| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies integers using classical arithmetic methods: primes, divisibility, modular arithmetic, congruences, arithmetic functions, Diophantine equations. Excludes analytic, algebraic, or geometric number theory unless tools reduce to elementary methods. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the discrete, integer-valued scale: divisibility structure, modular reduction, prime factorization, residue classes, integer sequences, basic Diophantine solutions. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Integers, primes, composite numbers, divisors, residues, congruence classes, arithmetic functions (φ, μ, τ, σ), integer sequences, Diophantine solutions. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Divisibility, primality, gcd/lcm structure, modular equivalence, multiplicative structure, order mod n, residue behavior, parity, factorization uniqueness. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Prime/composite numbers, congruence classes, multiplicative functions, arithmetic progressions, Diophantine types, residue systems. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Integer values, modulus values, residues, gcd/lcm values, arithmetic-function outputs, exponents in factorization, congruence parameters, Diophantine coefficients. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded by base modulus, prime-power decompositions, congruence parameters, divisor structure, arithmetic-function values, Diophantine parameter sets. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Assuming unique factorization in ℤ; assuming small moduli for computation; treating primes as “atomic units”; ignoring analytic distribution of primes; restricting Diophantine equations to manageable forms. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Breakdown in non-UFD rings; limitations in detecting large prime factors; failure of simple modular tools for higher Diophantine problems; difficulty with huge moduli or computationally hard factorizations. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes classical axioms of arithmetic; uniqueness of prime factorization; predictable modular behavior; well-defined arithmetic functions; existence of solutions governed by integer structure. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes integers carry enough structure for classification; congruence relations behave uniformly; modular equivalence preserves arithmetic behavior; prime-factorization structure underlies all arithmetic. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Modular arithmetic must agree with integer arithmetic; arithmetic functions must respect multiplicativity; divisibility rules must align with factorization; congruence properties must remain invariant. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment between gcd/lcm structure, modular relations, factorization, arithmetic functions, and Diophantine solvability; all must integrate into a coherent integer-based framework. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Prime occurrence, divisibility patterns, modular residues, parity behavior, periodicity in congruences, factorization structure, arithmetic-function behavior (φ, μ, σ, τ), basic Diophantine solvability. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Cannot detect analytic distribution of primes; sequences may hide deep structure; modular arithmetic cannot reveal hidden factorization; large-number behavior obscured by computational limits; undecidable Diophantine problems. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Integer size, modulus value, prime factor exponents, gcd/lcm values, residue classes, arithmetic-function outputs, Diophantine parameter counts. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Euclidean algorithm, modular arithmetic tools, divisibility tests, sieve methods (elementary forms), arithmetic-function computation, basic Diophantine techniques, prime-finding algorithms. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Divisibility, gcd/lcm, primes, congruence, residue class, multiplicative functions, modular inverse, Diophantine equation, Euler’s totient function, Möbius function. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Computing gcd/lcm; performing modular reduction; solving congruences; computing arithmetic functions; doing prime factorization (for small integers); constructing solutions to simple Diophantine equations. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Generating integer sequences; computing modular tables; constructing multiplication/addition tables mod n; collecting divisors; generating factorizations; testing Diophantine forms against sample values. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling integers from intervals; selecting moduli; sampling residue classes; sampling primes; sampling arithmetic-function values; sampling simple Diophantine solutions. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Integer lists, modular tables, divisor sets, factorization trees, arithmetic-function tables, residue-system listings, sample Diophantine-solution sets. |
| | Resolution | The granularity or precision with which data is captured. | Resolution depends on modulus size, integer range, factorization completeness, precision of arithmetic-function tables, and granularity of integer sampling. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying gcd computations; cross-checking modular reductions; checking arithmetic-function consistency; validating factorization results; confirming congruence solutions. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Computational overflow; incorrect gcd/lcm; modular-reduction mistakes; misfactorizations; parity errors; miscomputed arithmetic functions; false positive/negative Diophantine solutions. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Divisibility laws, modular-arithmetic relations, congruence behavior, prime-distribution local patterns, multiplicative-function laws, recurrence relations in integer sequences, Chinese Remainder patterns. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | gcd, lcm, residue class, parity, multiplicativity (for μ, φ, σ, τ), order mod n, quadratic residues, invariant Diophantine structures (e.g., invariant under congruence substitutions). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Euclidean algorithm driving gcd structure; modular reduction shaping congruence classes; factorization mechanisms determining arithmetic function outputs; Diophantine substitution patterns determining solvability. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | gcd reduction pathways; prime-factorization chains; modular-lifting pathways; descent arguments in Diophantine equations; iterative congruence solving; divisor-chain pathways. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Prime, composite, gcd, lcm, congruence, residue, modular inverse, Euler’s φ, Möbius μ, divisor function σ, quadratic residue, Diophantine equation, unit mod n. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Primes vs composites; residue classes mod n; multiplicative vs non-multiplicative functions; solvable vs unsolvable congruences; linear vs quadratic vs higher Diophantine forms; primitive roots vs nonexistence. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Congruence equations; gcd/lcm identities; Euler’s theorem; Fermat’s little theorem; Chinese Remainder isomorphisms; Pell-type equations; recurrence equations; parity identities. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Modular-arithmetic models; divisor-lattice models; recurrence-sequence models; Diophantine-solution sets; residue-system diagrams; prime-factorization trees. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Integers treated as atomic factorization units; simple congruence classes; prime-power moduli; reduced divisor lattices; idealized Diophantine forms (linear or Pell-type). |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Failures in non-unique factorization domains; Diophantine undecidability; computational hardness of factoring; breakdown of simple congruence techniques for higher-degree equations; limits of primality testing. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Modular arithmetic as a unifier; factorization + multiplicative functions; Diophantine frameworks; residue theory; inversion formulas (Möbius inversion); elementary group-theoretic structure in ℤ/nℤ. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to algebra (rings, groups), cryptography (modular arithmetic, RSA), combinatorics (integer partitions), coding theory (residues), dynamical systems (recurrence sequences), and logic (Diophantine decidability). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying congruence conditions, modifying moduli, altering factorization inputs, adjusting Diophantine parameters, and manipulating arithmetic functions to test arithmetic behavior under controlled integer changes. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural integer behavior: monitoring primality patterns, tracking modular residues, examining divisibility trends, observing arithmetic-function fluctuations, studying Diophantine solvability without altering inputs. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing congruences; verifying gcd/lcm identities; checking multiplicativity of arithmetic functions; validating Diophantine solutions; testing primality; verifying modular inverses and cyclic-group orders. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing gcd/lcm; repeating congruence checks with alternative moduli; reproducing primality tests; re-evaluating arithmetic-function values; rediscovering Diophantine solutions under varied samples. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: analyzing residue frequency distributions; comparing arithmetic-function growth; evaluating divisibility-pattern stability; assessing Diophantine solvability trends; comparing factorization depths. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing integer behaviors under different moduli; comparing factorization patterns; evaluating competing Diophantine formulations; comparing residue-class distributions; assessing efficiency of arithmetic algorithms. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Miscomputed gcd/lcm; incorrect modular reductions; errors in primality checks; incorrect factorization; mis-evaluated arithmetic functions; false Diophantine solutions; integer overflow or precision errors. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding biased integer sampling; preventing reliance on small moduli only; avoiding selective factorization cases; ensuring random or representative sampling for Diophantine tests; preventing algorithm-dependent distortions. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Independent review of congruence proofs, factorization methods, arithmetic-function identities, and Diophantine arguments; verification of primality and modular-arithmetic claims. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating methods for solving congruences; refining factorization approaches; modifying arithmetic-function definitions; revising Diophantine frameworks; adjusting integer-sampling strategies. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of integer ranges, modulus choices, factorization methods, primality-testing algorithms, arithmetic-function conventions, and Diophantine-solving assumptions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of computational limits; no concealment of counterexamples; proper attribution of classical theorems; avoidance of overstating results derived from incomplete factorizations or unverifiable Diophantine trials. |