Simulation Runtime

Computational Execution Sandbox, Invariant Preservation & Algorithmic Tractability

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Mathematics Root · Simulation Telemetry · Execution Runtime

Epistemic Boundary

The Simulation Runtime is an environment for computational stress tests. Simulations do not constitute empirical proof. They do not prove theorems or provide experimental confirmation. Simulations validate the algorithmic tractability of models and test whether structural invariants hold within defined numerical boundaries (ACTIVE_P4_SEALS = 0).

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1. Runtime Architecture

The runtime architecture strictly separates formal mathematical models from computational approximations:

1. Formal Definitions (M3/M4-CHECKED): Unambiguous logical relationships, categorical morphisms, and invariant constraints (28 Lean 4 verified theorems).
2. Computational Model: The algorithmic discretization of formal operators, translated to executable numerical kernels (Python, Rust, C++).
3. Simulation Candidate (M4-SIMULATION): A specific executable configuration with parameter boundary limits.
4. Parameter Space: The boundary conditions under which the candidate is evaluated.
5. Invariant Test: An automated check ensuring that state transitions preserve essential symmetries and topological invariants (${\partial }^2=0$).

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2. Status Classifications

Simulation candidates are classified across five operational states:

• planned: A mathematical model has been identified for simulation, but no computational translation exists.
• scaffolded: Initial interfaces, variable bounds, and failure conditions are drafted. The code is not yet executable.
• prototype: An executable simulation exists. It has successfully demonstrated specific invariant checks within a narrow parameter space.
• reproducible: The prototype runs predictably across standardized inputs and generates deterministic receipts ($p_{\text{null}}=0.62$).
• reviewed: The computational methodology has been reviewed against the formal model.

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3. Failure & Falsification Conditions

A simulation is considered to have failed (providing falsifying friction) if:

• A formally proven mathematical invariant is violated during state evolution.
• The computational resources required exceed polynomial tractability for a supposedly localized operation.
• The boundary conditions collapse inconsistently or exhibit numerical divergence (${ϵ}_{\text{drift}}>{10}^{-6}$).

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4. Prototype Traces

• Recursive Observer Knot CA — Cellular automaton verification of observer projection idempotence.
• KP Field Propagation — Elliptic-hyperbolic finite-difference wave propagation solver.
• Closure Field Dynamics — Field relaxation and numerical stability benchmarks.

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5. Canonical Continuations

Direction	Target Resource	Purpose
Simulation Analytics	Cross-Model Telemetry	Multi-model invariant compliance and diagnostic metrics
Simulation Lab	Canonical Simulation Atlas Hub	15-point typed simulation contract, solver families, and $p_{\text{null}}=0.62$ benchmark
Simulation Bridge	Formalization ↔ Simulation Bridge	Diagnostic mapping between Lean 4 proofs and discrete numerical solvers
Proof Gateway	Lean 4 Formalization Roadmap	28 machine-verified theorem records and lemma dependency DAGs

EXTERNAL REFERENCE