e-12 across platforms; the smallest stored excess above one is greater than 3.87e-7.

How the evidence is organized

Layer	Purpose	Authoritative entry point
Exact acceptance	Rebuilds the fixed QPs and checks every finite proof obligation	python/verify_all.py
Frozen data	Canonical inputs, exact orbits, verdicts, hashes, and Lyapunov data	certificates/
Implementation cross-checks	Independent signed-state, full-state, and MATLAB implementations for period 66	python/README.md, matlab/README.md
Process archive	Selected theory, experiments, failures, state files, and internal reviews	research-process/INDEX.md
Route accounting	Scope and telemetry for the Codex/Kimi comparison	provenance/README.md
Continuous integration	Clean-environment regeneration and artifact-stability gate	Exact certificate workflow

Agreement between implementations is an internal reproducibility cross-check, not external peer review.

Reading the research process

Use research-process/INDEX.md rather than browsing the archive chronologically:

• Codex / period 66: begin with the persistent task state and algebraic reductions, then follow Stages 43–46 from obstruction to numerical discovery, rationalization, exact replay, and precision audit.
• Kimi Code K3 / period 23: begin with START_GOAL.txt and RESEARCH_LOG.md, then follow the withdrawn routes, targeted instability experiments, period locking, and the exact certificate.

Archive labels matter:

• numerical_screen and proof_attempt are exploratory;
• withdrawn records a route that was rejected;
• theorem and exact_certificate are accepted only within their stated scope;
• review means an internal check, not external peer review.

Raw native session files, credentials, private configuration, local absolute paths, caches, and repetitive bulk outputs are intentionally excluded. A single privacy-redacted transcript of the user-visible Codex and Kimi Code K3 interactions is published separately as INTERACTION_TRANSCRIPT.md. The 168 retained files in research-process/ are covered by research-process/manifest.json and checked in CI.

MATLAB reproduction

The MATLAB verifier covers the period-66 instance and requires MATLAB R2025a, Symbolic Math Toolbox, and a valid license:

addpath("matlab")
result = verify_exact_cycle_matlab();
assert(result.valid)

Then compare the generated MATLAB JSON against the frozen Python fields:

python python/verify_matlab_certificate.py

See matlab/README.md for the class-based test and licensed GitHub Actions instructions.

Repository map

.
├── python/             exact Python verifiers and comparison drivers
├── matlab/             independent period-66 MATLAB verifier and tests
├── certificates/       frozen inputs and machine-readable certificates
├── research-process/   curated Codex and Kimi discovery archives
├── provenance/         comparison scope, accounting, and evidence boundaries
├── docs/               detailed reproducibility contract
├── paper/              compiled manuscript PDF
└── .github/            CI workflows and ownership rules

For script-level descriptions, see python/README.md. For the exact predicates, artifact meanings, runtime contract, and release checklist, see docs/REPRODUCIBILITY.md.

Scope of the Codex–Kimi comparison

The two routes independently reached exact counterexamples to the same proposed convergence principle, but they did not solve the same QP and were not matched in compute, tools, telemetry, stopping policy, or human intervention. The comparison is descriptive and endpoint-aligned; it cannot support a causal ranking of model speed, cost, or capability.

License

The software source code and machine-readable certificate data in this repository are licensed under the MIT License. The manuscript PDF under paper/ and prose research materials are excluded from the MIT License and remain all rights reserved unless a file states otherwise.

Release and citation status

Before creating an immutable archival release, the maintainers must freeze the author list, add CITATION.cff, create a tagged archive, assign a DOI, and make the manuscript and code-availability statement point to that same release.

Repository owner: ConanXu-math.