Formal Verification Certificate
All theorems formally verified in Lean 4 with zero sorry gaps.
Futamura-Adelic • Lean 4 + Mathlib • Apoth3osis Labs
The Central Question
Can AI alignment be made mathematically rigorous? This project grounds alignment in two structures: Futamura projections — partial evaluation of interpreters yields compilers, which yields compiler generators, giving a verified chain from specification to execution — and adelic coherence from algebraic number theory, where the restricted product condition ensures consistency across all local perspectives (archimedean and p-adic). The composition of these two structures yields organic alignment: a system is aligned if its behavior is Futamura-verified and coherent across all scales of abstraction.
The Three Futamura Projections
| Projection | Input | Output |
|---|---|---|
| 1st | mix(interpreter, program) | compiled program |
| 2nd | mix(mix, interpreter) | compiler |
| 3rd | mix(mix, mix) | compiler generator |
Key Verified Results
futamura_firstmix(interpreter, program) = compiled_program
Futamura.lean
futamura_secondmix(mix, interpreter) = compiler
Futamura.lean
futamura_thirdmix(mix, mix) = compiler_generator
Futamura.lean
restricted_product_well_formedAdelic restricted product satisfies coherence
Adelic.lean
organic_alignmentFutamura-verified + adelically coherent ⇒ aligned
Alignment.lean
“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.
A valid, original framing or conjecture
Formal argument with paper-level rigor
Connecting theory to observable outcomes
Working software the project depends on
Reproducible research with methodology and data
Machine-verified claim checked by a proof assistant
Foundational, load-bearing implementation
Connecting subsystems or knowledge domains end-to-end
Strategic direction & roadmaps
Writing, documentation & papers
Visual, UX & information design
Teaching, tutorials & workshops
Community, outreach & events
Partnerships, governance & policy
Translation, media & accessibility
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.
Contribution Certificates
Immutable contribution records per MENTAT-CA-001. Each certificate is cryptographically anchored with IPFS CIDs.
MENTAT Contribution Record
IDEA
Conceptual Contribution
CONTRIBUTION LEVEL: IDEA
Ontological EngineerOrganic Alignment via Partial Evaluation and Adelic Coherence
Contributor
Apoth3osis Labs
R&D Division
Core insight: AI alignment can be grounded in two mathematical structures working together. (1) Futamura projections — partial evaluation of interpreters yields compilers, then compiler generators. This gives a verified chain from specification to execution. (2) Adelic coherence — the restricted product condition from algebraic number theory ensures consistency across all local perspectives (archimedean + p-adic). Together: alignment is coherence across scales of abstraction.
MENTAT Contribution Record
THEORY
Mathematical Foundation
CONTRIBUTION LEVEL: THEORY
Ontological EngineerThree Projections and the Restricted Product
Contributor
Apoth3osis Labs
R&D Division
Complete framework: (1) First Futamura projection: mix(interpreter, program) = compiled_program, (2) Second projection: mix(mix, interpreter) = compiler, (3) Third projection: mix(mix, mix) = compiler_generator. (4) Adelic restricted product: a value is coherent iff its local components agree at all but finitely many primes, and the remaining components lie in the ring of integers. Organic alignment = the composition of Futamura verification with adelic coherence.
Builds Upon
MENTAT Contribution Record
PROOF
Formally Verified
CONTRIBUTION LEVEL: PROOF
Ontological EngineerLean 4 Formalization — Futamura-Adelic Organic Alignment
Contributor
Apoth3osis Labs
R&D Division
Machine-checked Lean 4 formalization of Futamura projections and adelic coherence. Proves all three projections correct, partial evaluation equivalence with compilation, restricted product well-formedness, and the organic alignment theorem: if a system is Futamura-verified and adelically coherent, it is aligned across all abstraction scales. All proved without sorry/admit.
Builds Upon
MENTAT Contribution Record
KERNEL
Computationally Verified
CONTRIBUTION LEVEL: KERNEL
Ontological EngineerFutamura-Adelic Verified Kernel
Contributor
Apoth3osis Labs
R&D Division
All theorems kernel-checked by Lean 4. Guard-no-sorry passes. Standard axioms only.
Builds Upon
MENTAT Contribution Record
BRIDGE
Cross-Level Connection
CONTRIBUTION LEVEL: BRIDGE
Ontological EngineerStandalone Repository + Alignment Case Studies
Contributor
Apoth3osis Labs
R&D Division
Published as standalone GitHub repository with case studies applying the Futamura-adelic framework to concrete alignment scenarios.
Builds Upon
Governed by MENTAT-CA-001 v1.0 · March 2026