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, Dr. 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).

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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:

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

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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.

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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.

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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: $$\text{card}(\mathbb{Z}/n\mathbb{Z}(\varepsilon_m)) = n^m$$

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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:

1. 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.
2. 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

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Axiomatic Purity Audit

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

Module	GAP Source	Theorems	sorry	External Axioms	Foundations
Stbc.lean	lib/stbc.gi	8	0	0	propext, Classical.choice, Quot.sound
Partitio.lean	lib/partitio.gi	6	0	0	propext, Classical.choice, Quot.sound
Zmodnze.lean	lib/zmodnze.gi	4	0	0	propext, Classical.choice, Quot.sound
Zmodnz.lean	lib/zmodnz.gi	17	0	0	propext, Classical.choice, Quot.sound
Total		35	0	0	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.

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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

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Authors & Citation

ITouch Systems Formal Verification Lab

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

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

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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.

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🏛️ Associated Formal Verification, World Records & Scientific Corpus

All monographs, formal Lean 4 specifications, and empirical datasets across the WERR & ITouch Systems Research Group are permanently archived on CERN Zenodo, arXiv, Elsevier SSRN, and Hugging Face.
(Primary links use CERN Zenodo Concept DOIs / Çatı DOI, which automatically resolve to the latest published version while preserving exact version-pinned reproducibility.)

#	Pillar / Monograph	Core Contribution & Focus	👑 Concept DOI (Always Latest)	📌 Latest Version DOI	arXiv / SSRN / Hugging Face & Verification Status	Primary Repository
01	Paper 1: FNS	Procedural Weight Derivation & Non-Linear Decision Boundaries (v3)	10.5281/zenodo.22774934	10.5281/zenodo.22867037	SSRN: 7659198 (Distributed) • Multi-Seed Pareto & Manifold Parity	pCwOrM/fractal-neural-synthesis
02	Paper 2: OED	Orbital Error Dynamics, Ephemeral Parameter Resonance & CD4+ Gating (v1)	10.5281/zenodo.22900464	10.5281/zenodo.22900465	arXiv:2609.30115 • 🤗 HF Paper (Verified) • TR 2026/016285	pCwOrM/fractal-neural-synthesis
03	Paper 3: WERR Map	Universal Natural Language Decision Map & Real-Time Edge Triage (v2)	10.5281/zenodo.22867425	10.5281/zenodo.22939253	arXiv:2609.25498 • 🤗 HF Paper • 🤗 HF Dataset	pCwOrM/werr • pCwOrM/answerr
04	Paper 4: Werracle	Sub-Cent Intra-Block AI Reflex Oracles & Flash-Loan Circuit Breakers (v1)	10.5281/zenodo.22942598	10.5281/zenodo.22942599	arXiv:2609.30719 • SSRN: 7626279 • 🤗 HF Paper	pCwOrM/werracle • pCwOrM/werralem
05	Paper 5: Formal Lean 4	Zero-Storage Procedural Synthesis: Lean 4 Verification & 40-Core Validation	10.5281/zenodo.22774934	10.5281/zenodo.22867037	arXiv:2609.33066 • 🤗 HF Paper • Lean 4 (10 Theorems, 0 sorry)	pCwOrM/werr • pCwOrM/werracle
06	Paper 6: GAP-Lean4	Machine-Checked Computational Group Theory: Schreier-Sims & BSGS Sifting (v2)	10.5281/zenodo.23092708	10.5281/zenodo.23156544	arXiv:2609.38492 • 🤗 HF Paper • Lean 4 (35 Theorems, 0 sorry)	pCwOrM/gap-lean4-port
07	Paper 7: Black Hole v5	Unitary Black Hole Page Curve via Dynamical Horizon Orbifolds & GKP Stabilizers (v5)	10.5281/zenodo.22961999	10.5281/zenodo.23072120	S(t=1) = 0.00 · Tr(ρ²) = 1.00 · R² = 0.9822 · 573× Stress Quench • Lean 4 (10 Theorems)	pCwOrM/fractal-neural-synthesis • pCwOrM/werr
08	Paper 8: WerrSoma	Bio-Synthetic Neuromorphic Interfacing with 158K-Neuron Drosophila Connectome (v1)	10.5281/zenodo.22996625	10.5281/zenodo.22996626	arXiv:submit/8161759 • Live 3D Portal (werrsoma.answerr.me) • TR 2026/016633	Lexovian/WerrSoma • pCwOrM/WerrSoma
09	Paper 9: WerreduR	Procedural Pedagogy: Resolving the Saturn School Disequilibrium Paradox (v2)	10.5281/zenodo.22999420	10.5281/zenodo.23128224	SSRN: 7641538 (Distributed) • Lean 4 (7 Theorems, 0 sorry) • Live Portal	jesmaat/WerreduR • pCwOrM/WerreduR
10	Paper 10: Schönhage	Bipartite Carrier Compression in Schönhage Trilinear Multiplication Tensors (v2)	10.5281/zenodo.23232541	10.5281/zenodo.23295370	arXiv:2610.14263 • 🤗 HF Paper • CrocSwap #226 (50d76881)	pCwOrM/integer-mult-bounds
11	Paper 11: Güneş Dili	Deterministic Morpho-Semantic Tokenization & Algebraic Morphotactics in Lean 4 (v1)	10.5281/zenodo.23003711	10.5281/zenodo.23003712	arXiv:submit/8248674 • Lean 4 (4 Theorems, 0 sorry) • Live Studio	pCwOrM/gunes-dili
12	Paper 12: Niemeier A₈³	Machine-Certified 24D Niemeier Lattice A₈³, K3 Hodge Duality & Umbral Moonshine (v1)	10.5281/zenodo.23324066	10.5281/zenodo.23324067	Lean 4 (0 sorry, 0 axioms) • χ(X) = 24 · order(G_glue) = 27 · ℓ = 9	pCwOrM/niemeier-a8-umbral-moonshine
13	Paper 13: GDT v3.0	Machine-Certified Lean 4 Formalization of the General Divisor Theorem (v2)	10.5281/zenodo.23316068	10.5281/zenodo.23425590	Lean 4 (81 Theorems, 0 sorry) • Companion v1: 10.5281/zenodo.23426323 (v1: 23426324)	pCwOrM/general-divisor-theorem
14	Paper 14: Elliptic Sieve	Multi-Layer Algebraic Sieve & Vélu Isogeny Engine: 230 Curves on NSF ICARM (v1)	10.5281/zenodo.23436282	10.5281/zenodo.23436283	NSF ICARM Leaderboard (user/114): 30x #1 Gold Records, 80 Top-3, 100% log N	pCwOrM/werr-elliptic-sieve
15	Paper 15: Empirical OEV	Empirical OEV & Liquidation Dynamics: Chainlink SVR on Base & Ethereum L1 (v2)	10.5281/zenodo.23623588	10.5281/zenodo.23642090	🤗 HF Dataset (1.16k+ views, 126 likes) • 94.4% Aggregate / 98.3% Median SVR Recapture	pCwOrM/werracle • pCwOrM/werrealm
HW	Edge Robotics Core	Zero-Memory Embedded C/C++11 System-1 Reflex Kernel (WerrCoreEdge.h v0.5.1)	10.5281/zenodo.22867425	10.5281/zenodo.22939253	ATmega328P / ESP32 / STM32 (0 Bytes Heap, <128B Stack, TR 2026/016633)	pCwOrM/WERRobotics

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🏆 Global Records, Formal Proofs, Curated Awesome Merges & On-Chain Infrastructure

1. Certified Mathematical Records & Formal Verification (0 sorry)

• 🥇 NSF ICARM Elliptic Curve Rank Leaderboard (elliptic-rank.icarm.cloud/user/114): 230 certified high-rank elliptic curves (#4308–#4537) over $\mathbb{Q}$ with 100% complete prime conductor factorization ($\log N$) up to Rank $\ge 19$ (#4323), achieving 30 First-Place (#1 Gold) Records, 80 Podium (Top-3), and 418 Top-10 placements across 43 categories (building upon the foundational fibration families of Elkies, Dujella, Rubinstein, and Klagsbrun; supported under NSF Grant DMS-2425401).
• 🥇 Schönhage Trilinear Multiplication Tensor Bound (arXiv:2610.14263 • CrocSwap/integer-mult-bounds#226): Constructive 15-round bipartite carrier matching at dimensions $h=23, 25$ eliminating 185 carrier roles and 377,890 physical wires ($W = 177,284,805 \to 176,906,915$), certifying $\kappa = 826,215,307 / 20,000,000,000,000 \approx 4.131076535 \times 10^{-5}$ ($1.377\times$ net gain over PR #49), formally incorporated into CrocSwap/integer-mult-bounds (commit 50d76881) and verified in Lean 4 (0 sorry).
• 📐 170+ Machine-Checked Theorems in Lean 4 (0 sorry, 0 unproven axioms):
  • General Divisor Theorem (GDT v3.0): 81 theorems/lemmas (0 sorry) formalizing Dan Hawkley's residue density framework (10.5281/zenodo.23316068).
  • Computational Group Theory (gap-lean4-port): 35 theorems (0 sorry) formalizing Charles Sims' Schreier-Sims stabilizer chains (GAP-0299), backtrack ordered partitions (GAP-0332), and modular ring inverses (GAP-0331) (arXiv:2609.38492).
  • Procedural Boundary Synthesis & EVM/Cairo Fixed-Point Bounds: 40+10 theorems (WerracleProof.lean, 0 sorry) verifying ZMod 9 ($\mathbb{Z}/9\mathbb{Z}$) principal ideal closure ($\mathcal{I}_3 = {0,3,6}$), Q16.16 overflow safety, and unitary Page curve S-matrix conservation (arXiv:2609.33066).
  • 24D Niemeier Lattice $A_8^3$, WerreduR & Güneş Dili: Complete formal specifications (0 sorry) across lattice unimodularity ($\chi=24, \ell=9$), horizon complementarity ($B(t)+S(t)=1$), and deterministic morphological parsing.

2. Curated Ecosystem Recognitions & Benchmark Evaluations

• Merged Upstream Pull Requests:
  • ✅ wbierbower/awesome-physics#16: Featured in Quantum Physics for unitary Black Hole Page Curve simulation.
  • ✅ 34j/best-of-lean4#292: Indexed in the curated Best of Lean 4 theorem proving directory.
  • ✅ yh-yao/awesome-edge-ai-agents#13 & #16: Featured for both werr / answerr and WerreduR edge reflex engines.
  • ✅ MartinuzziFrancesco/awesome-scientific-machine-learning#9 & #10: Featured under Model Discovery.
  • ✅ meta-introspector/lean-workers-union#2: Registered verified member (gap-lean4-port).
  • ✅ Jev Ecosystem & Agentic Benchmarks: Listed on awesome-jev#69 / madewithjev; 100% (49/49) on WindTunnel WebMCP (3.35 ms, 0 VRAM); 100% Tool Schema Adherence (0.38 ms) on Sierra Tau-Bench; 23.74 on JevBench v1.4.1 (7.32 ms); 38.0× speedup on Jevenator 2.

3. Sovereign On-Chain Deployments (Werrium / $WERR), Multi-Chain Testnets & Grant Pipelines

• 💎 Base L2 Mainnet (Chain ID: 8453) — Live & BaseScan Verified:
  • Werrium ($WERR) Token Contract: 0xFfa2c95b68B76af00FBD84295cBA5e5268a996d2 (Fixed 1,000,000,000 supply, No-Mint, EIP-2612 Gasless Permit, burnForATP metabolic compute quota & burnForReflex on-chain decision sealing • Optimism Superchain Token List PR #1391).
  • Gnosis Safe 2/3 Multisig Treasury: 0x7CC5fd367e210F4678Bcfca054cbe0F87291E355 (949,501,000 WERR locked).
  • WerrTreasuryTimelock (Vesting & Cliff): 0x839A6d7E726D60EA2B587e793aE36342297e8C63 (40,000,000 WERR locked).
  • Aerodrome Finance DEX Liquidity Pools (Base Mainnet): vAMM-WERR/USDC & vAMM-WERR/WETH.
• 🌐 Multi-Chain Testnets & Grant Tracks:
  • Starknet Foundation Seed Grant (In Screening): Cairo 2.x / Sierra WerracleTripodZMod9 (single felt252 seed, ~1,420 Cairo steps, Ekubo v2 LVR extension).
  • Ethereum Sepolia Testnet (Chain ID: 11155111): Werrium ($WERR) at 0xe6ecc9575B4D52642c55FCf64E8E2d3baa6e6EF7 & Werracle live liquidation arbitrage telemetry.
  • NEAR Aurora Testnet (Chain ID: 1313161555) & Bare-Metal EVM Node (Chain ID: 4242): Live Werracle.sol & WerracleFeeHook.sol deployments (radar.werracle.com).
  • Active Research Proposals: Flashbots MEV Research (FRP #205 & PR #206), Morpho DAO (MIP-81 OEV Observatory), Uniswap v4 / Unichain Hook Grants.