| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Includes the numerical simulation of physical systems using discretized or algorithmic representations of governing laws; covers classical mechanics, electromagnetism, quantum mechanics, statistical physics, fluid dynamics, plasma physics, materials physics, astrophysics, and multiscale modeling. Excludes purely analytical theory, experiments without computational components, and numerical work that does not involve physical systems. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates from atomic and molecular scales in quantum and molecular simulations to macroscopic astrophysical or continuum scales. Timescales range from attosecond wavefunction evolution to millions of simulated years for planetary or cosmological models. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Particles, fields, grid cells, lattice sites, wavefunctions, probability densities, solver variables, boundary conditions, numerical fluxes, discrete operators, and physical constants encoded in algorithms. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Energy, momentum, position, velocity, charge density, field amplitude, coupling constants, interaction potentials, numerical stability bounds, and discretization accuracy. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Numerical methods, discretization schemes, solver classes, boundary conditions, model types, physical regimes, and simulation architectures. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Position arrays, velocity fields, density fields, electric and magnetic fields, wavefunctions, temperature fields, stress tensors, distribution functions, and auxiliary solver variables such as residuals or timesteps. |
| | Parameterization | How variables encode and represent the system’s state. | States encoded by mesh resolution, timestep size, discretization order, physical parameters, coupling constants, potential functions, solver tolerances, and initial/boundary conditions. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Reduced dimensions (1D, 2D), coarse gridding, ideal boundary conditions, linearized equations, truncated interaction ranges, approximate potentials, continuum approximations, and ignoring subgrid or quantum effects when resolution is insufficient. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Valid when numerical resolution captures key physics, approximated interactions remain dominant, discretization errors stay controlled, solver stability holds, and omitted physics does not significantly alter system behavior. Breaks down when resolution is too low, when strongly nonlinear or quantum effects dominate, or when boundary conditions distort the solution. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes governing equations are correct, discretization methods approximate them faithfully, numerical solutions converge toward physical solutions, and computational artifacts remain separable from true system behavior. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes floating point arithmetic is sufficiently accurate, solver convergence implies physical fidelity, simplified models approximate real systems, and numerical noise does not meaningfully distort emergent dynamics. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires consistent coupling of discretized equations, solver algorithms, initial and boundary conditions, and physical models without contradictions between numerical and physical assumptions. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Entities, variables, and assumptions must integrate to form a unified computational framework linking physical laws, numerical methods, solver architectures, and simulation outcomes. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Observable signals include simulation output fields such as density, velocity, temperature, pressure, field strength, particle trajectories, correlation functions, energy spectra, solver residuals, convergence curves, and numerical stability patterns. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by numerical precision, floating point error, mesh resolution, timestep stability, sampling interval, solver accuracy, memory limits, and computational capacity for resolving multiscale behavior. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Uses meters, seconds, kilograms, joules, pascals, hertz, electron volts, nondimensional numbers, grid units, timestep counts, iteration counts, and code dependent normalized units. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Instruments are virtual: numerical solvers, mesh generators, particle trackers, field analyzers, Fourier analyzers, visualization tools, convergence monitors, diagnostics modules, and logging systems. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Terms such as convergence threshold, CFL condition, residual norm, error estimator, timestep limit, numerical diffusion coefficient, and mesh quality metrics are defined by simulation procedure standards. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Procedures include mesh refinement checks, timestep adaptation, solver iteration cycles, field sampling routines, ensemble runs, Fourier transforms, statistical averaging, error estimation, and diagnostic extraction at set intervals. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Data is gathered through periodic snapshot dumps, time series logging, particle sampling, grid scans, convergence monitoring, checkpointing, adaptive refinement output, and parallel data collection across compute nodes. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling rules include spatial grid sampling, temporal sampling based on solver stability, ensemble sampling for stochastic models, Monte Carlo sampling for statistical systems, and resolution dependent sampling of fine scale structures. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Data appears as multidimensional field arrays, particle lists, time series, spectra, correlation matrices, error fields, solver logs, mesh data, and derived diagnostics such as fluxes or energy distributions. |
| | Resolution | The granularity or precision with which data is captured. | Determined by mesh spacing, timestep size, solver order, numerical precision, available memory, parallelization scale, and discretization quality. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Calibration uses benchmark simulations, analytic solution comparisons, grid convergence studies, code verification suites, energy conservation tests, symmetry checks, and cross validation with independent solvers. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Errors arise from discretization, numerical diffusion, floating point rounding, aliasing, insufficient resolution, solver divergence, inaccurate boundary conditions, and instability in stiff or nonlinear regimes. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Stable patterns include numerical stability laws, convergence behavior, discretized conservation laws, scaling of computation time with resolution, turbulence spectra in simulations, wave propagation consistency, and reproducible emergent patterns from iterative solvers or particle ensembles. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Invariants include conserved mass, momentum, and energy in conservative schemes; symmetries preserved under appropriate discretization; constant norms in unitary quantum evolution schemes; and invariant mesh topology under structured grids. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Mechanisms arise from discretized physical interactions such as advection, diffusion, electromagnetic coupling, particle collisions, quantum evolution, or gravitational forces; also from numerical processes such as iterative relaxation, operator splitting, and error propagation. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Pathways include time stepping updates, iterative solver convergence, particle–field coupling cycles, mesh refinement cascades, energy transfer across scales in simulated turbulence, and integration of field equations through discretized operators. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Core terms include discretization, mesh topology, timestep stability, numerical diffusion, convergence, residual, operator splitting, boundary scheme, solver accuracy, and parallel scaling. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Classifies methods by numerical family (finite-difference, finite-volume, finite-element, spectral, particle-in-cell, Monte Carlo), by equation type (elliptic, parabolic, hyperbolic), by scale (atomistic, mesoscopic, continuum), and by solver type (explicit, implicit, hybrid). |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Includes discretized PDE forms, time-integration update equations, matrix–vector formulations, Hamiltonian evolution algorithms, stochastic update rules, finite-volume flux equations, Monte Carlo sampling rules, and iteration formulas for solver convergence. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Uses continuum fluid models, kinetic models, quantum lattice models, molecular dynamics models, N-body gravitational models, statistical physics models, turbulence models, lattice Boltzmann models, and hybrid multi-physics computational frameworks. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Idealizations include reduced dimensions, uniform grids, simplified boundary approximations, linearized equations, truncated interaction potentials, idealized force fields, approximate collision operators, and ignoring subgrid processes when computational limits require it. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Valid when mesh resolution captures governing scale, timestep meets stability rules, solver errors remain bounded, omitted interactions are secondary, and numerical diffusion does not distort physical behavior; breaks down for extreme nonlinearities, chaotic systems, stiff equations, or quantum regimes with high precision needs. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Integrates discretization theory, numerical analysis, solver methods, parallel computing, and physical modeling into unified simulation frameworks supporting multi-scale, multi-physics systems. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to computer science, applied mathematics, data science, engineering, plasma physics, materials science, astrophysics, fluid dynamics, quantum computing, and high-performance computing. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Experiments involve manipulating mesh resolution, timestep size, solver type, numerical scheme order, boundary conditions, initial conditions, physical parameters, and coupling strengths to isolate causal effects on stability, convergence, accuracy, and emergent physical behavior. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observational approaches monitor simulation output as it naturally evolves, tracking spontaneous instabilities, emergent patterns, chaotic behavior, turbulence cascades, or convergence trends without direct numerical perturbation. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Hypotheses tested by comparing simulation results to analytic solutions, benchmark problems, laboratory data, symmetry predictions, conservation laws, and known scaling behaviors across resolution or parameter sweeps. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Replication requires re-running simulations with different solvers, different mesh types, varied timesteps, independent codebases, and alternate initial conditions to confirm robustness and eliminate model-specific artifacts. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Methods include ensemble averaging, uncertainty quantification, regression of scaling relationships, spectral analysis of fields, statistical evaluation of convergence rates, Monte Carlo sampling, and stochastic interpretation of noisy or chaotic outputs. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Models compared by accuracy, stability across parameters, convergence behavior, computational efficiency, conservation properties, and compatibility with physical constraints or experimental data. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Errors arise from discretization, floating point roundoff, numerical diffusion, aliasing, insufficient resolution, instability of stiff solvers, divergence in chaotic regimes, and inaccurate boundary treatments. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Bias minimized through mesh refinement studies, solver cross-checking, blind analysis of numerical outputs, standardized benchmarking, stripping out nonphysical transients, and enforcing symmetry or conservation constraints. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Findings evaluated through code comparison projects, replication by independent teams, publication peer review, HPC benchmarking exercises, and cross validation against experimental or analytical results. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Theories updated when simulations reveal unexpected instabilities, anomalous scaling, unphysical divergences, or phenomena inconsistent with existing physical or numerical models, prompting refinement of discretization, closure models, or solver algorithms. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Requires complete disclosure of solver type, mesh structure, timestep rules, numerical tolerances, boundary conditions, convergence criteria, code versioning, hardware environment, and modeling assumptions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Requires reproducible simulation setups, honest reporting of limitations, proper handling of HPC resources, avoidance of selective output reporting, and adherence to community standards for data availability and code transparency. |