| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies rigorous properties of real numbers, limits, continuity, differentiation, integration, sequences, series, and convergence in real-valued settings. Includes metric spaces, measure theory, Lebesgue integration, function spaces, approximation theory, compactness, completeness, and real-variable techniques. Excludes complex-specific holomorphic phenomena and algebraic structures not expressible through limits or measure. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at pointwise, local, and global scales on ℝ or metric spaces: infinitesimal behavior (derivatives), neighborhood-scale limits, global convergence patterns, measure-theoretic sizes, topology of function spaces, and large-scale analytic structure. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Real numbers, sequences, functions, sets, open/closed sets, measurable sets, metrics, intervals, neighborhoods, derivatives, integrals, measures, σ-algebras, function spaces (e.g., Lᵖ), compact/connected sets. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Continuity, uniform continuity, differentiability, integrability, convergence (pointwise, uniform, almost everywhere), measurability, boundedness, completeness, compactness, total variation, Lipschitz behavior, monotonicity. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Metric spaces, normed spaces, measurable spaces, Lebesgue spaces (Lᵖ), sets of finite/infinite measure, absolutely continuous functions, bounded variation functions, Cᵏ classes, improper integrals, convergence modes, compact sets. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Function value at a point; sequence index; step size in approximations; limit parameter; measure of a set; norm of a function; oscillation over an interval; derivative/integral values; convergence rates; chosen ε and δ values. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded by ε–δ formulations; metric definitions; norms; measure functions; σ-algebra generators; partition refinements; integration approximations; modulus of continuity; functional parameters in Lᵖ norms. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Assuming continuity or differentiability when analyzing local behavior; using piecewise smooth approximations; restricting to bounded or compact domains; ignoring pathological sets; working with Riemann integrals instead of Lebesgue integrals in simple settings. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Simplifications fail for fractal sets, non-measurable sets, nowhere-differentiable functions, unbounded domains, divergent series, or functions requiring Lebesgue integration; ε–δ reasoning breaks down for discontinuous paths or non-metric spaces. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Real numbers form a complete ordered field; limits exist uniquely when they exist; measurable sets form σ-algebras; integrals respect countable additivity; differentiation and integration interact via fundamental theorems; compactness and completeness control convergence. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes classical logic; assumes standard topology on ℝ; assumes ability to quantify infinitesimal behavior via limits; assumes measurable sets suffice for analytic structure; assumes completeness and order structure are indispensable foundations. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Limit definitions must agree under different formulations; differentiation and integration must satisfy fundamental theorems; measure-theoretic and topological structures must align; convergence definitions must not contradict completeness; compactness principles must integrate coherently with metric structures. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony between topology, measure theory, integration, differentiation, functional analysis foundations, convergence modes, and completeness structure of ℝ; analytic results must respect algebraic and order properties of the real numbers. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Limits approaching finite or infinite values; rates of convergence of sequences/series; continuity/discontinuity behavior; differentiability or nondifferentiability; oscillation of functions; integrability properties; measure of sets; variation of functions; norm changes in Lᵖ spaces; convergence behaviors under different modes (pointwise, uniform, almost everywhere). |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Inability to directly observe infinitesimals; limits detectable only through approximations; difficulty distinguishing uniform vs. pointwise convergence numerically; inability to measure non-measurable sets; loss of precision near discontinuities or singularities; numerical instability near vertical tangents or stiff gradients. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | ε–δ tolerances; norms (Lᵖ norms, sup norm); derivative values; integral values; oscillation magnitude; measure of sets; convergence rate indicators; step size in approximations; partition mesh size; error bounds. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Limit-approximation algorithms; numerical differentiation/integration tools; measure-theoretic computation packages; norm calculators; sequence evaluators; partition-refinement engines; function-plotting and sampling tools; numerical ODE solvers (when used analytically). |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Limit defined by ε–δ conditions; continuity defined by limit preservation; derivative defined by limit of difference quotient; integral defined via Riemann or Lebesgue constructions; measurability defined via σ-algebra membership; convergence defined via real-number topology or measure-theoretic criteria. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Checking ε–δ conditions; constructing sequences and testing convergence; computing Riemann or Lebesgue integrals; approximating derivatives; computing measures via coverings; verifying uniform convergence via supremum norms; approximating Lᵖ norms; partition refinement. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Standardized sequence-testing routines; uniform sampling of function values; consistent partition refinement for integrals; structured limit-approximation procedures; controlled evaluation over compact sets; stepwise refinement for detecting discontinuities. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling function values on dense subsets; selecting representative sequences; sampling subsets for measure approximation; sampling behaviors near suspected discontinuities; sampling convergence behavior under different norms; selecting compact subsets for uniform tests. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Function-value lists; sequence tables; partition meshes; integral approximations; derivative approximations; measure-approximation arrays; Lᵖ norm tables; convergence diagnostics; error bounds; graphs/plots representing limiting behavior. |
| | Resolution | The granularity or precision with which data is captured. | Determined by sampling density; step size; partition granularity; numerical precision; approximation tolerance; number of terms in partial sums; ability to resolve rapid oscillations or sharp gradients; measure-approximation fineness. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Validating numerical integration via multiple methods; checking derivative estimates against symbolic derivatives; confirming convergence using several norms; cross-checking measure approximations; comparing limit approximations under decreasing tolerances; ensuring stability under refinement. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Numerical rounding errors; false convergence detection; oscillation under-sampling; failure to detect discontinuities; miscalculated integrals near singularities; derivative blow-up; measure-approximation error; instability near unbounded variation. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Limit laws (sum, product, quotient rules); continuity laws; differentiation rules (product, chain, quotient rules); integration laws (linearity, monotone convergence, dominated convergence); compactness and completeness principles; uniform convergence relations; measure-theoretic additivity; fundamental theorem linking differentiation and integration. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Completeness of ℝ; order structure; measure of sets (invariant under measurable equivalence); norms of functions; total variation; Lipschitz constants; oscillation on intervals; convergence types (pointwise vs uniform) preserved under transformations; invariance under isometries for metric structures. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Limit-taking mechanisms governing continuity and differentiability; integration as limit of sums; measure formation via outer measure and Carathéodory construction; differentiation emerging from local linearization; compactness driving convergence of sequences; completeness ensuring existence of limits; dominated-convergence and monotone-convergence governing integrability transitions. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Function → sequence of approximations → limit → continuity/differentiability conclusions; measurable set → measurable function → integration → convergence theorems; sequence → Cauchy property → limit via completeness; improper integral → comparison test → convergence judgment; function → derivative → antiderivative → integral via fundamental theorem. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Limit, continuity, uniform continuity, derivative, integral (Riemann/Lebesgue), measure, σ-algebra, measurable function, convergence modes (pointwise, uniform, almost everywhere, Lᵖ), completeness, compactness, connectedness, variation, Lipschitz condition. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Continuous vs discontinuous functions; differentiable vs nondifferentiable; Riemann-integrable vs Lebesgue-integrable; measurable vs nonmeasurable sets; bounded vs unbounded functions; absolutely continuous vs singular; functions of bounded variation; Lᵖ spaces; Cᵏ classes. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Limit definitions via ε–δ; derivative: (f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}); Riemann integral: limit of Riemann sums; Lebesgue integral: (\int f,d\mu = \sup{\int s,d\mu : s \le f, s \text{ simple}}); measure additivity: (\mu(\cup A_i)=\sum \mu(A_i)); dominated convergence: (\lim \int f_n = \int \lim f_n). |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Metric-space models; interval-based models; function-space models (Lᵖ, C[a,b], BV); measurable-space models; graphical limit models; diagrammatic ε–δ structures; convergence-diagram models; refinement-partition models for integrals. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Continuous and differentiable functions; compact domains; smooth approximations; Riemann integrability; sequences with regular convergence behavior; ignoring measure-zero pathologies; assuming boundedness or monotonicity. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Simplifications fail for nowhere-differentiable functions; fractal sets; highly oscillatory or unbounded functions; non-measurable sets; functions requiring Lebesgue integration; divergence issues; loss of compactness; measure-theoretic irregularities. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Fundamental theorem of calculus (bridge between differentiation and integration); measure theory unifying integration and size; convergence theorems unifying limit processes; functional analysis connecting real analysis to operator theory; topology providing unifying language for continuity and compactness. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | PDEs and ODEs; probability theory via measure and integration; physics through continuous models; signal processing (Fourier analysis); optimization (convexity, continuity); economics (integral equations, dynamic systems); geometry (curves, surfaces via real-variable methods). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying ε–δ tolerances; modifying step sizes for numerical approximations; altering partitions in Riemann sums; adjusting sampling density for function evaluation; perturbing functions to test continuity/differentiability stability; modifying measure approximations via covering choices; testing convergence under different norms or metrics. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural sequence limits; monitoring uncontrolled convergence behavior; tracking continuity or discontinuity at fixed points; observing derivative-like behavior via difference quotients; letting integral approximations evolve without altering function inputs; observing measure behavior under nested coverings. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing continuity via ε–δ conditions; verifying convergence (pointwise, uniform, Lᵖ) by norms or supremum distances; checking differentiability via limit of difference quotients; testing integrability via Riemann vs. Lebesgue criteria; validating bounded variation or absolute continuity; verifying measure additivity. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing limits with smaller tolerances; repeating integrals with refined partitions; recalculating derivatives using alternative approximations; replicating convergence tests with different sequences; validating measure approximations with independent coverings; repeating uniform-convergence tests with varied sample sets. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Estimating convergence rates; analyzing error decay in numerical integration; comparing oscillation statistics across intervals; assessing derivative stability under perturbation; evaluating distribution of function values; analyzing variation or Lipschitz estimates via sampled data. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing Riemann vs Lebesgue integrability; contrasting modes of convergence; comparing numerical vs analytic derivative computations; comparing behavior of functions under different norms (L¹, L², L∞); contrasting compact vs non-compact domain behavior; comparing different metric-space models. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying rounding errors; under-resolving oscillatory functions; incorrect detection of convergence; miscalculated integrals due to coarse partitions; instability in derivative approximations; measure estimation errors; propagation of numerical inaccuracies in iterative approximations. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding biased sampling of “smooth” regions while ignoring singularities; ensuring balanced sampling over domain; avoiding reliance on a single numerical method; controlling for machine precision effects; using uniform criteria across different convergence or integrability tests. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing ε–δ proofs; checking correctness of convergence arguments; auditing numerical integration/derivation against analytic results; confirming measure calculations; evaluating consistency of approximation methods; rechecking compactness or completeness claims. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating convergence criteria; refining integrability definitions; modifying measure constructs; correcting derivative approximations; integrating new theorems on convergence, regularity, or measure; revising functional assumptions for pathological cases. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of tolerances, sampling densities, partition choices, norm definitions, convergence criteria, and computational limitations; clear articulation of assumptions regarding measurability, boundedness, and domain structure. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of analytic failures or divergence; acknowledging numerical instabilities; ensuring reproducibility; avoiding misleading interpretations of limited-domain sampling; maintaining rigor in limit, measure, and convergence claims. |