Formal Verification Certificate
All theorems formally verified in Lean 4 + Mathlib with zero sorry gaps.
Homology Nucleus • Lean 4 + Mathlib • Apoth3osis Labs
The Central Question
Do computable normal forms for homology agree with abstract algebraic quotients? Computational homology uses XOR Gaussian elimination to pick canonical representatives. Abstract algebra uses quotient modules Zk/Bk. This project proves they are the same: the canonical selector is idempotent, the induced quotient equals the Mathlib Submodule quotient, and the whole construction carries a genuine Mathlib Nucleus structure.
Key Results
repr(repr(z)) = repr(z)Canonical representative is idempotent
repr x = repr y ↔ x - y ∈ BₖAgrees with Mathlib Submodule quotient
Nucleus instanceIdempotent, inflationary, order-preserving closure
F₂ RREF deterministicPivot tracking ensures unique canonical form
“Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them.”
Frank Herbert, Dune
A janitor who proves a theorem outranks a tenured professor who publishes noise.
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A valid, original framing or conjecture
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Contribution Certificates
Immutable contribution records per MENTAT-CA-001. Artifacts are content-addressed and pinned to IPFS.
MENTAT Contribution Record
IDEA
Conceptual Contribution
CONTRIBUTION LEVEL: IDEA
Ontological EngineerComputable Representatives as Algebraic Quotients
Contributor
Apoth3osis Labs
R&D Division
Core insight: the canonical representative selector for F₂ chain complexes — computed via deterministic XOR Gaussian elimination — is not merely an algorithm but induces exactly the classical quotient Zₖ/Bₖ. The selector carries a genuine Mathlib Nucleus structure, proving that computable normal forms align with abstract algebraic quotients.
MENTAT Contribution Record
THEORY
Mathematical Foundation
CONTRIBUTION LEVEL: THEORY
Ontological EngineerF₂ Linear Algebra Meets Mathlib Submodule Quotients
Contributor
Apoth3osis Labs
R&D Division
Complete framework: (1) F₂ matrices with deterministic RREF and pivot tracking, (2) canonical representative selector that is idempotent, (3) proof that the induced quotient equals Mathlib’s Submodule quotient: repr x = repr y ↔ x - y ∈ Bₖ, (4) the quotient carries a genuine Mathlib Nucleus structure — idempotent, inflationary, order-preserving.
Builds Upon
MENTAT Contribution Record
PROOF
Formally Verified
CONTRIBUTION LEVEL: PROOF
Ontological EngineerLean 4 + Mathlib Formalization
Contributor
Apoth3osis Labs
R&D Division
Machine-checked Lean 4 formalization with Mathlib dependencies. F₂ matrix arithmetic, deterministic RREF, canonical representatives, idempotence proof, quotient equivalence with Submodule quotients, and Nucleus structure construction. All proved without sorry/admit.
Builds Upon
MENTAT Contribution Record
KERNEL
Computationally Verified
CONTRIBUTION LEVEL: KERNEL
Ontological EngineerHomology Nucleus Verified Kernel
Contributor
Apoth3osis Labs
R&D Division
All theorems kernel-checked by Lean 4. Mathlib compatibility verified. Guard-no-sorry passes on all modules.
Builds Upon
MENTAT Contribution Record
BRIDGE
Cross-Level Connection
CONTRIBUTION LEVEL: BRIDGE
Ontological EngineerStandalone Repository + Interactive Proof Maps
Contributor
Apoth3osis Labs
R&D Division
Published as standalone GitHub repository with interactive 2D/3D proof maps (UMAP visualizations of the proof dependency graph), comprehensive documentation, and integration with the HeytingLean project.
Builds Upon
Governed by MENTAT-CA-001 v1.0 · March 2026