Simulation Atlas

Computational Surrogates, Lattice Solvers & Empirical Boundary Governance

Spine Position

Mathematics Root · Computational Surrogates, Numerical Lattice Solvers & Null Benchmarks

Public Status Boundary. All simulations hosted in this Atlas function strictly as numerical surrogates and computational testbeds. They evaluate algorithmic stability, invariant preservation, and discrete convergence under declared boundary conditions. They do not constitute analytical mathematical proofs, physical experiments, or externally confirmed fundamental laws of nature.

$$\boxed{{\displaystyle \mathbf{\text{Simulation}}\ne \mathbf{\text{Experiment}}}}$$$$\boxed{{\displaystyle \mathbf{\text{Numerical convergence}}\ne \mathbf{\text{physical truth}}}}$$

Figure 4.3 — Computation vs Empirical Boundary: Strict separation between numerical simulation surrogates and experimental physical measurement.

Explicitly demarcates numerical solvers, observables, and null models from empirical physical measurement and experimental comparison.

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

---

1. The Canonical Simulation Contract Schema

To eliminate ambiguous claims, every simulation record in the corpus adheres to a formal typed contract:

[SIMULATION CONTRACT SCHEMA]
├── simulationId: String (Unique canonical identifier)
├── targetTheory: String (FFE | KP | FQFT | Realica)
├── mathematicalObject: String (Discrete operator or field representation)
├── equations: String (Governing differential / difference system)
├── parameters: Object (Typed parameter configuration)
├── initialConditions: String (Boundary & initial state vectors)
├── numericalMethod: String (Finite difference, Crank-Nicolson, Spectral)
├── resolution: String (Lattice size N, time-step dt)
├── randomSeed: Integer (Deterministic reproduction seed)
├── observable: String (Computed metric or invariant residual)
├── nullModel: String (Baseline comparator or permutation test)
├── uncertaintyClass: String (Numerical discretization error bound)
├── reproductionCommand: String (CLI command to reproduce run)
└── status: SimulationStatusEnum

Simulation Status Taxonomy:

Status Tier	Definition	Evidentiary Weight
TOY_MODEL	Simplified low-dimensional system demonstrating conceptual mechanics.	Illustrative only; no physical scaling.
NUMERICAL_SANITY_CHECK	Basic verification of algebraic consistency and non-divergence.	Software sanity check.
COMPUTATIONAL_BENCHMARK	High-precision numerical test of invariant preservation or operator bounds.	Algorithmic benchmark.
CALIBRATED_RETRODICTION	Parameter-fitted reproduction of known experimental values.	Fitted baseline; zero prospective degrees of freedom.
NULL_TEST	Adversarial statistical test against scrambled or surrogate datasets.	Null hypothesis evaluation.
HELD_OUT_VALIDATION	Prediction evaluated on data not used during model tuning.	Empirical cross-validation.
PROSPECTIVE_TEST	Pre-registered numerical prediction awaiting experimental detector data.	Prospective empirical falsifier.

---

2. Canonical Simulation Families

2.1 Fabric Field Equation (FFE) Family

• SIM-FFE-01: KKT Constrained Toy Model
  • Status: TOY_MODEL
  • System: Finite 2D discrete manifold with quadratic action $S_{\text{dyn}}(u)=\frac{1}{2}{u}^{T}Hu$ and linear constraint $Cu=0$. Verifies multiplier convergence $ϵ<{10}^{-7}$.
• SIM-FFE-02: Graph Incidence Cycle Conservation
  • Status: COMPUTATIONAL_BENCHMARK
  • System: Boundary operator ${\partial }_2$ on 64-node simplicial complex. Verifies algebraic zero-divergence ${\partial }_1{\partial }_2=0$.

2.2 KP-Field Operator Family

• SIM-KP-01: Finite Operator Positivity Benchmark
  • Status: NUMERICAL_SANITY_CHECK
  • System: Discrete graph Laplacian ${\mathcal{L}}_{\text{KP}}$ on $N=256$ lattice. Computes lowest eigenvalue ${\lambda }_0=0,{\lambda }_1>0$.
• SIM-KP-02: Resolvent Residual Solver
  • Status: COMPUTATIONAL_BENCHMARK
  • System: Preconditioned conjugate gradient solver for $({\mathcal{L}}_{\text{KP}}+\lambda I)v=f$. Computes residual $\parallel ({\mathcal{L}}_{\text{KP}}+\lambda I)v-f{\parallel }_2<{10}^{-10}$.
• SIM-KP-03: Parabolic Dissipation Dynamics
  • Status: TOY_MODEL
  • System: Time-evolution ${\partial }_t\varphi =-{\mathcal{L}}_{\text{KP}}\varphi +\mathcal{N}(\varphi )$. Demonstrates energy decay $dE/dt\le 0$.

2.3 Fractal Quantum Field Theory (FQFT) Family

• SIM-FQFT-01: K6 Spectral Sanity Check
  • Status: NUMERICAL_SANITY_CHECK
  • System: Complete graph $K_6$ Laplacian spectrum $\{0,6,6,6,6,6\}$. Confirms degenerate eigenvalue structure.
• SIM-FQFT-02: Lepton Mass Ratio Calibrated Retrodiction
  • Status: CALIBRATED_RETRODICTION
  • Truth Boundary: Calibrated to 3 charged lepton masses ($e,\mu ,\tau$). $N_{\text{data}}=3,N_{\text{tuned}}=3,{\text{DOF}}_{\text{residual}}=0$. No predictive claim beyond calibration.
• SIM-FQFT-03: Three-Band Null Ensemble
  • Status: NULL_TEST
  • Truth Boundary: Permutation test across 10,000 synthetic spectra. Empirical null p-value $p_{\text{null}}=0.62$. Prospective prediction counts: $P_3=0,P_4=0,P_5=0,P_6=0$.
• SIM-FQFT-04: Parameter Identifiability Analysis
  • Status: COMPUTATIONAL_BENCHMARK
  • System: Hessian singular value decomposition on parameter manifold. Evaluates parameter cross-correlation and sloppy spectrum.

---

3. Legacy Computational Testbeds

Historical testbeds preserved with strict status boundaries:

• FQFT Scale Transition Model — Status: REVIEW_REQUIRED
• Closure Field Dynamics — Status: REVIEW_REQUIRED
• Invariant Preservation Check — Status: REVIEW_REQUIRED
• KP Field Closure Dynamics — Status: REVIEW_REQUIRED

---

4. Canonical Descent & Reproduction

To reproduce all canonical simulation benchmarks locally:

npm run sim:benchmark

Primary Inverse-Problem & Identifiability Anchors. The mathematical analysis of rank-deficient Jacobians and ill-posed parameter estimation follows:

• Aster, R. C., Borchers, B., & Thurber, C. H. (2018). Parameter Estimation and Inverse Problems. Elsevier. DOI: 10.1016/C2016-0-03820-2.
• Gutenkunst, R. N., et al. (2007). Universally Sloppy Parameter Sensitivities in Systems Biology Models. PLOS Comput. Biol., 3(10), e189. DOI: 10.1371/journal.pcbi.0030189.

---

5. Canonical Continuations

Direction	Target Resource	Purpose
Formal Proofs	Lean 4 Formalization Roadmap →	28 machine-verified theorem records across 9 formal modules
Falsifiability Criteria	Falsifiability Index & Negative Boundaries →	Explicit failure conditions and parameter identifiability limits
Proof Governance	Proof Governance & Verification Scale →	Six-stage M0–M5 verification and publication lifecycle
Manuscript D	Verified Publications & Research Manuscripts →	Negative identifiability theorems and standalone LaTeX packages