| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies vector spaces and linear transformations over fields or rings. Includes matrices, determinants, eigenvalues/eigenvectors, inner products, orthogonality, linear systems, bases, dimension, diagonalization, spectral theory, and canonical forms. Excludes nonlinear transformations except when linearized; excludes general module theory except when specializing to vector spaces. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at algebraic, geometric, and computational scales: finite-dimensional and infinite-dimensional spaces; coordinate systems; linear mappings; matrix representations; continuous vs. discrete vector structures. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Vectors, scalars, matrices, linear maps, bases, subspaces, eigenvalues, eigenvectors, linear systems, inner products, orthonormal sets, projections, canonical forms. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Linearity (additivity + scalar multiplication), dimension, independence, spanning, invertibility, orthogonality, norms, rank, determinant, spectral properties, singularity, stability under linear combinations. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Finite-dimensional vs infinite-dimensional vector spaces; Euclidean vs abstract vector spaces; inner-product spaces; normed spaces; orthogonal/unitary spaces; subspaces; direct sums; column/row spaces; eigenspaces; matrix algebras. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Current vector coordinates; selected basis; matrix entries; rank; determinant; eigenvalue/eigenvector selection; projection coefficients; decomposition parameters (SVD, QR, Jordan). |
| | Parameterization | How variables encode and represent the system’s state. | Encoding via coordinate systems, matrices, bases, transformation matrices, Gram–Schmidt orthogonalization, spectral decompositions, change-of-basis matrices, block decompositions. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Assuming finite-dimensional contexts; treating bases as orthonormal; ignoring numerical instability; assuming exact arithmetic; focusing on diagonalizable matrices; treating infinite-dimensional spaces via finite truncations. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down in infinite-dimensional settings; numerical instability affects orthogonality; non-diagonalizability requires Jordan form; approximate methods fail for ill-conditioned systems; exact arithmetic assumptions break in floating-point. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Vector addition and scalar multiplication obey axioms; linear mappings preserve structure; basis existence; dimension well-defined; inner-product axioms hold when introduced; matrix multiplication consistently encodes linear composition. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes existence of bases (requires choice in general); assumes linear approximations meaningfully describe phenomena; assumes fields allow solutions of linear systems; assumes coordinate representation is faithful. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Linear maps must preserve addition and scalar multiplication; matrix representations must agree with chosen bases; eigenstructures must align with characteristic polynomials; orthogonality must match inner-product definition; decomposition theorems must interoperate correctly. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony between vector-space axioms, matrix algebra, inner-product structures, spectral theory, coordinate geometry, and computational linear algebra (algorithms, stability). |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Behavior of vectors under linear transformation; row/column dependencies; solvability of linear systems; matrix rank changes; eigenvalue/eigenvector structure; orthogonality patterns; projection behavior; determinant changes under operations; stability of decompositions (QR, SVD). |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by numerical precision; inability to compute exact eigenvalues for large matrices; instability in near-singular systems; inability to directly observe infinite-dimensional structure; limitations of floating-point arithmetic; ill-conditioning hiding true rank. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Vector norms; matrix norms; determinant magnitude; rank value; condition number; eigenvalue magnitude; angle between vectors; projection coefficients; singular values; numerical error bounds. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Gaussian elimination solvers; LU/QR/SVD decomposition tools; eigenvalue algorithms; Gram–Schmidt orthogonalizers; matrix calculators; numerical linear algebra libraries (LAPACK, BLAS); symbolic algebra systems (Mathematica, Maple). |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Rank defined via dimension of row or column space; determinant defined via multilinear alternating form; eigenvalues/eigenvectors defined via solving (Ax=\lambda x); orthogonality defined via inner product; projection defined via orthogonal decomposition; condition number defined via operator norm ratios. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Row reduction; computing determinants; solving linear systems; performing decompositions (QR, LU, SVD); computing eigenvalues/eigenvectors; orthogonalizing bases; projecting vectors; computing pseudoinverses; estimating condition numbers. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Standardized matrix-generation methods; controlled sampling of random matrices; structured eigenvalue experiments; systematic basis construction; repeated row-reduction tests; consistent use of pivot strategies; performing decompositions under fixed tolerances. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling random vectors; sampling matrices from various distributions (uniform, Gaussian); selecting subspaces; sampling eigenvalue problems; sampling ill-conditioned or sparse matrices; sampling orthonormal sets. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Matrices; vectors; row-reduced echelon forms; eigenvalue/eigenvector lists; decomposition outputs (U, Σ, V*); projection vectors; pseudoinverses; residuals for linear systems; condition number tables. |
| | Resolution | The granularity or precision with which data is captured. | Determined by precision of floating-point arithmetic; granularity of decomposition outputs; stability of eigenvalue approximations; tolerance thresholds in algorithms; resolution limits in detecting near-linear dependence. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Validating row-reduction results across algorithms; cross-checking eigenvalue computations; verifying decompositions by reconstruction; comparing projections under different bases; checking stability via perturbation tests; ensuring norm and inner-product accuracy. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Numerical rounding errors; pivoting instabilities; loss of orthogonality in Gram–Schmidt; incorrect rank detection; eigenvalue drift; decomposition inaccuracies; sensitivity to conditioning; algorithmic failures on singular or near-singular matrices. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Linearity (superposition principle); dimension laws; rank–nullity theorem; eigenvalue–eigenvector relationships; orthogonality relations; invariance of determinant under row operations (up to sign); spectral decomposition laws; canonical-form relations (Jordan, diagonalization). |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Dimension; rank; determinant (up to units); eigenvalues; singular values; norms; orthogonality; trace; characteristic/minimal polynomials; invariants under similarity transformations; subspace dimensions; condition number (in numerical settings). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Linear transformations inducing predictable geometric effects (rotation, reflection, projection, scaling); matrix multiplication encoding composition; eigenvalue mechanisms governing long-term behavior; orthogonalization via Gram–Schmidt; decomposition mechanisms revealing structure (QR, LU, SVD). |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Matrix → row reduction → solution/x-space classification; matrix → eigenvalue computation → diagonalization or Jordan form; vector set → Gram–Schmidt → orthonormal basis; transformation → decomposition → analysis (e.g., A = UΣV* in SVD). |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Vector, scalar, basis, dimension, linear independence, span, subspace, rank, kernel, image, determinant, eigenvalue, eigenvector, projection, norm, inner product, linear operator, similarity, orthonormal set, decomposition. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Finite vs infinite dimensional spaces; Euclidean vs general inner-product spaces; diagonalizable vs non-diagonalizable operators; symmetric/Hermitian vs general matrices; normal vs non-normal matrices; singular vs nonsingular operators; orthogonal/unitary transformations. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Ax = b; A = PDP⁻¹ (diagonalization); A = PJP⁻¹ (Jordan form); A = QR; A = UΣV* (SVD); det(A) formulas; rank–nullity: dim(V)=rank(A)+nullity(A); projection: proj_u(v)=((v·u)/(u·u))u; eigenvalue equation: Av=λv. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Matrix models of linear transformations; geometric models (vectors in ℝⁿ); decomposition diagrams; coordinate-change diagrams; subspace lattices; operator-spectral models; row-reduced echelon form as structural representation. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Orthogonal bases; diagonalizable matrices; truncated infinite-dimensional settings; idealized exact arithmetic; symmetric/Hermitian operator assumptions; simplified canonical forms; ignoring conditioning. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Idealizations fail in defective matrices (non-diagonalizable); infinite dimensions require functional analysis; numerical instability breaks orthogonality; ill-conditioned systems distort decomposition accuracy; non-normal matrices violate many geometric intuitions. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Spectral theory unifying operators and decompositions; functional calculus linking matrices to polynomials; singular value decomposition connecting geometry and analysis; duality theory (row/column spaces); unification with multilinear algebra via tensor spaces. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Physics (quantum operators); engineering (signal processing via SVD, PCA); computer science (machine learning, optimization, numerical algorithms); statistics (covariance matrices, regression); geometry (transformations, projections); economics (input–output models). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating matrices (changing entries, sparsity, conditioning); altering bases; varying decomposition methods (QR, LU, SVD); modifying norms; perturbing linear systems; testing effects of similarity transforms; adjusting vector sets to test independence or orthogonality. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural behavior of transformations under fixed matrices; monitoring stability of eigenvalues; observing rank changes under row operations; watching projections evolve; tracking norms and angles; monitoring convergence of iterative solvers without altering the underlying system. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing linear independence; verifying that transformations are linear; checking rank–nullity relationships; validating orthogonality; testing diagonalizability; checking decomposition correctness (A=UΣV*, A=PJP⁻¹, A=QR); validating numerical solutions by residual norms. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing decompositions under different algorithms; repeating row-reduction; verifying eigenvalue computations via multiple methods; performing the same projection under different bases; recomputing condition numbers under varied perturbations. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing distributions of singular values across random matrices; evaluating average conditioning; assessing error growth in numerical solutions; comparing convergence rates of iterative solvers; analyzing stability across matrix families. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing dense vs sparse matrix models; contrasting numerical vs symbolic methods; comparing decomposition methods (QR vs SVD vs LU); comparing eigenvalue algorithms; contrasting different norm-induced geometries; comparing bases and coordinate systems. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying rounding errors; detecting breakdowns in orthogonality; diagnosing rank misidentification; quantifying residuals in linear systems; identifying instability in eigenvalue computations; tracking algorithmic drift in iterative solvers; diagnosing ill-conditioning. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding biased sampling of “nice” matrices; ensuring diverse conditioning levels; preventing dependence on a single numerical library; controlling for basis choice; balancing dense vs sparse examples; avoiding overfitting of observations to well-behaved matrices. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing correctness of decompositions; auditing eigenvalue/eigenvector results; cross-checking numerical conclusions across libraries; re-evaluating rank and independence claims; validating projections; reviewing perturbation analyses. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating decomposition methods; refining stability assumptions; modifying numerical algorithms; revising classification of matrices (normal, orthogonal, diagonalizable) based on new findings; adjusting tolerance thresholds; integrating improvements in numerical conditioning theory. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of matrix formats, algorithms used, tolerance parameters, pivot strategies, decomposition settings, norm choices, and floating-point limitations; clear statement of numerical uncertainty. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest error reporting; avoiding concealment of instability; ensuring reproducibility; clearly stating algorithmic limitations; acknowledging pathological matrices; maintaining rigor in linear-algebraic claims. |