Formally Verified Lean 4 Port of GAP: Computational Discrete Algebra

📄 Official Published arXiv Preprint: arXiv:2609.38492 [cs.LO] │ Direct PDF │ DOI: 10.48550/arXiv.2609.38492
Title: "Machine-Checked Computational Group Theory in Lean 4: Operational Schreier-Sims Stabilizer Chains, BSGS Sifting, and Backtrack Ordered Partitions"
Authors: Volkan Dağlı, Zerrin Dağlı, Dağhan Dağlı • Zenodo Release DOI: 10.5281/zenodo.23045504

Note

🛡️ Phased Verification Roadmap & Isolation Boundary

Phase 1 & Phase 2 (Completed & Published): Fully verified across 35 machine-checked theorems with 0 sorry, 0 admit, and 0 external axioms (covering Schreier-Sims stabilizer chains lib/stbc.gi, ordered partition refinements lib/partitio.gi, cyclotomic extension rings lib/zmodnze.gi, and constructive Bézout modular inverses lib/zmodnz.gi). Independently audited via gokujo 1.0.0 (PR #5, Lake / Nix Lean 4.28.0) with witness report in telemetry/gokujo-check-2026-10-06.json. Formally published on arXiv:2609.38492 and Zenodo (DOI: 10.5281/zenodo.23045504).
Phase 3 (Active Isolated Reserve): Permutation group orders, orbit-stabilizer systems, and conjugacy structures are strictly maintained in an isolated staging reserve (docs/PHASE3_STRATEGIC_ROADMAP.md & docs/PHASE3_PATTERN_LANGUAGE_SPEC.md) in active collaborative alignment with Mike DuPont (@jmikedupont2).

This repository provides machine-checked formal verifications in Lean 4 / Mathlib for the core computational discrete algebra algorithms and representations of the GAP System (Groups, Algorithms, Programming).

Developed by the ITouch Systems Formal Verification Lab (Volkan Dagli [@pCwOrM] & Family).

What's New in Release v0.2.0 (Phase 2 Milestone)

Release v0.2.0 advances beyond static residue structures into verified operational computational group theory and backtrack combinatorial algorithms, formalizing three major modules from GAP 4's core library with 0 sorry, 0 admit, and 0 external axioms:

GAP-0299 (lib/stbc.gi): Schreier-Sims Stabiliser Chains & Transversal Invariants
GAP-0332 (lib/partitio.gi): Backtrack Ordered Partitions & Cell Refinement Invariants
GAP-0332 (lib/zmodnze.gi): Rings $\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)$ of Cyclotomic Extensions
GAP-0331 (lib/zmodnz.gi): Modular Residue Methods, Invertibility & Extended Euclidean GCD

1. Stabiliser Chains & Schreier-Sims Algorithms (`lib/stbc.gi`)

Module: RequestProject.Gap.Library.Stbc
GAP Reference: lib/stbc.gi (by Heiko Theißen and Ákos Seress)

Faithfully models GAP's stabiliser chain hierarchy, transversal trees, sifting reductions, and Base & Strong Generating Set (BSGS) membership testing:

Core Structures:
- GAP.Stbc.StabLevel: Models each chain level $(\beta_i, \Delta_i, S_i, u_i)$ with base point, basic orbit, generators, and inverse transversal representatives.
- GAP.Stbc.StabChain: The complete descending stabiliser chain $[G^{(1)}, G^{(2)}, \dots, G^{(k)}]$.
Operational Algorithms:
- siftOneLevel: Single-level coset reduction $g \mapsto u_y^{-1} \cdot g$.
- siftFull / siftedPermutation: Full multi-level Schreier sifting across the base.
- membershipTestKnownBase: Group membership decision procedure using sifting.
- extendSchreierPoint: Transversal tree extension step.
Verified Theorems (0 sorry):
- siftOneLevel_fixes_basePoint: Proves that single-level sifting strictly fixes the base point $\beta_i$.
- siftFull_fixes_all_basePoints: Proves that a fully sifted element fixes every base point in the base sequence $(\beta_1, \dots, \beta_k)$.
- siftedPermutation_mem_subgroup_iff: Invariant preservation during coset reduction.
- membershipTestKnownBase_sound: Proves that if membership test returns true, then $g \in G$.
- membershipTestKnownBase_iff_mem: Soundness and completeness equivalence for subgroup membership.
- extendSchreierPoint_invariant: Invariant preservation of the transversal tree under tree extension.

2. Ordered Partitions for Backtrack Searching (`lib/partitio.gi`)

Module: RequestProject.Gap.Library.Partitio
GAP Reference: lib/partitio.gi (by Heiko Theißen)

Models ordered partitions and cell refinement operations fundamental to GAP's permutation group backtrack search, partition backtracks, and automorphism computation:

Core Structures:
- GAP.Partitio.OrderedPartition: Partition of $\Omega$ into ordered, non-empty, pairwise disjoint cells.
- fixcells: Identifies cells consisting of a single fixed point (size 1).
- splitCellByPred: Core cell-splitting primitive underlying SplitCell and IsolatePoint.
Verified Theorems (0 sorry):
- splitCellByPred_disjoint: Formally proves that cell splitting always yields mutually disjoint subcells.
- splitCellByPred_union: Proves exact element conservation (the union of subcells equals the original cell).
- splitCellByPred_length_sum: Proves cardinality conservation $|C_{yes}| + |C_{no}| = |C|$.
- mem_splitCellByPred_iff: Characterizes exact predicate-driven membership in split cells.

3. Cyclotomic Extension Rings $\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)$ (`lib/zmodnze.gi`)

Module: RequestProject.Gap.Library.Zmodnze
GAP Reference: lib/zmodnze.gi (by Alexander Konovalov)

Formalizes elements and arithmetic of GAP's cyclotomic extension rings $\mathbb{Z}/n\mathbb{Z}(\varepsilon)$, where $\varepsilon^m = 1$:

Core Structures:
- ZmodnZepsObj: Formal representation via coefficient vectors Fin m → GAP.ZModnZObj n.
- AddCommGroup and convolution group-ring multiplication mulOp.
Verified Theorems (0 sorry):
- card_eq: Machine-checks GAP's exact Size formula: $$\text{card}(\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)) = n^m$$

4. $\mathbb{Z}/n\mathbb{Z}$ Modular Residue Methods (`lib/zmodnz.gi`)

Module: RequestProject.Gap.Library.Zmodnz
GAP Reference: lib/zmodnz.gi (by Thomas Breuer)

Dual verification architecture combining abstract mathematical isomorphism with operational executable algorithms:

Semantic Model Isomorphism: Canonical bijection ZModnZObj n ≃ ZMod n, deriving CommRing and Field (for prime $n$), with unit theorem IsUnit a ↔ a.val.Coprime n.
Constructive Executable Algorithms: Verified isUnitExec and constructive inverseOpExec via the Extended Euclidean Algorithm (Nat.gcdA Bézout coefficients) without non-constructive choice:
```
theorem inverseOpExec_correct (a : ZModnZObj n) (h : isUnitExec a = true) :
    ∃ inv : ZModnZObj n, inverseOpExec a = some inv ∧ mulExec a inv = oneExec
```

Axiomatic Purity Audit

Every theorem in this repository has been audited with #print axioms:

Module	GAP Source	Theorems	Foundations
`Stbc.lean`	`lib/stbc.gi`	8	`propext, Classical.choice, Quot.sound`
`Partitio.lean`	`lib/partitio.gi`	6	`propext, Classical.choice, Quot.sound`
`Zmodnze.lean`	`lib/zmodnze.gi`	4	`propext, Classical.choice, Quot.sound`
`Zmodnz.lean`	`lib/zmodnz.gi`	17	`propext, Classical.choice, Quot.sound`
Total		35	Standard Lean 4 Core

🛡️ Independent Telemetry Verification Gate (PR #5):
Independently verified by Mike DuPont (@jmikedupont2) via gokujo 1.0.0 (lean-worker), backend Lake with Nix-pinned Lean 4.28.0 (store 75c91mcn).
Results: 8/8 files built, 223 declarations scanned, 100 axioms audited, 0 holes, 0 sorries. Gate: PASS (exit 0). See witness artifact at telemetry/gokujo-check-2026-10-06.json.

Build Instructions

# Clone the repository
git clone https://github.com/pCwOrM/gap-lean4-port.git
cd gap-lean4-port

# Build the entire verified library
lake build RequestProject

Authors & Citation

ITouch Systems Formal Verification Lab

Volkan Dagli (@pCwOrM) & Family
Research Lab: Mersin / Istanbul, Turkey

Correspondence: ask@answerr.me | pcworm@pcworm.net

Community, Collaboration & Standards

Zulip Channel: Join real-time technical discussions on our Zulip realm at solfunmeme.zulipchat.com (Streams: #general > greetings, #general > architecture).
Shared Terminology Standard: Collaborative formal verification standard defined in Shared Terminology Guide v2 (DuPont–Dağlı Specification).
Rung 0–5 Verification Ledger: Formal accounting of chunks, anchors, pre/post/frame contracts, and degree qualifiers in docs/RUNG_LEDGER.md.
Distributed Architecture: Dual-engine Lean 4 + eBPF/Nix architecture in docs/ARCHITECTURE.md (Formalizing the joint Rung 0–5 verification ladder).
Rung 0–5 Verification Bridge & Worker Packets: Automated bridge generator (tools/bridge_generator.py) and consolidated witness report (tasks/bridge_witness_report.json) generating turnkey harmonic.gap-worker-job/1 packets for Mike DuPont's lean-worker and aristotle-cli-rs.

🏛️ Associated Formal Verification & Scientific Corpus

This formal verification engine serves as the mathematical and algebraic pillar for the broader zero-storage procedural AI ecosystem:

Makale / Sütun	Başlık / Katkı	En Son Sürüm DOI	Çatı / Konsept DOI	arXiv / Doğrulama Durumu	Birincil Depo
Paper 1: MFNS	Mandelbrot Fractal Neural Synthesis	`10.5281/zenodo.22867037`	`10.5281/zenodo.22774934`	Açık Bilim Önbaskısı (Revizyonda)	`pCwOrM/mandelbrot-fractal-neural-synthesis`
Paper 2: OED	Orbital Error Dynamics (OED v3)	`10.5281/zenodo.22900465`	`10.5281/zenodo.22896855`	`arXiv:2609.30115` • HF Paper	`pCwOrM/mandelbrot-fractal-neural-synthesis`
Paper 3: WerreduR	Procedural Fractal Pedagogy (v4.0)	`10.5281/zenodo.23128224`	`10.5281/zenodo.22999420`	Hedef: Q1 AIED / CAEAI • Canlı Simülatör	`jesmaat/WerreduR` (Mirror: `pCwOrM/WerreduR`)
Paper 4: BH Page	Black Hole Page Curve & Quantum Gravity (v5)	`10.5281/zenodo.23072120`	`10.5281/zenodo.22961999`	CERN Record 22978460 • Stabilizer Kodları	`pCwOrM/mandelbrot-fractal-neural-synthesis`
Paper 5: Formal	Lean 4 40-Core Gauntlet (v2)	`10.5281/zenodo.22983889`	`10.5281/zenodo.22974543`	`arXiv:2609.33066` • HF Paper	`pCwOrM/werr` & `werracle`
Paper 6: WerrSoma	Drosophila Connectome (158K Nöron, v1.1)	`10.5281/zenodo.23072929`	`10.5281/zenodo.22996625`	`arXiv:submit/8161759` [on hold] • 3D Portal	`Lexovian/WerrSoma` (Mirror: `pCwOrM/WerrSoma`)
Paper 7: Werracle	Sub-Cent Intra-Block EVM AI Oracle (v1)	`10.5281/zenodo.22942599`	`10.5281/zenodo.22942598`	`arXiv:2609.30719` • HF Paper	`pCwOrM/werracle` & `werralem`
Paper 8: Schönhage	Tensor Taşıyıcı Sıkıştırma ($W=176M$)	`10.5281/zenodo.23268580`	`10.5281/zenodo.23268579`	`arXiv:2610.14263` (PDF • DOI: `10.48550/arXiv.2610.14263` • `submit/8207241` [published]) • CrocSwap #226 (Incorporated) • Contributors • Audit `50d76881`	`pCwOrM/integer-mult-bounds`
Paper 9: Güneş Dili	Deterministik Morfoloji & 53 Koma	`10.5281/zenodo.23273006`	`10.5281/zenodo.23273005`	Lean 4 Doğrulanmış • Canlı Portal	`pCwOrM/gunes-dili`
Paper 10: GDT	Genel Bölen Teoremi & Kalan Koşullanması	`10.5281/zenodo.23316070`	`10.5281/zenodo.23316070`	Lean 4 Doğrulanmış (2.287 Kontrol PASS)	`pCwOrM/general-divisor-theorem`
Paper 11: Niemeier	24D Niemeier $A_8^3$ & K3 Umbral Moonshine	`10.5281/zenodo.23324066`	`10.5281/zenodo.23324066`	Lean 4 Sertifikalı (0 sorry, 0 aksiyom) • Kayıt 23324067	`pCwOrM/niemeier-a8-umbral-moonshine`
Motor: WERR	Zero-VRAM Gauntlet & JevBench Intake	`10.5281/zenodo.22939253`	`10.5281/zenodo.22867425`	`arXiv:2609.25498` • HF Paper • HF Dataset	`pCwOrM/werr`
Platform: answerr	Reflex AI & Canlı Çalışma Alanı	Canlı: `answerr.me`	-	Çift Bilişsel Refleks Platformu & API	`pCwOrM/answerr`
Ayrık Alg	GAP-Lean4 Biçimsel Doğrulanmış Port (35 Teorem)	`10.5281/zenodo.23045504`	`10.5281/zenodo.23045503`	`arXiv:2609.38492` • HF Paper	`pCwOrM/gap-lean4-port`
Robotik: WERRobotics	Sıfır-Bellek Arduino & Gömülü Edge AI Çekirdeği (`v0.5.1`)	`10.5281/zenodo.22939253`	`10.5281/zenodo.22867425`	ATmega328P / ESP32 / STM32 (`0 Byte Heap`, `TR 2026/016633`)	`pCwOrM/WERRobotics`

Citation

If you build upon or reference this formal verification, please cite:

@article{dagli_2026_gap_lean4_arxiv,
  author       = {Da{\u{g}}l{\i}, Volkan and Da{\u{g}}l{\i}, Zerrin and Da{\u{g}}l{\i}, Da{\u{g}}han},
  title        = {{Machine-Checked Computational Group Theory in Lean 4: Operational Schreier-Sims Stabilizer Chains, BSGS Sifting, and Backtrack Ordered Partitions}},
  journal      = {arXiv preprint arXiv:2609.38492 [cs.LO]},
  year         = {2026},
  month        = sep,
  doi          = {10.48550/arXiv.2609.38492},
  url          = {https://arxiv.org/abs/2609.38492},
  note         = {Zenodo DOI: 10.5281/zenodo.23045504; Mathlib 4 Compatible, 35 Theorems, 0 sorry}
}

@software{dagli_2026_gap_lean4,
  author       = {Dağlı, Volkan and Dağlı, Zerrin and Dağlı, Dağhan},
  title        = {{Machine-Checked Computational Group Theory in Lean 4: Operational Schreier-Sims Stabilizer Chains, BSGS Sifting, and Backtrack Ordered Partitions}},
  month        = sep,
  year         = 2026,
  publisher    = {Zenodo},
  version      = {v0.2.0},
  doi          = {10.5281/zenodo.23045504},
  url          = {https://doi.org/10.5281/zenodo.23045504}
}

Name		Name	Last commit message	Last commit date
.github/workflows		.github/workflows
RequestProject		RequestProject
docs		docs
estimation		estimation
tasks		tasks
telemetry		telemetry
tools		tools
.gitignore		.gitignore
ARISTOTLE_SUMMARY.md		ARISTOTLE_SUMMARY.md
LICENSE		LICENSE
README.md		README.md
lake-manifest.json		lake-manifest.json
lakefile.toml		lakefile.toml
lean-toolchain		lean-toolchain

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Formally Verified Lean 4 Port of GAP: Computational Discrete Algebra

🛡️ Phased Verification Roadmap & Isolation Boundary

What's New in Release v0.2.0 (Phase 2 Milestone)

1. Stabiliser Chains & Schreier-Sims Algorithms (`lib/stbc.gi`)

2. Ordered Partitions for Backtrack Searching (`lib/partitio.gi`)

3. Cyclotomic Extension Rings $\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)$ (`lib/zmodnze.gi`)

4. $\mathbb{Z}/n\mathbb{Z}$ Modular Residue Methods (`lib/zmodnz.gi`)

Axiomatic Purity Audit

Build Instructions

Authors & Citation

Community, Collaboration & Standards

🏛️ Associated Formal Verification & Scientific Corpus

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Formally Verified Lean 4 Port of GAP: Computational Discrete Algebra

🛡️ Phased Verification Roadmap & Isolation Boundary

What's New in Release v0.2.0 (Phase 2 Milestone)

1. Stabiliser Chains & Schreier-Sims Algorithms (lib/stbc.gi)

2. Ordered Partitions for Backtrack Searching (lib/partitio.gi)

3. Cyclotomic Extension Rings $\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)$ (lib/zmodnze.gi)

4. $\mathbb{Z}/n\mathbb{Z}$ Modular Residue Methods (lib/zmodnz.gi)

Axiomatic Purity Audit

Build Instructions

Authors & Citation

Community, Collaboration & Standards

🏛️ Associated Formal Verification & Scientific Corpus

Citation

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1. Stabiliser Chains & Schreier-Sims Algorithms (`lib/stbc.gi`)

2. Ordered Partitions for Backtrack Searching (`lib/partitio.gi`)

3. Cyclotomic Extension Rings $\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)$ (`lib/zmodnze.gi`)

4. $\mathbb{Z}/n\mathbb{Z}$ Modular Residue Methods (`lib/zmodnz.gi`)