| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies equations involving derivatives of unknown functions, including ordinary differential equations (ODEs), partial differential equations (PDEs), systems of differential equations, and boundary/initial value problems. Includes existence/uniqueness theory, stability, qualitative analysis, spectral methods, variational principles, distributional solutions, and numerical approximations. Excludes algebraic equations, discrete dynamical systems (except as approximations), and stochastic processes unless incorporated within deterministic DE frameworks. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at local (infinitesimal derivative-based) scales, intermediate dynamical scales (trajectories, flows), and global scales (long-time behavior, steady states, asymptotic regimes). PDEs operate on spatial/temporal domains, manifolds, and higher-dimensional continua. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Unknown functions; derivatives (first, higher-order, partial); vector fields; flows; operators (Laplace, divergence, gradient, Hessian, etc.); boundary and initial data; Green’s functions; fundamental solutions; weak/distributional solutions; nonlinear operators; semigroups of operators. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Linearity vs nonlinearity; order of equation; ellipticity, parabolicity, hyperbolicity; regularity/smoothness; stability; uniqueness; existence; blow-up behavior; conservation laws; dissipation; invariants; symmetry properties; well-posedness. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | ODEs (linear, nonlinear); autonomous vs non-autonomous systems; PDE classes (elliptic, parabolic, hyperbolic); boundary-value problems; initial-value problems; weak formulations; variational PDEs; spectral PDEs; systems of ODE/PDE; coupled field equations. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Time variable t; spatial variables x; state vector y(t); gradients; Laplacians; divergence values; initial/boundary parameterization; regularity norms (H¹, Cᵏ, Lᵖ); stability exponents; spectral values; semigroup evolution parameters. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded via coefficients (variable or constant); domain geometry; boundary conditions; forcing terms; operator coefficients (diffusion rate, wave speed); nonlinearity parameters; initial-data norms; discretization step sizes in numerical schemes. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Linearization around equilibria; assuming smooth coefficients; separating variables; treating domains as simple geometric shapes; ignoring irregular boundary effects; using constant coefficients; assuming small amplitude; using weak/variational formulations to bypass lack of smoothness. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down for strong nonlinearity; shocks or discontinuities; rough domains or coefficients; chaotic regimes; singularity formation; solutions outside classical regularity; high-dimensional blow-up; PDEs requiring distributional or measure-valued solutions. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Differentiability (classical or weak); deterministic evolution laws; continuity of operators; applicability of existence/uniqueness theorems; well-posedness in suitable function spaces; stability under small perturbations; conservation/dissipation embedded in operator structure. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes time/space are continuous; assumes derivatives encode true physical or abstract change; assumes domains allow differential structure; assumes classical calculus tools generalize to weak settings; assumes representability via operator-theoretic frameworks (semigroups, variational forms). |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Differential operators must align with geometric/analytic structure; boundary and initial conditions must be compatible; weak and classical solutions must agree when regular enough; stability and uniqueness must not contradict existence claims; conserved quantities must match governing equations. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony between ODE/PDE theory, functional-analytic frameworks, harmonic analysis, numerical approximation schemes, variational principles, geometric structures of domains, and physical or abstract conservation/dissipation laws. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Trajectories of ODE solutions; steady states; oscillations; blow-up events; diffusion and wave propagation; heat dissipation profiles; shock formation; boundary-layer behavior; PDE solution surfaces; eigenfunctions of differential operators; time-series evolution from numerical solvers. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited ability to resolve steep gradients; numerical difficulty near singularities; inability to track infinite-time behavior directly; resolution limits for high-dimensional PDEs; aliasing in spectral methods; stiffness obscuring true dynamics; inability to observe weak/distributional behavior directly. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Amplitude; frequency; time constants; spatial length scales; derivative magnitudes; energy norms (L², H¹); stability exponents (Lyapunov-type); step sizes (Δt, Δx); residual norms; spectral values (eigenvalues). |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Numerical ODE solvers (Runge–Kutta, multistep); PDE solvers (finite element, finite difference, spectral); derivative approximators; mesh generators; eigenvalue solvers; stability analyzers; shock-capturing schemes; boundary-condition evaluators; variational solvers. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Solution defined as function satisfying differential equation (classically or weakly); stability defined via sensitivity to initial data; blow-up defined as finite-time divergence; derivative approximations defined via finite differences; residual defined as operator applied to approximate solution; well-posedness defined by existence–uniqueness–continuity. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Solving ODE IVPs; solving PDE boundary-value problems; computing numerical derivatives; assembling discrete operators; performing stability tests; computing residuals; iterating implicit/explicit time steps; computing energy norms; detecting shocks or steep gradients. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Uniform or adaptive time-stepping; structured or adaptive spatial grids; controlled domain discretization; refinement and coarsening cycles; systematic parameter sweeps (e.g., diffusion coefficient, forcing strength); standardized initial/boundary condition selection; sampling PDE solutions along slices or surfaces. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling solution trajectories at discrete times; sampling spatial PDE data on grids; sampling eigenmodes for spectral decomposition; sampling near boundaries or singularities; parameter sampling to study bifurcations; sampling across random initial conditions. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Time series; solution grids; contour maps; surface plots; phase portraits; eigenvalue/eigenvector tables; residual error arrays; finite-element mesh data; coefficient arrays from spectral expansions; weak-solution diagnostics. |
| | Resolution | The granularity or precision with which data is captured. | Determined by mesh density, time-step size, solver tolerance, smoothness of solution, spectral truncation level, numerical precision, and regularity of coefficients. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Cross-validating numerical solutions using independent solvers; refining meshes/time steps until convergence; comparing with analytical solutions when available; validating residual norms; checking stability region of time-stepping methods; verifying boundary-condition enforcement; checking conservation/dissipation properties. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Truncation error; round-off error; instability from stiffness; aliasing in spectral methods; boundary-layer resolution failure; incorrect shock capturing; discretization artifacts; weak-solution ambiguity; error accumulation in long-time integration. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Existence–uniqueness laws (Picard–Lindelöf); superposition for linear equations; conservation/dissipation laws for PDEs; maximum/minimum principles; comparison principles; invariant manifolds in ODEs; energy inequalities; finite propagation speed in hyperbolic PDEs; smoothing effects in parabolic PDEs; elliptic regularity laws. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Energy norms; mass/charge integrals; momentum; Hamiltonians; Lyapunov functions; invariants under flows; divergence-free constraints; conserved fluxes; eigenvalues of operators; symmetry groups; topological degree; PDE-specific invariants (vorticity, entropy). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Derivatives encoding instantaneous rate-of-change; operators (Laplace, divergence, gradient) generating diffusion, wave propagation, or potential fields; nonlinearities creating shocks or blow-up; semigroup evolution generating time dynamics; boundary conditions shaping spatial behavior; variational principles producing Euler–Lagrange PDEs; spectral mechanisms governing stability or oscillation. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | ODE: vector field → flow map → trajectory → stability analysis; PDE: operator + boundary data → weak formulation → solution regularity → asymptotics; Linear PDE: eigenfunction expansion → modal evolution → long-time behavior; Nonlinear PDE: perturbation → bifurcation → pattern formation or blow-up; Variational PDE: functional → minimizer → Euler–Lagrange equation. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Derivative, flow, trajectory, operator, elliptic/parabolic/hyperbolic classification, boundary-value problem, initial-value problem, fundamental solution, weak solution, viscosity solution, Green’s function, semigroup, spectrum, stability, bifurcation, integral kernel. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Linear vs nonlinear; autonomous vs non-autonomous; first-order vs higher-order; ODE vs PDE vs systems; elliptic/parabolic/hyperbolic PDEs; steady-state vs time-dependent; local vs global solutions; classical vs weak/distributional; well-posed vs ill-posed; dissipative vs conservative systems. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | ODE: (y’ = f(t,y)); PDE: (u_t = \Delta u) (heat), (u_{tt} = \Delta u) (wave), (-\Delta u = f) (Poisson); Green’s function formulas; weak formulation integrals; semigroup evolution (u(t)=e^{tA}u_0); eigenfunction expansions; divergence form operators; conservation laws (\partial_t \rho + \nabla\cdot J = 0). |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Phase portraits; flow diagrams; PDE solution surfaces; finite-element meshes; spectral decomposition models; operator matrices; Fourier-mode models; variational-energy landscapes; bifurcation diagrams; fundamental-solution convolution models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Linearization around equilibria; smooth coefficients; constant-coefficient PDEs; rectangular or spherical domains; ignoring nonlinearities; assuming global boundedness; assuming compatibility of boundary data; finite-dimensional Galerkin truncations; treating solutions as classical when only weak solutions exist. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Failure under strong nonlinearity; shock formation; singularities/blow-up; irregular coefficient regimes; rough domains; turbulence; chaotic dynamics; infinite-dimensional instabilities; PDEs requiring measure-valued or distributional solutions; breakdown of uniqueness or regularity. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Semigroup theory unifying linear PDE evolution; spectral theory linking ODE/PDE to eigenvalue problems; variational principles unifying PDEs with optimization; dynamical-systems theory unifying ODE behavior; conservation laws unifying physics and PDE; functional analysis linking PDE to operator theory; geometric PDE frameworks. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Physics (mechanics, electromagnetism, fluid dynamics); engineering (control, signal propagation, heat transfer); biology (reaction–diffusion systems); economics (dynamic systems); geometry (geometric flows); probability (stochastic PDEs as limits of deterministic ones). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying initial conditions, boundary conditions, forcing terms, or coefficients; modifying domain geometry; adjusting discretization scales (Δt, Δx); linearizing around equilibria; introducing controlled perturbations; switching between explicit/implicit schemes; testing shock-capturing methods; altering PDE operator types (diffusion, advection, reaction). |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural system evolution without intervention: tracking spontaneous formation of gradients, oscillations, or shocks; observing large-time asymptotics; monitoring diffusion spreading or wave propagation; watching stability or instability unfold; observing qualitative patterns (limit cycles, attractors). |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing existence/uniqueness assumptions; verifying stability hypotheses; checking regularity claims; validating conservation laws; testing numerical schemes for convergence; verifying blow-up criteria; testing operator coercivity in variational formulations; applying comparison principles; checking eigenvalue predictions. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Re-running simulations with different solvers; refining meshes until convergence plateaus; recomputing eigenvalues/eigenfunctions with independent algorithms; replicating energy decay tests; repeating bifurcation analyses; confirming long-time behavior across time-step variations; re-evaluating shocks with different capturing schemes. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Estimating convergence rates of numerical solutions; analyzing energy dissipation rates; evaluating distributions of derivatives or gradients; assessing statistical regularity of turbulence-like PDE behavior; measuring sensitivity to initial conditions; estimating error norms; comparing trajectories across parameter sweeps. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing linear vs nonlinear models; contrasting explicit vs implicit time-stepping; comparing finite difference vs finite element vs spectral methods; contrasting weak vs classical formulations; comparing reduced (ODE) models to full PDEs; evaluating trade-offs among stability, accuracy, and computational cost; comparing approximate vs exact analytic solutions. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting truncation errors; round-off errors; CFL violations in hyperbolic PDEs; instability from stiffness; aliasing in spectral methods; numerical diffusion or dispersion; incorrect boundary-condition enforcement; misidentification of blow-up; divergence due to coarse resolution. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding bias toward “nice” smooth solutions; sampling across diverse initial data; testing multiple geometries; avoiding solver bias (one algorithm’s strengths dominating interpretation); balancing fine/coarse meshes; checking behavior near singularities; avoiding assumptions of linearity when nonlinear effects dominate. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing mathematical proofs of stability or regularity; auditing numerical evidence for convergence; verifying well-posedness assumptions; cross-comparing PDE/ODE reduction arguments; evaluating approximation quality; rechecking energy or mass conservation; validating analytic/numeric agreement. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating existence/uniqueness criteria; refining stability classifications; modifying blow-up conditions; adjusting numerical schemes; incorporating new operator-theoretic results; revising turbulence or pattern-formation models; improving boundary/initial condition formulations; updating scaling laws. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Disclosing all solver settings, discretization scales, tolerances, grid geometries, convergence thresholds, and assumptions about regularity; reporting failures or instabilities; documenting conditions for which solutions could not be obtained. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of non-convergence or blow-up; avoiding claims beyond solver resolution; ensuring reproducibility; clarifying approximation limitations; acknowledging uncertainty in long-time or high-dimensional PDE predictions; avoiding selective reporting of stable cases only. |