Formalization Roadmap

Lean 4 Proof Gateway, Lemma DAGs & Machine Verification Milestones

Spine Position

Mathematics Root · Interactive Theorem Proving, Lean 4 DAGs & Proof Governance

Public Status Boundary. This roadmap tracks active interactive theorem proving campaigns in Lean 4. It enforces strict separation between paper proof sketches, syntactic source code, and verified machine proofs. No theoretical proposition is promoted to verified status without an explicit, machine-checked Lean 4 certificate without sorry.

$$\boxed{{\displaystyle \mathbf{\text{Proof sketch}}\ne \mathbf{\text{machine proof}}}}$$$$\boxed{{\displaystyle \mathbf{\text{Machine compilation}}\ne \mathbf{\text{physical validation}}}}$$$$\boxed{{\displaystyle \mathbf{\text{Imported theorem}}\ne \mathbf{\text{authorial theorem}}}}$$

Figure 4.2 — Formal Proof Promotion Pipeline: Multi-stage formal verification workflow from conjecture to compiler-accepted Lean 4 artifacts and clean environment reproduction.

Illustrates the rigorous formalization trajectory from claim and proof sketch to compiler acceptance, machine verification, and reproducibility.

Credit: Ivan Pasev / GILC Research·CC BY-NC-SA 4.0·SCHEMATIC

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1. The Eight-Stage Formalization Ladder

Every mathematical theorem candidate in the Science of Fabric Reality ascends through eight formal promotion states:

Stage	Status Identifier	Operational Meaning	Validation Gate
1	UNFORMALIZED	Natural-language conjecture or physical intuition.	Informal motivation recorded in research notes.
2	STATEMENT_DRAFTED	Rigorous mathematical proposition typed with explicit domains and boundaries.	Semantic boundary review in Mathematics Root.
3	PROOF_SKETCH	Pen-and-paper analytical derivation outlining necessary lemma dependencies.	Peer review within the SFR Review Portal.
4	LEAN_SOURCE_PRESENT	Lean 4 source file with typed axioms, structures, and theorem stubs.	Source file tracked in Lean 4 module suite.
5	COMPILES	Lean 4 project builds cleanly without syntax or type errors.	Lake toolchain build clean (lake build).
6	MACHINE_VERIFIED	Complete formal proof closed without sorry or unverified axioms.	Verified by Lean 4 kernel (M4–M5 maturity).
7	DEPENDENCY_VERIFIED	Full transitive lemma dependency DAG audited for circularity.	Strict lemma DAG & Mathlib compatibility check.
8	REPRODUCIBLE	Fully reproducible build checked from a fresh, clean environment.	Deterministic CI verification pass.

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2. Priority Theorem Records & 28 Machine-Verified Proofs

The formalization campaign tracks 28 machine-verified proofs across 9 Lean 4 modules:

Theorem Identifier	Carrier Domain	Lean Module	Current Stage	Verification Target
THM-INV-COMP-01	Admissible Morphisms	Fabrica.InvariantEngineering	MACHINE_VERIFIED	Preservation of local admissibility under composite state transition (thm_state_local_admissibility_composition).
THM-INV-INV-01	Invariant Groupoids	Fabrica.InvariantEngineering	MACHINE_VERIFIED	Admissibility preservation under inverse state transition (thm_state_local_admissibility_inversion).
THM-FABRICA-CYCLE-01	Discrete Topology	Fabrica.InvariantEngineering	MACHINE_VERIFIED	Linear subspace closure of boundary-free 1-cycles (thm_one_cycle_linear_combination).
THM-FABRICA-CYCLE-02	Discrete Topology	Fabrica.InvariantEngineering	MACHINE_VERIFIED	Zero boundary on 1-cycle linear combinations (thm_one_cycle_zero_boundary).
THM-OBS-IDEMPOTENT-01	Observer Monads	Fabrica.ObserverKnot	MACHINE_VERIFIED	Composite idempotence $(P_1\circ P_2{)}^2=P_1\circ P_2$ for commuting observer projectors (thm_obs_idempotent_composition).
THM-TRACE-RECIPROCITY-01	Trace Pairing	Fabrica.TraceReciprocity	MACHINE_VERIFIED	Symmetric bilinear involution $⟨x,y{⟩}_{\tau }⟺⟨y,x{⟩}_{\tau }$ (thm_trace_involution).
THM-TRACE-DUAL-EVAL-01	Trace Pairing	Fabrica.TraceReciprocity	MACHINE_VERIFIED	Bilinear trace dual symmetry TraceDual T y x ↔ TraceDual T x y (thm_trace_dual_symmetry).
THM-TRACE-REFL-01	Trace Pairing	Fabrica.TraceReciprocity	MACHINE_VERIFIED	Reflexive trace state self-duality TraceDual T x x (thm_trace_dual_reflexive).
THM-FFE-STATIONARITY-01	Constrained Dynamics	Fabrica.FFE	MACHINE_VERIFIED	Coupled Euler-Lagrange dynamical stationarity and exact foliation constraint satisfaction $C(u)=0$ (thm_ffe_variational_stationarity).
THM-FFE-HOMOGENEOUS-01	Constrained Dynamics	Fabrica.FFE	MACHINE_VERIFIED	Homogeneous source stationarity triviality under zero source (thm_ffe_homogeneous_triviality).
THM-FFE-SUPERPOSITION-01	Constrained Dynamics	Fabrica.FFE	MACHINE_VERIFIED	Multiplier pointwise linear superposition (thm_ffe_multiplier_superposition).
THM-FFE-KKT-UNIQUE-01	Discrete Optimization	Fabrica.FFE	MACHINE_VERIFIED	Finite-dimensional KKT uniqueness and trivial kernel $(u,\mathrm{\Lambda })=(0,0)$ under strictly positive $H$ and full-rank $C^{\ast }$ (thm_finite_ffe_kkt_zero_nullity).
THM-FQFT-RESOLVENT-02	Operator Theory	Fabrica.FQFT	MACHINE_VERIFIED	Two-sided resolvent operator invertibility and state reconstruction (thm_resolvent_two_sided_inverse).
THM-FQFT-RESOLVENT-ID-01	Operator Theory	Fabrica.FQFT	MACHINE_VERIFIED	Algebraic First Resolvent Identity $R_1(A_2(R_2\psi ))-R_1(A_1(R_2\psi ))=R_1\psi -R_2\psi$ (thm_first_resolvent_identity_algebraic).
THM-FQFT-DIRICHLET-EL-01	Variational Analysis	Fabrica.FQFT	MACHINE_VERIFIED	Dirichlet energy scaling lower bound under geometric dilation (thm_dirichlet_scaling_lower_bound).
THM-FQFT-COVARIANCE-01	Scale Dynamics	Fabrica.FQFT	MACHINE_VERIFIED	Transfinite scale covariance commutation in scale-flow spaces (thm_scale_covariance_algebraic).
THM-PGP-IDENTITY-01	Gauge Theory	Fabrica.PGP	MACHINE_VERIFIED	Identity gauge transformation invariance $F({\text{id}}_{\mathrm{\Phi }}\phi )=F(\phi )$ (thm_pgp_gauge_identity).
THM-PGP-COMPOSITION-01	Gauge Theory	Fabrica.PGP	MACHINE_VERIFIED	Gauge group composite action invariance preservation (thm_pgp_gauge_composition).
THM-PGP-ORBIT-01	Gauge Theory	Fabrica.PGP	MACHINE_VERIFIED	Functional invariance across gauge orbit iterations (thm_pgp_gauge_orbit_invariance).
THM-MONAD-LEFT-ID-01	Epistemic Monads	Fabrica.ObserverMonad	MACHINE_VERIFIED	Observer monad categorical left identity $(\text{unit}(a)\gg =f)\equiv f(a)$ (thm_monad_left_identity).
THM-MONAD-RIGHT-ID-01	Epistemic Monads	Fabrica.ObserverMonad	MACHINE_VERIFIED	Observer monad categorical right identity $(m\gg =\text{unit})\equiv m$ (thm_monad_right_identity).
THM-MONAD-ASSOC-01	Epistemic Monads	Fabrica.ObserverMonad	MACHINE_VERIFIED	Observer monad sequential observation associativity (thm_monad_associativity).
THM-FABRICON-REFLEXIVE-01	Fabricon Relationality	Fabrica.Fabricon	MACHINE_VERIFIED	Fabricon conjugate source-destination reciprocity (thm_fabricon_reflexive_coupling).
THM-FABRICON-INVOLUTION-01	Fabricon Relationality	Fabrica.Fabricon	MACHINE_VERIFIED	Fabricon double conjugation identity involution (thm_fabricon_conjugate_involution).
THM-FABRICON-SYMMETRY-01	Fabricon Relationality	Fabrica.Fabricon	MACHINE_VERIFIED	Symmetric coupling bidirectional reciprocity (thm_fabricon_coupling_symmetry).
THM-STABILIZATION-IDEMP-01	Infinite Stabilization	Fabrica.Realica	MACHINE_VERIFIED	Realica stabilization fixed-point operator idempotence (thm_stabilization_fixed_point_idempotence).
THM-STABILIZATION-CONV-01	Infinite Stabilization	Fabrica.Realica	MACHINE_VERIFIED	Realica single-step stabilization sequence closure (thm_stabilization_state_convergence).
THM-REALICA-CONSERVE-01	Infinite Stabilization	Fabrica.Realica	MACHINE_VERIFIED	Realica partition potential invariance conservation (thm_realica_partition_conservation).

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3. Audited Proof Registry Telemetry

The current state of formal artifacts in the repository is audited directly from filesystem compiler records:

[LEAN 4 COMPILER AUDIT — REPOSITORY TRUTH]
├── Verified Lean Modules: 9 (Fabrica.ObserverKnot, Fabrica.InvariantEngineering, Fabrica.TraceReciprocity, Fabrica.FFE, Fabrica.FQFT, Fabrica.PGP, Fabrica.ObserverMonad, Fabrica.Fabricon, Fabrica.Realica)
├── Fully-Closed Machine Proofs: 28
├── Formal Definitions: 42
├── Unproven Axioms / Sorry: 0
├── Compilation Status: 100% VERIFIED
└── Lean Kernel Diagnostic: PASS

Formalization Law: Zero Synthetic Claims

No mathematical proposition is classified as MACHINE_VERIFIED from narrative prose alone. Every verified status requires a compiled .lean artifact without sorry and does not substitute for empirical validation (ACTIVE_P4_SEALS = 0).

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4. Canonical Continuations

Direction	Target Resource	Purpose
Axioms Register	Constitutional Axioms & Object Register →	Source-backed primitives and quarantined formal proposals
Proof Governance	Proof Governance & Verification Scale →	Six-stage M0–M5 publication and verification standards
Theorem Candidates	Theorem Candidates Registry →	Candidate theorems across spectral graph and operator theory
Simulation Lab	Simulation Laboratory & Numerical Atlas →	Discretized numerical solvers, null tests ($p_{\text{null}}=0.62$), and receipts