Project Aleph-Omega is an experimental formal-methods research project for generating finite semantic path-equivalence theorem artifacts in Lean/Mathlib from Python witness records.
The project represents formal systems as semantic objects, satisfaction-preserving translations as morphisms, finite morphism chains as paths, and pointwise agreement of composite translations/model maps as sufficient evidence for quotient-level path equality.**
Project Aleph-Omega studies one focused mathematical question:
When a statement is translated from one formal system into another, what semantic content is preserved?
The repository combines a Python finite-computation layer, a Lean theorem-proving layer, a standalone Lake project scaffold, concrete finite Lean examples, quotient-category-style structure, manuscript documentation, and continuous integration for the formal stack.
Project Aleph-Omega builds a finite semantic framework for studying satisfaction preservation.
It defines:
- formal systems with models, sentences, and satisfaction relations,
- translations between systems,
- morphisms that preserve satisfaction,
- failure cases where preservation breaks,
- equivalence classes of morphisms,
- quotient morphisms,
- quotient-style composition,
- a standalone Lean quotient-category-like structure,
- concrete finite Lean examples,
- a Python computational analogue for finite experiments.
A preservation morphism has two parts:
sentence translation: φ ↦ F(φ)
model map: m ↦ F(m)
The key preservation condition is:
if source model m satisfies source sentence φ,
then target model F(m) satisfies translated sentence F(φ).
Symbolically:
A ⊨ φ implies F(A) ⊨ F(φ)
The Lean Formalization is located at:
formal/lean/AlephOmegaCore.lean
The Lean core includes:
FormalSystemPreservationMorphismidentity_preserves_satisfactioncomposition_preserves_satisfactionMorphismEquivalentQuotientMorphism- quotient composition
- quotient identity laws
- quotient associativity
AlephOmegaQuotientCategory- concrete finite systems:
TwoSystem,RenamedTwoSystem, andThirdTwoSystem - concrete nontrivial preservation morphisms
- a concrete preservation chain
- quotient integration for the concrete chain
The strongest careful Lean claim is:
Project אω contains a Lean-checked concrete finite preservation pipeline across three explicit formal systems, including nontrivial preservation morphisms, a nontrivial composition chain, and integration of that chain into a standalone quotient-category-like structure.
The Python Layer is located in:
src/rigor/
It implements finite computational analogues of the formal ideas, including:
- finite logical universes,
- finite institution-like systems,
- bridge translations,
- satisfaction preservation checkers,
- failure taxonomies,
- theorem inventories,
- quotient morphism analogues,
- quotient category checks,
- Lean/Python correspondence reports,
- manuscript generation utilities.
The Python layer should be read as a finite computational laboratory, not as a full proof assistant.
The Lean formalization is also packaged as a standalone Lake project:
formal/aleph_omega_lake/
Build it with:
./scripts/check_lake.sh
To verify the full formal stack locally, run:
./scripts/check_formal_stack.sh
This checks:
- the primary Lean formalization,
- Lake synchronization,
- the Lake project build,
- the Python test suite.
The repository includes GitHub Actions CI:
.github/workflows/formal-stack.yml
The workflow verifies the Lean, Lake, synchronization, and Python formal stack on pushes and pull requests to main.
The project includes a research manuscript layer:
docs/project_aleph_omega_manuscript.md
docs/manuscript_theorem_inventory.md
docs/manuscript_figures.md
docs/manuscript_front_matter.md
docs/manuscript_completion_report.md
These documents explain the project’s definitions, theorem flow, architecture, contribution boundaries, and limitations.
Important reviewer-facing documents:
docs/formal_claim_upgrade.md
docs/lean_formalization_index.md
docs/concrete_lean_completion_report.md
docs/correspondence_completion_report.md
docs/lean_packaging_completion_report.md
docs/manuscript_completion_report.md
Project אω is:
- a finite formal-methods research artifact,
- a semantic preservation laboratory,
- a Lean-supported quotient-category prototype,
- a bridge between finite computation and proof-assistant formalization,
- a serious mathematical-computation project.
Project אω is not yet:
- a universal theory of institutions,
- a proof about all logics,
- a full Mathlib
Categoryinstance, - a complete Lean verification of every Python function,
- a solved open problem,
- a field-changing theorem.
Those are future directions, not current claims.
Run the Python test suite:
python3 -m pytest
Check the Lean core:
./scripts/check_lean.sh
Build the Lake project:
./scripts/check_lake.sh
Run the full formal stack:
./scripts/check_formal_stack.sh
formal/
lean/
AlephOmegaCore.lean
aleph_omega_lake/
lakefile.lean
AlephOmega.lean
AlephOmega/
AlephOmegaCore.lean
src/
rigor/
finite semantic and formal-reporting modules
tests/
Python test suite
docs/
manuscript, theorem inventory, reports, and reviewer documentation
scripts/
Lean, Lake, sync, and formal-stack verification scripts
The most accurate public description is:
Project אω is a finite institution-inspired, Lean-supported research framework for studying satisfaction preservation under semantic translation. It includes a Python computational layer, a Lean-checked quotient-category-style formal core, concrete finite Lean examples, and a manuscript package that separates formal claims from computational analogues and explicit non-claims.
Add a license before public release if one is not already present.
A reviewer-facing map of the documentation package is available at:
docs/public_release_index.md
This index tells readers where to start, which documents contain the exact claims, and how to verify the formal stack.
A reviewer-facing setup and verification guide is available at:
docs/quickstart.md
The fastest complete verification command is:
./scripts/check_formal_stack.sh
A reviewer-facing verification status page is available at:
docs/verification_status.md
It separates Lean-checked results, Python-tested computational results, CI-checked infrastructure, and explicit non-claims.
The public release package is summarized at:
docs/public_release_completion_report.md
The next research-strengthening track begins with Mathlib integration feasibility:
docs/mathlib_integration_feasibility.md
This document analyzes what would be required to upgrade the standalone quotient-category-like Lean structure into a real Mathlib-compatible category instance.
Project אω now includes a separate experimental Mathlib Lake project for future category-theory integration.
New location:
formal/aleph_omega_mathlib/
Checker:
./scripts/check_mathlib_scaffold.sh
Careful claim:
Project אω now has a separate experimental Mathlib scaffold. This is not yet a Mathlib Category instance for the main Aleph-Omega formalization.
Project אω now includes an experimental Mathlib project with a real smoke-test Category instance.
Artifact:
formal/aleph_omega_mathlib/AlephOmegaMathlib/CategorySmokeTest.lean
Careful claim:
Project אω can now define and build a real Mathlib
Categoryinstance in its experimental Mathlib scaffold. This is not yet the Aleph-Omega quotient category instance.
Project אω now includes an experimental Mathlib Category instance for formal systems and satisfaction-preserving morphisms.
Artifact:
formal/aleph_omega_mathlib/AlephOmegaMathlib/FormalSystemCategory.lean
Careful claim:
Project אω now has a real Mathlib category instance for the direct preservation-morphism structure. This is not yet the quotient category instance.
Project אω now includes a completion report for its first serious Mathlib category-theory milestone.
Artifact:
docs/mathlib_direct_category_completion_report.md
Careful claim:
Project אω now contains an experimental Mathlib project with a real Category instance whose objects are formal systems and whose morphisms are satisfaction-preserving morphisms. This is still not yet the quotient category instance.
Project אω now includes a technical blueprint for upgrading the direct Mathlib preservation-morphism category into a quotient category.
Artifact:
docs/mathlib_quotient_category_blueprint.md
Careful claim:
Project אω now has a detailed technical plan for a real Mathlib quotient category whose morphisms are equivalence classes of satisfaction-preserving morphisms. The quotient category instance itself is the next target.
Project אω now includes an experimental Mathlib quotient category prototype.
Artifact:
formal/aleph_omega_mathlib/AlephOmegaMathlib/QuotientFormalSystemCategory.lean
Careful claim:
Project אω now contains an experimental Mathlib quotient category prototype whose morphisms are quotient classes of satisfaction-preserving morphisms. This should still be reviewed carefully before being treated as final.
Project אω now includes a completion report for the experimental Mathlib quotient category prototype.
Artifact:
docs/mathlib_quotient_category_completion_report.md
Careful claim:
Project אω now contains an experimental Mathlib quotient category prototype whose morphisms are quotient classes of satisfaction-preserving morphisms, with representative-independent composition and a real Mathlib
Categoryinstance. It remains an experimental prototype pending cleanup and expert review.
Project אω now includes a correspondence report connecting the original standalone Lean core to the experimental Mathlib quotient category prototype.
Artifact:
docs/mathlib_correspondence_report.md
Careful claim:
Project אω now has a documented correspondence between its standalone Lean quotient-category core and its experimental Mathlib quotient category prototype. The two tracks are not yet definitionally unified.
Project אω now ports the concrete three-system preservation chain into the experimental Mathlib quotient-category track.
Artifact:
formal/aleph_omega_mathlib/AlephOmegaMathlib/ConcreteChain.lean
Careful claim:
Project אω now has a concrete three-system preservation chain inside the experimental Mathlib quotient-category prototype.
Project אω now includes a completion report for the PhD-level Mathlib strengthening track.
Artifact:
docs/mathlib_strengthening_completion_report.md
Careful claim:
Project אω now has an experimental Mathlib category-theory track with a direct category, a quotient category prototype, representative-independent quotient composition, and a concrete three-system preservation chain.
Project אω now begins the Python-to-Lean finite model export track.
Artifact:
docs/lean_export_blueprint.md
Careful claim:
Project אω now has a technical blueprint for exporting finite Python semantic systems into Lean finite formal systems.
Project אω now includes its first Python-to-Lean exporter for finite formal systems.
Artifacts:
src/rigor/lean_finite_system_exporter.py
formal/generated/ExportedTinySystem.lean
docs/lean_finite_system_exporter.md
Careful claim:
Project אω now has a Python-to-Lean exporter that generates a finite formal system with Lean-checkable satisfaction facts.
Project אω now includes a Python-to-Lean exporter for finite satisfaction-preserving morphisms.
Artifacts:
src/rigor/lean_morphism_exporter.py
formal/generated/ExportedTinyMorphism.lean
docs/lean_morphism_exporter.md
Careful claim:
Project אω now has a Python-to-Lean exporter that generates Lean-checkable finite satisfaction-preserving morphisms.
Project אω now includes a script that regenerates Python-to-Lean exports and checks them with Lean.
Artifact:
scripts/check_generated_lean_exports.sh
docs/generated_lean_export_verification.md
Command:
./scripts/check_generated_lean_exports.sh
Careful claim:
Project אω now has reproducible Python-generated Lean finite-system and preservation-morphism exports verified by Lean.
Project אω now verifies Python-generated Lean exports inside the main formal-stack gate.
Artifacts:
scripts/check_formal_stack.sh
scripts/check_generated_lean_exports.sh
tests/test_rigor_formal_stack_generated_exports.py
Careful claim:
Project אω now includes reproducible Python-generated Lean exports as part of its official formal verification pipeline.
Project אω now includes a completion report for the Python-to-Lean finite export pipeline.
Artifact:
docs/lean_export_completion_report.md
Careful claim:
Project אω now has a Python-to-Lean export pipeline that generates finite Lean formal systems and finite satisfaction-preserving morphisms from Python data, then verifies the generated Lean artifacts inside the formal-stack gate.
Project אω now begins the track for moving Python-generated Lean artifacts into the experimental Mathlib category-theory project.
Artifact:
docs/mathlib_export_integration_blueprint.md
Careful claim:
Project אω now has a precise plan for moving Python-generated Lean artifacts into its experimental Mathlib category-theory track.
Project אω now has a Python exporter that generates finite FormalSystem artifacts directly inside the experimental Mathlib category-theory track.
Artifacts:
src/rigor/mathlib_finite_system_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/ExportedTinyMathlibSystem.lean
docs/mathlib_finite_system_exporter.md
Careful claim:
Project אω now has a Python exporter that generates finite FormalSystem artifacts directly inside the experimental Mathlib category-theory track.
Project אω now has a Python exporter that generates finite PreservationMorphism artifacts directly inside the experimental Mathlib category-theory track.
Artifacts:
src/rigor/mathlib_morphism_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/ExportedTinyMathlibMorphism.lean
docs/mathlib_morphism_exporter.md
Careful claim:
Project אω now has a Python exporter that generates finite PreservationMorphism artifacts directly inside the experimental Mathlib category-theory track.
Project אω now includes a script that regenerates Python-produced Mathlib-track exports and verifies them through the experimental Mathlib Lake build.
Artifacts:
scripts/check_generated_mathlib_exports.sh
docs/generated_mathlib_export_verification.md
Command:
./scripts/check_generated_mathlib_exports.sh
Careful claim:
Project אω now has a reproducible script that regenerates Python-produced Mathlib finite-system and preservation-morphism exports and verifies them through the experimental Mathlib Lake build.
Project אω now verifies Python-generated Mathlib exports inside the main formal-stack gate.
Artifacts:
scripts/check_formal_stack.sh
scripts/check_generated_mathlib_exports.sh
tests/test_rigor_formal_stack_generated_mathlib_exports.py
Careful claim:
Project אω now includes reproducible Python-generated Mathlib finite-system and preservation-morphism exports as part of its official formal verification pipeline.
Project אω now includes a completion report for the Python-to-Mathlib export pipeline.
Artifact:
docs/mathlib_export_completion_report.md
Careful claim:
Project אω now has a Python-to-Mathlib export pipeline that generates finite FormalSystem and PreservationMorphism artifacts directly inside the experimental Mathlib category-theory track and verifies them through the formal-stack gate.
Project אω now begins the generated quotient-category export track.
Artifact:
docs/generated_quotient_export_blueprint.md
Careful claim:
Project אω now has a precise plan for generating quotient-category artifacts from Python-produced Mathlib preservation morphisms.
Project אω now has a Python exporter that generates quotient-category wrapper artifacts for Python-produced Mathlib preservation morphisms.
Artifacts:
src/rigor/mathlib_quotient_wrapper_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/ExportedTinyMathlibQuotient.lean
docs/mathlib_quotient_wrapper_exporter.md
Verification update:
scripts/check_generated_mathlib_exports.sh now regenerates and verifies the generated quotient wrapper.
Careful claim:
Project אω now has a Python exporter that generates quotient-category wrapper artifacts for Python-produced Mathlib preservation morphisms and verifies them through the generated Mathlib export checker.
Project אω now generates a quotient-category composition theorem inside the experimental Mathlib track.
Artifacts:
src/rigor/mathlib_quotient_composition_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/ExportedTinyMathlibQuotientComposition.lean
docs/mathlib_quotient_composition_exporter.md
Careful claim:
Project אω now generates a finite quotient-category composition theorem inside the experimental Mathlib category-theory track.
Project אω now includes a completion report for the generated quotient-category export path.
Artifact:
docs/generated_quotient_export_completion_report.md
Careful claim:
Project אω now has a Python-generated Mathlib quotient export path that produces quotient-category wrapper artifacts and a finite quotient-category composition theorem, then verifies them through the generated Mathlib checker and formal-stack gate.
Project אω now includes a reviewer-facing index of generated Lean and generated Mathlib artifacts.
Artifact:
docs/generated_lean_artifact_index.md
Careful claim:
Project אω now has a documented index of its generated Lean and generated Mathlib artifacts, including standalone finite systems, Mathlib preservation morphisms, quotient wrappers, quotient composition theorems, and verification scripts.
Project אω now includes a completion report for the generated quotient-category pipeline.
Artifact:
docs/generated_quotient_category_completion_report.md
Careful claim:
Project אω now has a generated quotient-category pipeline: Python-generated Mathlib preservation morphisms are wrapped into quotient morphism classes, composed in a generated quotient-category example, and verified through the Mathlib Lake build and formal-stack gate.
Project אω now begins the generated finite semantic lab track.
Artifact:
docs/generated_semantic_lab_blueprint.md
Careful claim:
Project אω now has a precise plan for scaling generated Mathlib quotient-category artifacts into a small finite semantic laboratory.
Project אω now includes a Python data model for a generated finite semantic lab.
Artifact:
src/rigor/generated_semantic_lab_model.py
docs/generated_semantic_lab_model.md
Careful claim:
Project אω now has a Python data model for a generated finite semantic lab containing multiple finite systems, preservation morphisms, and composable chains.
Project אω now includes a completion report for the generated finite semantic lab.
Artifact:
docs/generated_semantic_lab_completion_report.md
Careful claim:
Project אω now has a generated finite semantic lab: Python defines multiple finite systems, preservation morphisms, and composable chains, then exports them into a generated Mathlib artifact with quotient wrappers and quotient-category composition theorems.
Project אω now includes a reviewer-facing artifact index for the generated finite semantic lab.
Artifact:
docs/generated_semantic_lab_artifact_index.md
Careful claim:
Project אω now has a reviewer-facing index of the generated finite semantic lab, including four generated systems, three generated preservation morphisms, quotient morphism classes, and two generated quotient-category composition theorems.
Project אω now includes a final report closing the generated finite semantic lab track.
Artifact:
docs/generated_semantic_lab_final_report.md
Careful claim:
Project אω now contains a generated finite semantic lab: Python defines multiple finite systems, preservation morphisms, and composable chains, then exports them into the experimental Mathlib quotient-category track with quotient wrappers and quotient-category composition theorems.
Project אω now begins the semantic lab expansion track.
Artifact:
docs/semantic_lab_expansion_blueprint.md
Careful claim:
Project אω now has a precise plan for expanding its generated finite semantic lab beyond chains into multiple named finite semantic diagram families.
Project אω now begins the theorem-strengthening track for nontrivial quotient path equivalence.
Artifact:
docs/nontrivial_quotient_path_equivalence_blueprint.md
Careful claim:
Project אω now has a precise theorem plan for proving nontrivial quotient path equivalence by pointwise translation and model-map equality, rather than by bare definitional equality.
Project אω now includes a Python data model for a generated finite diamond diagram.
Artifact:
src/rigor/generated_diamond_diagram_model.py
docs/generated_diamond_diagram_model.md
Careful claim:
Project אω now has a Python data model for a generated finite diamond diagram whose two distinct paths agree pointwise on models and sentence translations.
Project אω now exports a generated finite diamond diagram into Mathlib with a theorem-backed quotient path equivalence proof.
Artifacts:
src/rigor/diamond_diagram_mathlib_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/DiamondDiagram.lean
docs/diamond_diagram_mathlib_exporter.md
Careful claim:
Project אω now exports a generated finite diamond diagram into Mathlib and proves that its two source-to-target paths are equal as quotient morphisms by pointwise translation and model-map equivalence, using PreservationEquivalent and Quotient.sound rather than bare rfl.
Project אω now includes a completion report for the generated theorem-backed diamond diagram.
Artifact:
docs/nontrivial_diamond_theorem_completion_report.md
Careful claim:
Project אω now contains a generated theorem-backed diamond diagram: two generated source-to-target paths are proved equal as quotient morphisms by pointwise translation equality, pointwise model-map equality, PreservationEquivalent, and Quotient.sound, rather than by bare definitional equality.
Project אω now includes a report summarizing the theorem-backed semantic lab milestone.
Artifact:
docs/theorem_backed_semantic_lab_report.md
Careful claim:
Project אω now contains a theorem-backed generated semantic lab component: a generated finite diamond diagram whose two source-to-target paths are proved equal as quotient morphisms through pointwise translation equality, pointwise model-map equality, PreservationEquivalent, and Quotient.sound.
Project אω now begins the reusable theorem-schema track for quotient path equality.
Artifact:
docs/reusable_diamond_proof_schema_blueprint.md
Careful claim:
Project אω now has a precise plan for abstracting its generated diamond path-equality proof into a reusable quotient morphism equality theorem schema.
Project אω now includes a reusable Mathlib theorem schema for quotient morphism equality from pointwise equality.
Artifacts:
formal/aleph_omega_mathlib/AlephOmegaMathlib/ReusableQuotientPathEquality.lean
docs/reusable_quotient_path_equality.md
Careful claim:
Project אω now has a reusable Mathlib theorem schema proving quotient morphism equality from pointwise translation and model-map equality.
Project אω now refactors the generated diamond path-equality proof to use the reusable quotient morphism equality theorem schema.
Artifacts:
src/rigor/diamond_diagram_mathlib_exporter.py
docs/diamond_refactor_to_reusable_theorem.md
Careful claim:
Project אω now generates diamond path-equality proofs through a reusable quotient morphism equality theorem schema rather than rebuilding the quotient proof directly inside each generated file.
Project אω now includes a completion report for the reusable quotient path equality theorem schema.
Artifact:
docs/reusable_theorem_schema_completion_report.md
Careful claim:
Project אω now contains a reusable Mathlib theorem schema for quotient path equality: if two preservation morphisms agree pointwise on sentence translations and model maps, then their quotient morphism classes are equal; the generated diamond diagram now uses this reusable theorem.
Project אω now demonstrates reuse of its quotient path equality theorem schema across multiple generated diamond diagrams.
Artifacts:
src/rigor/second_diamond_mathlib_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/SecondDiamondDiagram.lean
docs/second_diamond_reusable_theorem.md
Careful claim:
Project אω now demonstrates reuse of its quotient path equality theorem schema across multiple generated diamond diagrams.
Project אω now includes a final report for the reusable quotient path equality theorem-schema track.
Artifact:
docs/reusable_theorem_schema_final_report.md
Careful claim:
Project אω now contains a reusable Mathlib theorem schema for quotient morphism equality from pointwise translation and model-map equality, and demonstrates that schema across multiple generated finite diamond diagrams.
Project אω now begins the general path-chain equality track, extending beyond diamond diagrams toward longer generated path equivalence proofs.
Artifact:
docs/general_path_chain_equality_blueprint.md
Careful claim:
Project אω now has a precise plan for moving from reusable diamond path equality to reusable longer path-chain equality.
Project אω now includes a generated theorem-backed equality proof for two longer composed path chains.
Artifacts:
src/rigor/long_path_chain_mathlib_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/LongPathChain.lean
docs/long_path_chain_equality_example.md
Careful claim:
Project אω now generates a theorem-backed equality proof for two longer composed path chains using the reusable quotient path equality theorem schema.
Project אω now includes a completion report for the long path-chain equality track.
Artifact:
docs/path_chain_equality_completion_report.md
Careful claim:
Project אω now generates theorem-backed equality proofs for longer composed path chains, using the reusable quotient path equality theorem schema.
Project אω now includes a final report for the longer generated path-chain equality track.
Artifact:
docs/path_chain_equality_final_report.md
Careful claim:
Project אω now generates theorem-backed equality proofs for longer composed path chains, using the reusable quotient path equality theorem schema.
Project אω now begins the multi-diagram theorem reuse track, organizing generated diagrams that invoke the same reusable quotient equality theorem schema.
Artifact:
docs/multi_diagram_theorem_reuse_blueprint.md
Careful claim:
Project אω now has a precise plan for demonstrating theorem-schema reuse across multiple generated diagram shapes.
Project אω now indexes multiple generated diagram artifacts that reuse the same quotient path equality theorem schema.
Artifact:
docs/multi_diagram_theorem_reuse_index.md
Careful claim:
Project אω now indexes multiple generated diagram artifacts that reuse the same quotient path equality theorem schema.
Project אω now includes a completion report for theorem-schema reuse across multiple generated diagram shapes.
Artifact:
docs/multi_diagram_theorem_reuse_completion_report.md
Careful claim:
Project אω now demonstrates theorem-schema reuse across multiple generated diagram shapes: two generated diamonds and one generated long path-chain.
Project אω now includes a final report for theorem-schema reuse across multiple generated diagram shapes.
Artifact:
docs/multi_diagram_theorem_reuse_final_report.md
Careful claim:
Project אω now demonstrates theorem-schema reuse across multiple generated diagram shapes: two generated diamonds and one generated long path-chain.
Project אω now includes a local verification readiness audit covering core generated theorem artifacts, verification scripts, and final reports.
Artifact:
docs/verification_readiness_audit.md
Careful claim:
Project אω now has a local verification readiness audit covering the core generated theorem artifacts, verification scripts, and final reports.
Project אω now includes a documented final verification command sequence for local release review.
Artifact:
docs/final_verification_command_report.md
Careful claim:
Project אω now has a documented final verification command sequence for local release review.
Project אω now includes a final verification status report separating local readiness, network-sensitive checks, and honest release boundaries.
Artifact:
docs/final_verification_status_report.md
Careful claim:
Project אω is locally release-ready in structure: it has generated theorem artifacts, reusable theorem-schema infrastructure, verification scripts, and final reports. Full verification still requires running the Python and Lean/Lake checks on the current machine or CI.
Project אω now includes a complete final-verification reporting layer: readiness audit, command report, status report, and phase-level summary.
Artifact:
docs/final_verification_phase_report.md
Careful claim:
Project אω now has a complete final-verification reporting layer: readiness audit, command report, status report, and phase-level summary.
Project אω now begins a new technical layer focused on generalized finite path-equivalence: representing, verifying, and generating theorem-backed Lean artifacts for arbitrary finite path pairs.
Artifact:
docs/generalized_finite_path_equivalence_blueprint.md
Careful claim:
Project אω now has a technical blueprint for a generalized finite path-equivalence layer that can represent, verify, and generate theorem-backed Lean artifacts for arbitrary finite path pairs.
Project אω now includes Python-side generalized finite path objects for representing finite path composites and checking path-equivalence witnesses.
Artifacts:
src/rigor/generalized_finite_path_model.py
docs/generalized_finite_path_model.md
Careful claim:
Project אω now has a Python-side generalized finite path model capable of representing finite path composites and checking path-equivalence witnesses.
Project אω now includes a generalized finite path verifier for checking composability, endpoint agreement, pointwise translation agreement, and pointwise model-map agreement for finite path-equivalence witnesses.
Artifacts:
src/rigor/generalized_finite_path_verifier.py
docs/generalized_finite_path_verification_report.md
Careful claim:
Project אω now has a generalized finite path verifier that checks composability, endpoint agreement, pointwise translation agreement, and pointwise model-map agreement for arbitrary finite path-equivalence witnesses.
Project אω now includes a generalized finite path witness family showing that the Python-side framework can represent and verify multiple finite path-equivalence shapes, including diamonds, long chains, and mixed-length path pairs.
Artifact:
docs/generalized_finite_path_witness_family.md
Careful claim:
Project אω now has a generalized finite path witness family showing that the Python-side framework can represent and verify multiple finite path-equivalence shapes, including diamonds, long chains, and mixed-length path pairs.
Project אω now includes a theorem-export plan for turning generalized finite path-equivalence witnesses into theorem-backed Lean/Mathlib artifacts.
Artifact:
docs/generalized_finite_path_theorem_export_plan.md
Careful claim:
Project אω now has a concrete theorem-export plan for turning generalized finite path-equivalence witnesses into theorem-backed Lean/Mathlib artifacts.
Project אω now includes a completion report for the generalized finite path-equivalence layer.
Artifact:
docs/generalized_finite_path_equivalence_completion_report.md
Careful claim:
Project אω now has a generalized finite path-equivalence layer: Python-side path objects, composite behavior, equivalence witnesses, verifier reports, witness families, and a theorem-export plan toward Lean/Mathlib artifacts.
Project אω now begins the generalized witness-to-Lean theorem export layer.
Artifact:
docs/generalized_witness_to_lean_export_blueprint.md
Careful claim:
Project אω now has a blueprint for exporting generalized finite path-equivalence witnesses into theorem-backed Lean/Mathlib artifacts.
Project אω now exports a generated Lean theorem artifact indexing generalized finite path-equivalence witnesses over existing theorem-backed generated diagrams.
Artifacts:
src/rigor/generalized_witness_lean_theorem_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/GeneralizedWitnessFamily.lean
docs/generalized_witness_lean_theorem_exporter.md
Careful claim:
Project אω now exports a generated Lean theorem artifact indexing generalized finite path-equivalence witnesses over existing theorem-backed generated diagrams.
Project אω now generates a new direct generalized witness Lean theorem artifact for a four-step finite path-equivalence proof.
Artifacts:
src/rigor/direct_generalized_witness_lean_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/DirectGeneralizedWitness.lean
docs/direct_generalized_witness_lean_exporter.md
Careful claim:
Project אω now generates a new direct generalized witness Lean theorem artifact for a four-step finite path-equivalence proof.
Project אω now includes a completion report for the generalized witness-to-Lean export layer.
Artifact:
docs/generalized_witness_export_completion_report.md
Careful claim:
Project אω now has a generalized witness-to-Lean export layer: it can index generalized witness examples as Lean theorem artifacts and generate a direct four-step finite path-equivalence theorem using reusable quotient path equality infrastructure.
Project אω now includes a final report for the generalized witness-to-Lean theorem export layer.
Artifact:
docs/generalized_witness_export_final_report.md
Careful claim:
Project אω now has a generalized witness-to-Lean theorem export layer: it indexes generalized witness examples as Lean theorem artifacts and directly generates a four-step finite path-equivalence theorem using reusable quotient path equality infrastructure.
Project אω now begins the parameterized witness-to-Lean theorem export phase.
Artifact:
docs/parameterized_witness_to_lean_export_blueprint.md
Careful claim:
Project אω now has a blueprint for a parameterized witness-to-Lean theorem export pipeline driven by Python PathEquivalenceWitness records.
Project אω now includes a reusable Lean theorem layer for finite path-composite equality.
Artifacts:
src/rigor/parameterized_path_composite_theorem_layer_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/ParameterizedPathCompositeTheorem.lean
docs/parameterized_path_composite_theorem_layer.md
Careful claim:
Project אω now includes a reusable Lean path-composite equality theorem layer for generated finite path-equivalence witnesses.
Project אω now includes a metadata normalizer that converts Python PathEquivalenceWitness records into Lean-safe theorem export metadata.
Artifacts:
src/rigor/parameterized_witness_export_metadata.py
docs/parameterized_witness_export_metadata.md
Careful claim:
Project אω now has a parameterized metadata layer for converting finite path-equivalence witnesses into Lean-safe theorem export metadata.
Project אω now includes a parameterized exporter that generates Lean theorem artifacts from normalized finite path-equivalence witness metadata.
Artifacts:
src/rigor/parameterized_witness_lean_exporter.py
formal/aleph_omega_mathlib/AlephOmegaMathlib/Generated/ParameterizedWitnessFamily.lean
docs/parameterized_witness_lean_exporter.md
Careful claim:
Project אω now has a parameterized exporter that generates Lean theorem artifacts from normalized finite path-equivalence witness metadata.
Project אω now includes a report documenting the batch export of finite path-equivalence witness family records into a generated Lean theorem artifact.
Artifact:
docs/parameterized_witness_batch_export_report.md
Careful claim:
Project אω now batch-exports finite path-equivalence witness family records into a generated Lean theorem artifact using normalized witness metadata.
Project אω now includes a verification integration report for the parameterized witness-to-Lean exporter.
Artifact:
docs/parameterized_export_verification_integration_report.md
Careful claim:
Project אω now integrates its parameterized witness-to-Lean exporter into the generated Lean artifact and verification pipeline.
Project אω now includes a main theorem report explaining the parameterized finite path-equivalence theorem-generation architecture.
Artifact:
docs/parameterized_main_theorem_report.md
Careful claim:
Project אω now has a parameterized theorem-generation architecture for finite path-equivalence witnesses: Python witness records are normalized into generated Lean systems, morphisms, path composites, pointwise equality proofs, and quotient path equality theorems.
Project אω now includes a final research-level report for the parameterized witness-to-Lean theorem export pipeline.
Artifact:
docs/parameterized_export_research_final_report.md
Careful claim:
Project אω now has a parameterized witness-to-Lean theorem export pipeline: finite Python path-equivalence witnesses are normalized into generated Lean systems, morphisms, path composites, pointwise equality proofs, and quotient path equality theorems.