<_RESEARCH/PROJECTS

Tensor Logic + Homology

VERIFIED0 SORRY5 MENTAT CERTSLean 4

>_VERIFICATION.SEAL

Formal Verification Certificate

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

0 SORRY

Tensor Logic + Homology • Lean 4 + Mathlib • Apoth3osis Labs

❓

The Central Question

Can logic programs compute homology? This project shows that Datalog-style programs evaluated over F₂ (XOR) semantics naturally encode chain complexes. Simplicial data enters as facts, boundary maps emerge from rules, and fixpoint iteration yields cycle and boundary spaces. Gaussian elimination over GF(2) then extracts Betti numbers. The entire pipeline — from TSV input to topological invariants — is formally verified in Lean 4.

>PIPELINE

Simplicial TSV         Input: vertex/edge/triangle facts
       |
  AST Parser            Logic program construction
       |
  ┌────┼────┐
  │    │    │
Bool  Fuzzy  F₂        Three evaluation modes
  │    │    │
  └────┼────┘
       |
Chain Complex           F₂ boundary maps: ∂ₖ
       |
Gauss Elim (GF2)       RREF, pivot tracking
       |
Betti Numbers           rank(Zₖ) - rank(Bₖ)

Three Evaluation Modes

Boolean

Standard Datalog: true/false fixpoint iteration. Classical deductive database semantics.

Fuzzy

Real-valued [0,1] propagation. Soft inference for probabilistic reasoning.

F₂ (XOR)

Parity semantics encoding homological information. Chain complexes via XOR boundary maps. This mode computes Betti numbers.

Resources

GHGitHub Repository IDXAll Research Projects

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>_MENTAT.JOIN

“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.

Not as a slogan. As a structural fact of how the network operates. The only currency that matters is the quality of your contribution, measured not by committee but by mathematics.

ONTOLOGICAL ENGINEER8 designations

IDEA

A valid, original framing or conjecture

THEORY

Formal argument with paper-level rigor

APPLICATION

Connecting theory to observable outcomes

CODE

Working software the project depends on

EXPERIMENT

Reproducible research with methodology and data

PROOF

Machine-verified claim checked by a proof assistant

KERNEL

Foundational, load-bearing implementation

BRIDGE

Connecting subsystems or knowledge domains end-to-end

NOETIC ENGINEER8 designations

VISIONARY

Strategic direction & roadmaps

NARRATOR

Writing, documentation & papers

DESIGNER

Visual, UX & information design

EDUCATOR

Teaching, tutorials & workshops

CULTIVATOR

Community, outreach & events

DIPLOMAT

Partnerships, governance & policy

INTERPRETER

Translation, media & accessibility

SENTINEL

Ethics, review & quality assurance

Every accepted contribution receives a MENTAT Contribution Record — cryptographically signed, IPFS-pinned, permanently yours. No committee decides your worth. The type checker does.

APPLY TO MENTAT EXPLORE PROJECTSMESH-ENCRYPTED NETWORK FOR TRUSTED AUTONOMOUS TRANSACTIONS

>_MENTAT.CERTIFICATES

Contribution Certificates

Immutable contribution records per MENTAT-CA-001. Artifacts are content-addressed and pinned to IPFS.

MENTAT-CA-001|MCR-TLH-001

2026-01-13

MENTAT Contribution Record

IDEA

Conceptual Contribution

CONTRIBUTION LEVEL: IDEA

Ontological Engineer

Logic Programs as Homology Computations

Contributor

Apoth3osis Labs

R&D Division

Core insight: Datalog-style logic programs over F₂ (XOR) provide a natural computational model for chain complexes. Facts encode simplices, rules encode boundary maps, and fixpoint iteration computes cycle/boundary spaces — yielding Betti numbers without explicit matrix construction.

MENTAT · Mesh-Encrypted Network for Trusted Autonomous TransactionsImmutable · Content-Addressed · Tamper-Proof

MENTAT-CA-001|MCR-TLH-002

2026-01-13

MENTAT Contribution Record

THEORY

Mathematical Foundation

CONTRIBUTION LEVEL: THEORY

Ontological Engineer

Three-Mode Evaluation: Boolean, Fuzzy, and F₂

Contributor

Apoth3osis Labs

R&D Division

Complete framework with three semantics for the same logic programs: (1) Boolean — standard Datalog with true/false, (2) Fuzzy — real-valued [0,1] fixpoints for soft inference, (3) F₂ — XOR semantics where parity encodes homological information. The F₂ mode connects to chain complexes via Gaussian elimination over GF(2).

Builds Upon

MCR-TLH-001

MENTAT · Mesh-Encrypted Network for Trusted Autonomous TransactionsImmutable · Content-Addressed · Tamper-Proof

MENTAT-CA-001|MCR-TLH-003

2026-01-13

MENTAT Contribution Record

PROOF

Formally Verified

CONTRIBUTION LEVEL: PROOF

Ontological Engineer

Lean 4 Formalization — AST, Evaluation, Chain Complexes

Contributor

Apoth3osis Labs

R&D Division

Machine-checked Lean 4 formalization of the entire pipeline: Abstract Syntax Trees for logic programs, three evaluation modes (Boolean/Fuzzy/F₂), chain complex construction from simplicial data, Gaussian elimination over F₂, and Betti number computation. All proved without sorry/admit.

Builds Upon

MCR-TLH-001MCR-TLH-002

MENTAT · Mesh-Encrypted Network for Trusted Autonomous TransactionsImmutable · Content-Addressed · Tamper-Proof

MENTAT-CA-001|MCR-TLH-004

2026-01-19

MENTAT Contribution Record

KERNEL

Computationally Verified

CONTRIBUTION LEVEL: KERNEL

Ontological Engineer

Tensor Logic Verified Kernel

Contributor

Apoth3osis Labs

R&D Division

All theorems kernel-checked by Lean 4. Guard-no-sorry passes on all modules. Standard axioms only.

Builds Upon

MCR-TLH-003

MENTAT · Mesh-Encrypted Network for Trusted Autonomous TransactionsImmutable · Content-Addressed · Tamper-Proof

MENTAT-CA-001|MCR-TLH-005

2026-01-19

MENTAT Contribution Record

BRIDGE

Cross-Level Connection

CONTRIBUTION LEVEL: BRIDGE

Ontological Engineer

Standalone Repository + Curry-Howard Pipeline

Contributor

Apoth3osis Labs

R&D Division

Published as standalone GitHub repository. The Curry-Howard pipeline connects to PyTorch for tensor-based fixpoint iteration, enabling differentiable topology and physics-informed neural networks from formally verified homology.

Builds Upon

MCR-TLH-003MCR-TLH-004

MENTAT · Mesh-Encrypted Network for Trusted Autonomous TransactionsImmutable · Content-Addressed · Tamper-Proof

Governed by MENTAT-CA-001 v1.0 · March 2026

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