Apoth3osis
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Homology Nucleus

VERIFIED0 SORRY5 MENTAT CERTSLean 4 + Mathlib
>_VERIFICATION.SEAL
FORMALLY VERIFIED • LEAN 4 • MACHINE-CHECKED • APOTH3OSIS∀∃⊢→∧∨¬≡QED0 SORRY

Formal Verification Certificate

All theorems formally verified in Lean 4 + Mathlib with zero sorry gaps.

0 SORRY

Homology Nucleus • Lean 4 + Mathlib • Apoth3osis Labs

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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 ∈ Bₖ

Agrees with Mathlib Submodule quotient

Nucleus instance

Idempotent, inflationary, order-preserving closure

F₂ RREF deterministic

Pivot tracking ensures unique canonical form

Resources

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