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