Formalization <-> Simulation Bridge
Bidirectional Traceability Between Mathematical Objects and Discrete Numerical Solvers
Spine Position
Mathematics Root · Formalization ↔ Simulation Diagnostic Mapping
Public Status Boundary. This bridge connects abstract formal mathematical specifications (F-class objects, candidate theorems, Lean 4 proofs) to discrete computational testbeds (finite-difference solvers, lattice automations, and operator resolvent packages). Computational simulation outputs stress-test mathematical invariants; they do not replace formal mathematical proofs or assert empirical confirmation.
1. Epistemic Architecture & Diagnostic Role
┌─────────────────────────────────────────────────────────────────────────────┐
│ EPISTEMIC BRIDGE BOUNDARY │
├─────────────────────────────────────────────────────────────────────────────┤
│ • Formal Layer: Defines algebraic structures, Dirichlet forms, and theorems │
│ • Simulation Layer: Discretizes state spaces to stress-test stability │
│ • Bridge Role: Maps invariant preservation to numerical residual bounds │
│ • Hard Rule: Simulation outputs NEVER prove mathematical theorems │
└─────────────────────────────────────────────────────────────────────────────┘The Formalization ↔ Simulation Bridge establishes explicit bidirectional mapping between abstract formalization layers and executable numerical engines. It enables researchers to visually verify which mathematical invariants are actively stress-tested and identify orphaned formal objects or ungrounded simulations.
2. Interactive Bridge Matrix
3. Structural Demarcation
3.1 Formal Mathematical Layer
The foundation consists of F-class formal objects (axioms, definitions, invariants, and Lean 4 machine-verified lemmas). These objects define the rigorous variational and topological constraints that any physical state transition must obey.
3.2 Discrete Simulation Layer
The execution layer consists of computational solvers and lattice engines (e.g., KP-field elliptic-hyperbolic solvers, hydrogenic TDSE solvers, and FQFT metric measure testbeds). These prototypes generate quantitative residual bounds for empirical falsification and numerical consistency checks.
3.3 Mapped Bridges vs Open Gaps
- Mapped Bridges: Signify that an active, versioned data pipeline exists connecting a formal invariant to an automated testbed (e.g.,
THM-FABRICA-CYCLE-01→ 1-Cycle simplicial boundary checks). - Open Gaps: Highlight theoretical frontiers where a candidate theorem lacks a numerical solver, or where a computational benchmark requires formal invariant axiomatization.
4. Canonical Continuations
| Direction | Node | Route | Focus / Purpose |
|---|---|---|---|
| Results Ledger | Scientific Results Ledger | /science/results | 35 machine-verified Lean 4 theorems and negative null benchmark ( |
| Reproducibility | Computational Reproducibility | /science/reproducibility | RFC 8785 JCS verification receipts and reproducible Lean 4 proof environment |
| Proof Gateway | Lean 4 Formalization Roadmap | /04-mathematics/formalization-roadmap | 35 machine-verified theorem records and lemma dependency DAGs |
| Formalization Batch 1 | Lean 4 Module Specifications | /04-mathematics/formalization-batch-1 | Direct Lean 4 code specifications across all 11 verified modules |
| Simulation Lab | Discrete Simulation Atlas | /04-mathematics/simulations/ | High-order finite-difference solvers, null tests ( |
| Theorem Register | Theorem Candidates Registry | /04-mathematics/theorem-candidates | Formalization candidate statements and Lean 4 lemma anchors |
| Falsifiability Map | Falsifiability Index & Negative Theorems | /04-mathematics/falsifiability-index | Quantitative falsification criteria and singular Fisher information bounds |