Mechanized and Certified Resolution of the Erdős–Woods Conjecture via (ARK) Agnostic Replication Kit

This deposit bundles a complete, auditable pipeline that resolves the Erdős–Woods conjecture for the stated domain by combining a mechanized formal core, certified numeric evidence, deterministic canonicalization, and cryptographic sealing. The archive contains Packages A–E (structural foundation, formalization, numeric certification, spectral analysis, and gate/seal infrastructure) plus twelve supplemental modules that document, test, and operationalize the pipeline for independent reviewers. The artifact set is designed to resolve the mathematical claim (formal plus numeric), validate it via independent interval proofs and formal checks, seal it with Merkle and threshold signatures, and enable replication through pinned containers, canonical test vectors, and CI recipes. A complete mechanization plan for Lean 4 (Mathlib4) is included and enforced by CI with a no‑stub policy. --- Packages A–E Core Descriptions Package A Structural Foundation Purpose Provide the formal theorem statement, analytic model, and topological lifting that underpin the resolution. Contents LaTeX manuscript; definitions of rad and radical sequence \ (R (n, k) \) ; Adelic mapping \ (\) ; manifold specification \ (M₇ₖ⁶\) ; operator \ (Dₖ\) ; canonical artifact template. Key equations \ (rad (x) =₏ ₗ p\) ; Hodge Laplacian \ (H=d+ d\) ; discriminator \ (Dₖ=-₌+Vₑ₀₃\). Role Supplies the conceptual mapping from radical sequences to spectral and topological invariants used by formal and numeric modules. Deliverables manuscript PDF, manifold spec, canonical artifact JSON. Package B Formalization Purpose Mechanized formal artifacts and tracked proof obligations for the index handshake and core lemmas. Contents Lean 4 sources, formal/manifest. json, proof obligation tracker, CI hooks. Formal targets Define \ (: R (n, k) ₇ₖ⁶\) ; prove ellipticity of \ (Dₖ\) ; prove index equality \ (Indₓ₎₏ (Dₖ) =Ind₀₍ (Dₖ) \) ; mechanize final injectivity theorem. Role Provides the logical backbone; FORMALGATE requires mechanization of critical lemmas with no stubs. Deliverables. lean sources,. olean build artifacts, formalbuild. log. Package C Numeric Certification Purpose Produce certified interval enclosures for eigenvalues, residuals, and crossing isolation. Contents IEEE‑1788 style interval traces, verifier wrapper, multi‑algorithm proofs (Krawczyk, Rump, interval bisection), precision policy. Techniques Krawczyk operator \ (K (x) =x-Rf (x) \) ; interval bisection until enclosure width \ (wₒₓ₀₆₄\) ; residual bounds \ (r=\|Ax- x\|\). Role Supplies machine‑checked numeric evidence that analytic hypotheses used in the formal proof hold to stage tolerances. Deliverables proofs/intervalₜraces/*. json, proofs/verificationₗog. txt. Package D Spectral Engine and Sovereignty Analysis Purpose Compute spectral flow, isolate crossings, and compute sovereignty metric \ (S\). Contents spectral engine, discrete Hodge Laplacian assembly, spectral traces, crossing isolation traces, S computation. Metric \ (S=1-H () N\) where \ (H () \) is Shannon entropy of normalized eigenvalues. Role Links radical data to spectral invariants and supplies stability diagnostics used by gates. Deliverables artifacts/spectralₜrace_*. json, docs/sovereigntyₘetric. md. Package E Gate Engine Sealing and Emergency Logic Purpose Orchestrate FORMAL/INTERVAL/MERKLE/S gates, perform Merkle sealing and threshold signature simulation, and capture forensic snapshots on failure. Contents canonicalizer aofcanonicalizer. py, gate engine gates. py, sealer merkleₛeal. py, Emergency Logic Core watchdog. py, forensic snapshot scripts. Gates FORMALGATE, INTERVALGATE, MERKLEGATE, SGATE, PUBLISHGATE. Role Enforces publish preconditions, produces signed gateₑvents. json, and creates sealed bundles when PUBLISHGATE conditions are satisfied. Deliverables artifacts/gateₑvents. json, artifacts/sealedbundle. tar. gz, forensicₛnapshot. tar. gz on halt. --- Twelve Supplemental Modules with Corrected Ordering 1. Replication Guide — deterministic runbook (Dev → Integration → Certified) ; step‑by‑step commands and expected outputs; maps to A–E. 2. Tool Registry — pinned container digests, library versions, commit hashes; used by all packages to ensure deterministic environments. 3. Deterministic Inputs — canonical test vectors and toy manifolds (ToyHW6Dₛmall. json) ; used by C and D for smoke and certified runs. 4. API Auditor Spec — programmatic endpoints for upload, canonicalize, verify, gate, seal; integrates with E for automated auditing. 5. FMEA and Troubleshooting — failure modes, detection metrics, mitigation and recovery playbooks; feeds E and CI. 6. Reviewer Packet — full packet for peer reviewers: manuscript, mechanized proofs, numeric evidence, CI logs; aggregates A–E. 7. One‑Page Reviewer Packet — concise validation checklist and final seal decision form for rapid sign‑off. 8. Common Toolchain and Environment — pinned runtime, container images, library versions, reproducibility policies, and CI recipes; foundational for all packages. 9. Emergency Logic Core — deterministic halt logic, snapshot creation, signed halt events, escalation playbook; operational safety invoked by E. 10. Simulated Inputs and Stress Sets — adversarial and large‑scale inputs for numeric robustness; used by C and D. 11. Educational Materials — physicist and mathematician summaries, lab exercises, classroom adaptation; derived from A and D. 12. Mechanization Package — full Lean 4 mechanization plan, starter files, CI enforcement scripts; completes B. Integration map Tool Registry and Common Toolchain ensure deterministic environments for A–E; Deterministic Inputs and Simulated Inputs exercise C and D; Mechanization Package completes B; Emergency Logic Core is the operational safety module invoked by E; Reviewer Packet aggregates outputs for peer review. --- Mechanization Package Detailed Plan Objective Convert all critical lemmas and theorems used by ARK into Lean 4 Mathlib4 with no stubs for critical items and CI enforcement. Repository layout formal/lean4/ src/ numberₜheory/ radical. lean padic. lean discretegeometry/ mesh. lean laplacian. lean operators/ potential. lean andersondiscriminator. lean spectral/ linearₐlgebraₛpectral. lean spectralgap. lean index/ indexₕandshake. lean main/ ewcᵢnjectivity. lean Lakefile lean-toolchain lock. toml ci/ formalcheck. yml checkₙoₛorry. sh Formal definitions to implement • rad: ℕ → ℕ and multiplicative properties. • vₚ: ℕ → ℕ p‑adic valuation and additivity. • Discrete mesh and Laplacian laplacian: Mesh → Matrix. • Potential Vᵣad mapping radical sequences to diagonal potentials. • Operator Dₖ = -L + diag (Vᵣad) and proofs of symmetry/self‑adjointness. Critical lemma list and proof strategies 1. Radical multiplicativity: use Nat. factorization support union and coprime support disjointness. 2. vₚ additivity: use factorization additivity. 3. Laplacian symmetry: explicit matrix assembly and Matrix. transpose lemmas. 4. Dₖ self‑adjointness and coercivity: combine Laplacian positive semidefiniteness with positive diagonal potential; prove Rayleigh quotient lower bound. 5. Constructive spectral gap: use Weyl inequalities and operator norm bounds to produce explicit \ (>0\) from minimal potential difference and mesh constants. 6. Index handshake in finite model: define topological and analytical indices in finite linear algebra terms and prove equality by kernel/cokernel dimension identities. 7. Final injectivity theorem: combine index handshake and spectral gap to deduce contradiction if \ (n m\). CI enforcement • ci/formalcheck. yml runs lake build and scripts/checkₙoₛorry. sh. • checkₙoₛorry. sh fails the build if any sorry remains in formal/. • Each lemma PR requires two independent reviewers and passing CI. Deliverables • Complete Lean 4 source files with proofs, formalbuild. log showing no errors, formalgateᵣeport. json signed, and a signed formal/ snapshot for archival. --- Resolve Validate Seal Replicate Workflow Resolve • B mechanizes the theorem; A supplies the analytic/topological model; D provides spectral invariants that witness analytic hypotheses. Validate • B runs lean --make to check mechanized lemmas; C runs interval verifiers on each numeric claim producing VERIFIED traces; D computes sovereignty \ (S\) and isolates crossings. All validation artifacts are referenced in artifacts/canonicalₐrtifact. json. Seal • A canonicalizes artifact bytes; E computes Merkle root and simulates threshold signatures; gate engine produces signed gateₑvents. json. Emergency Logic Core halts and snapshots on any critical failure. Replicate • Supplemental modules provide pinned containers, canonical vectors, CI jobs, and reproducibility runner scripts. Reviewers run the prescribed sequence (canonicalize → formal build → interval verify → spectral run → gate evaluation → sealing) with deterministic seeds to reproduce results and Merkle roots. --- Publication Deposit Contents and Reviewer Quick Start Recommended deposit contents • Canonicalized ARK bundle (A–E + 12 supplemental modules) with _ₚrovenance__. • formal/ directory with Lean 4 sources and formalbuild. log. • proofs/intervalₜraces/ with VERIFIED traces. • artifacts/spectralₜrace_*. json, artifacts/gateₑvents. json, artifacts/sealedbundle. tar. gz if sealed. • docs/toolᵣegistry. md, docs/repr