| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies geometric spaces defined by polynomial equations over fields, rings, or schemes; includes varieties, schemes, morphisms, divisors, cohomology, moduli, and birational geometry. Excludes arbitrary topological spaces or analytic manifolds unless endowed with algebraic structure. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates from local algebraic neighborhoods (specs of local rings) to global varieties and schemes; spans affine, projective, and higher-dimensional spaces; works over arbitrary base fields and rings. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Varieties, schemes, spectra of rings, morphisms, ideals, coordinate rings, divisors, line bundles, sheaves, cohomology classes, moduli points, function fields. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Dimension, smoothness/singularity, irreducibility, integrality, degree, cohomological invariants, birational type, local ring structure, positivity properties (ampleness, nefness). |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Affine varieties, projective varieties, schemes, morphisms, divisors, line bundles, coherent sheaves, moduli spaces, birational classes, function fields. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Polynomial coordinates, ideal generators, morphism components, divisor coefficients, local ring parameters, cohomology values, field characteristics. |
| | Parameterization | How variables encode and represent the system’s state. | Parameterized through coordinate charts (affine patches), projective coordinate systems, ideal generators, local trivializations of bundles, moduli parameters, and deformation families. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Assume algebraically closed fields; work in characteristic zero; idealize smooth varieties; ignore pathological singularities; use genericity assumptions; treat schemes of finite type over a field as manageable. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down in positive characteristic pathologies, non-Noetherian rings, wild singularities, non-separated schemes, or when cohomological finiteness fails. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes foundational commutative algebra, polynomial behavior, sheaf-theoretic structure, local–global principles, existence of spectra, and compatibility of morphisms with ring maps. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes polynomial equations fully encode geometry; schemes generalize varieties without loss of structure; cohomology reflects geometric properties; birational equivalence captures essential classification. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires compatibility between schemes, morphisms, sheaves, and cohomology; algebraic operations must reflect actual geometric transformations; coordinate changes must preserve structure. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment of local algebra with global geometry, sheaf behavior across open covers, cohomological data with geometric invariants, and ring–space duality across the entire categorical framework. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Zero-loci of polynomials, intersections, singularities, fiber behavior, divisor interactions, cohomology dimensions, deformation patterns under parameter changes. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Unable to detect transcendental structure; limits of polynomial-only representation; resolution issues near singularities; failure of schemes over non-Noetherian bases; inability to capture analytic geometry without additional structure. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Degree, dimension, multiplicity, intersection number, cohomological dimension, characteristic of base field, rank of bundles, valuations. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Coordinate rings, ideals, Gröbner bases, sheaf cohomology tools, resolution algorithms, valuation theory instruments, moduli-parameter techniques. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of varieties, schemes, morphisms, divisors, line bundles, cohomology groups, regular functions, singularities, moduli points. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Computing ideals; generating Gröbner bases; resolving singularities; calculating cohomology; determining divisor classes; computing intersections; forming fiber products; assembling affine covers. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Constructing affine/projective patches; gluing schemes; computing local rings; building deformation families; measuring fibers; establishing birational models. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling points within affine patches; sampling fibers of morphisms; selecting representative divisors; sampling charts of moduli spaces; testing local rings at various points. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Polynomial systems; ideals; coordinate rings; Gröbner bases; intersection tables; cohomology-class data; divisor coefficient lists; patchwise scheme charts. |
| | Resolution | The granularity or precision with which data is captured. | Controlled by polynomial degree; Gröbner-basis refinement; precision of symbolic computations; fineness of affine covers; granularity of divisor or cohomology decompositions. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying ideal membership; checking Gröbner basis correctness; confirming affine-patch gluing consistency; validating cohomology computations; checking intersection multiplicities. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Algebraic miscalculations; incorrect Gröbner bases; faulty singularity-resolution steps; misidentified divisors; cohomology miscounts; inconsistent scheme gluing; moduli misclassification. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Polynomial relations defining varieties; behavior of ideals under elimination; intersection theory rules; cohomological long exact sequences; functorial behavior under morphisms; Zariski closure laws. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Dimension, degree, genus, Kodaira dimension, Picard group, divisor class group, Chern classes, cohomology groups, birational invariants, numerical invariants (intersection numbers). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Schemes built from ring spectra; morphisms induced by ring homomorphisms; divisors generating line bundles; vanishing theorems driving cohomology behavior; birational contractions and blow-ups modifying geometry. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Resolution-of-singularities pathways; forming affine/projective covers; performing blow-ups; computing cohomology via Čech or derived functors; degeneration pathways in families of varieties; birational transformations. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Varieties, schemes, morphisms, divisors, line bundles, cohomology, moduli, birational equivalence, function fields, local rings, sheaves, blow-ups, fibers, base change. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Affine vs. projective vs. proper varieties; smooth vs. singular varieties; irreducible vs. reducible; rational, unirational, and general type varieties; moduli classifications (curves, surfaces, bundles). |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Polynomial equations; ideal descriptions; Gröbner basis relations; divisor and line-bundle relations; cohomology exact sequences; blow-up formulas; intersection-form equations. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Affine varieties, projective varieties, schemes of finite type, toric varieties, elliptic curves, higher-genus curves, surface models, moduli spaces, deformation families. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Smooth varieties; varieties over algebraically closed fields; constant-coefficient polynomial systems; models ignoring wild singularities; toric varieties as combinatorially idealized spaces. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Pathologies in positive characteristic; failure of resolution in general settings; non-Noetherian schemes; singularities too wild to resolve; cohomology non-finite; moduli non-separatedness. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Scheme theory; cohomological and derived frameworks; intersection theory; moduli theory; birational classification (Mori theory); functorial viewpoint via representable functors. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to number theory (Diophantine geometry), topology (cohomology and Chern classes), complex analysis (algebraic varieties as complex manifolds), mathematical physics (string theory, mirror symmetry), and combinatorics (toric geometry). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying defining polynomials, altering coefficients or base fields, modifying ideals, deforming varieties, introducing blow-ups/blow-downs, changing line bundles or divisors to test geometric behavior. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing geometric behavior without modifying formulas: tracking singularities, watching fibers under morphisms, monitoring cohomology dimensions, studying intersection behavior across natural families. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing smoothness via Jacobian criteria; checking dimension via Krull dimension; verifying ideal membership; validating cohomology computations; testing birational equivalence or moduli stability. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing Gröbner bases; reproducing intersection numbers using independent algorithms; validating local ring computations; checking cohomology across different covers; re-performing blow-up sequences. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical/geometric analogues: analyzing generic fiber behavior, cohomological vanishing patterns, frequency of singularities in families, distribution of divisor classes in moduli, stability of intersection numbers. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing schemes via birational type, cohomology, singularities, divisor theory, moduli-point behavior, polynomial complexity, or ideal structure; comparing different compactifications or models of the same object. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting incorrect Gröbner bases, faulty ideal computations, misidentified singularities, incorrect divisor intersections, cohomology miscounts, gluing inconsistencies between affine patches. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding bias toward smooth or simple varieties; ensuring fields and coefficients are not chosen to force desirable outcomes; preventing selective sampling of “nice” fibers; avoiding reliance on symmetric or toric examples alone. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Community verification of singularity resolutions, cohomology proofs, divisor-intersection calculations, moduli constructions, equivalence of birational models, and scheme-gluing arguments. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Revising classification of varieties; updating moduli definitions; refining birational geometry frameworks; adjusting cohomology theories; redefining divisor classes in response to new theorems or counterexamples. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of field assumptions, ideal-generation methods, Gröbner-basis algorithms, cohomology computation methods, divisorial conventions, and moduli-parameter definitions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Accurate reporting of computations; explicit acknowledgement of characteristic-dependent pathologies; proper attribution of geometric constructions; avoidance of overstating birational or cohomological claims. |