# Methods & Domains This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. [Catalogue presentation and source-binding data](https://ocme.evemisslab.com/data/i18n/catalogue.en.json) ## Domain registry (classification is not complete coverage) - Foundations and logic (foundations_logic) — The foundational structures of mathematical language, axioms, sets, types, provability and formal reasoning. - Arithmetic and number systems (arithmetic_number_systems) — Numbers, operations, extensions of number systems and basic quantitative structures. - Algebra (algebra) — Operational structures, equations, symmetry and algebraic systems. - Geometry (geometry) — Shape, space, distance, angle and geometric structures. - Topology (topology) — Continuity, neighborhoods, convergence and properties preserved under continuous deformation. - Analysis (analysis) — Limits, continuity, differentiation, integration and infinite processes. - Number theory (number_theory) — Integers and their arithmetic structures. - Combinatorics and discrete mathematics (combinatorics_discrete) — Finite or countable structures, counting, graphs and discrete relations. - Probability and statistics (probability_statistics) — Randomness, distributions, inference and uncertainty in data. - Dynamical systems (dynamical_systems) — Systems whose states evolve through time or iteration. - Differential equations (differential_equations) — Relations and evolution described by derivatives. - Numerical and computational mathematics (numerical_computational) — Approximation algorithms, error, stability and computational implementations. - Optimization and operations research (optimization_operations_research) — Finding the best decisions under constraints. - Mathematical physics (mathematical_physics) — Mathematical structures and methods in physical theories. - Mathematical biology (mathematical_biology) — Mathematical models of living systems. - Mathematics of economics and finance (economics_finance) — Mathematical models of economic, risk and financial systems. - Mathematics of computer science (computer_science_mathematics) — Mathematical foundations of algorithms, computation, information and formal languages. - Mathematics education (mathematics_education) — Mathematical learning, teaching, assessment and curricula. - History, philosophy and methodology (history_philosophy_methodology) — The history of mathematical ideas, standards of validity and problem-solving methods. - Formalized mathematics (formalized_mathematics) — Expressing, checking and replaying mathematics with proof assistants and formal systems. ## Methods & Domains ## Direct proof Derive the conclusion directly from known conditions through valid reasoning. ## When to use it - The conclusion follows step by step from definitions or known theorems. ## Basic steps - Expand the conditions. - Select relevant definitions or theorems. - Derive the conclusion step by step. ## Common failures - Silently using an unproved proposition. ## Proof by contradiction Assume the conclusion fails and derive a contradiction. ## When to use it - Negating the conclusion makes a conflict with known conditions easier to obtain. ## Basic steps - Negate the goal. - Derive consequences. - Obtain a contradiction. ## Common failures - The contradiction is unrelated to the original assumption. ## Mathematical induction Establish an infinite family of propositions through base cases and an inductive step. ## When to use it - Propositions are organized by natural numbers or a recursive structure. ## Basic steps - Prove the base case. - State the inductive hypothesis. - Prove the next step. ## Common failures - Using the inductive hypothesis outside its permitted scope. ## Construction Explicitly build a mathematical object satisfying the conditions. ## When to use it - An existence problem asks for a concrete object. ## Basic steps - Analyze the constraints. - Propose a candidate construction. - Verify every condition. ## Common failures - Showing an example without covering the general case. ## Invariant Find a quantity preserved by an operation or evolution. ## When to use it - States change repeatedly while some properties may be preserved. ## Basic steps - Define the states. - Propose a candidate quantity. - Prove preservation. - Compare the initial and final states. ## Common failures - The proposed quantity is not actually preserved. ## Symmetry Use equivalence under transformations to reduce the cases of a problem. ## When to use it - Several cases are equivalent under a transformation. ## Basic steps - Identify the transformation. - Prove the problem is preserved. - Reduce to representative cases. ## Common failures - Mistaking approximate symmetry for exact symmetry. ## Extremal principle Choose a largest, smallest or earliest object to force a structural conclusion. ## When to use it - An extreme element exists in a finite set or a well-ordered structure. ## Basic steps - Choose an extreme object. - Analyze its constraints. - Derive the conclusion. ## Common failures - The extremum does not exist or its attainment has not been established. ## Pigeonhole principle Infer a collision when there are more objects than containers. ## When to use it - The problem has finitely many categories and an imbalance in counts. ## Basic steps - Define the objects and containers. - Compare their counts. - Infer a collision. ## Common failures - The container categories are not mutually exclusive. ## Diagonalization Construct an object differing from every listed item in at least one position. ## When to use it - The goal is non-enumerability or escaping a given family. ## Basic steps - Assume an enumeration. - Choose a difference for each item. - Construct the new object. ## Common failures - The new object is outside the permitted space. ## Compactness Reduce an infinite or global problem to finite or convergent data. ## When to use it - Local information is needed to obtain a global conclusion. ## Basic steps - Choose a compact structure. - Extract finite or convergent data. - Transfer the property. ## Common failures - The compactness hypotheses are not satisfied. ## Localization Study local data first, then examine how it can be glued into a global result. ## When to use it - Local behavior can control the global problem. ## Basic steps - Choose a local scale. - Prove the local property. - Check the gluing. ## Common failures - Local solutions cannot be glued globally. ## Linearization Replace a nonlinear problem by a linear one at a local or approximation level. ## When to use it - First-order local behavior dominates. ## Basic steps - Choose an expansion point. - Take the linear principal part. - Estimate the remainder. ## Common failures - Ignoring the range in which the remainder estimate applies. ## Coordinates Translate geometric or abstract relations into coordinates and algebraic expressions. ## When to use it - Convenient coordinates can be chosen for the geometric relations. ## Basic steps - Choose a coordinate system. - Translate the objects. - Work algebraically. - Translate the meaning back. ## Common failures - The coordinate choice introduces a degeneracy. ## Generating function Encode sequence or combinatorial data as a formal power series. ## When to use it - A recurrence, counting problem or convolution structure is present. ## Basic steps - Construct the generating function. - Transform the relations. - Solve the relations. - Extract coefficients. ## Common failures - Confusing convergence with formal operations. ## Fourier methods Decompose functions into frequency components to analyze operations and structure. ## When to use it - Translation, periodicity or convolution structure is present. ## Basic steps - Choose a transform. - Move to the frequency domain. - Handle frequency components. - Apply the inverse transform. ## Common failures - Ignoring function-space and convergence hypotheses. ## Probabilistic method Use random selection to prove existence of a structure or estimate its properties. ## When to use it - Direct construction is difficult but average behavior can be controlled. ## Basic steps - Define a random model. - Estimate probabilities or expectations. - Infer existence. ## Common failures - The random model does not cover the target objects. ## Numerical approximation Use finite-precision algorithms to approximate a continuous or infinite problem. ## When to use it - An analytic solution is unavailable or too costly. ## Basic steps - Choose a discretization. - Analyze the error. - Perform the computation. - Check stability. ## Common failures - Treating an approximation as an exact proof. ## Computational experiment Use programs to generate cases and explore patterns or candidate counterexamples. ## When to use it - Many cases are needed or manual exploration is difficult. ## Basic steps - Formalize the computable problem. - Fix the environment. - Execute and record. - Analyze the limitations. ## Common failures - Presenting finite testing as a universal proof. ## Counterexample search Systematically search an explicit domain for an input that falsifies a proposition. ## When to use it - The proposition can be checked on a finite or sampled domain. ## Basic steps - Define the declared domain. - Build the decision procedure. - Execute the search. - Preserve the limitations. ## Common failures - Treating not finding a counterexample as a proof. ## Formal verification Use a kernel checker in an explicit formal system to verify an exact statement. ## When to use it - A machine-replayable universally quantified statement is needed. ## Basic steps - Define the formal model. - Build the semantic bridge. - Compile the proof. - Reject placeholders. ## Common failures - The formal statement and encyclopedia semantics are not fully aligned.