「theorem」の検索結果

Analyze prime numbers using primality tests, factorization algorithms, prime distribution analysis, and sieve methods. Covers trial division, Miller-Rabin, Sieve of Eratosthenes, and the Prime Number Theorem. Use when determining whether an integer is prime or composite, finding prime factorizations, counting or listing primes up to a bound, or investigating prime properties within a number-theoretic proof or computation.

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Use when a task needs the judgment of a Database Architect — choosing a normalization level matched to the actual workload, resolving a CAP theorem tradeoff (consistency vs. availability) for a distributed data store, selecting a sharding/partition key aligned with the dominant query pattern, validating an indexing strategy against real query plans, or deriving RPO/RTO backup and replication requirements from actual business impact.

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Audit an existing mathematical, logical, algorithmic, or technical proof from its stated conclusion backward as a bounded AND/OR graph of exact obligations. Use to reconstruct conclusion dependencies, determine whether a proof-supplied route establishes the conclusion, or expose hidden assumptions, unmet side conditions, cycles, overly strong steps, or silent route switches. For a source-mapped Lamport rendering use convert-lamport, and for forward hierarchy, construct, and scope conformance use forward-lamport before this skill. This does not decide full theorem-level entailment by searching for a different proof; do not turn the audit into a linear reversed proof, exhaustive proof search, or replacement proof.

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Audit an existing Lamport-style hierarchical mathematical proof forward for validity and whether its hierarchy, scope, dependencies, side conditions, and proof constructs discharge the stated theorem. Use when the central question concerns a top-down proof with numbered levels, subproofs, ASSUME/PROVE, SUFFICES, PICK, CASE, DEFINE, Q.E.D., or prose equivalents. For an ordinary proof that first needs source-preserving hierarchy, use convert-lamport; for conclusion-first AND/OR obligation reconstruction, use reverse-lamport. Do not use to generate or replace a proof, or to audit Lean, Coq, Isabelle, or other proof-assistant source.

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Test-to-requirement coverage mapping. Reads plan acceptance criteria and test files, produces a coverage gap report showing which requirements have no tests. Use during Phase 2 (Test) or Phase 4 (Review) to verify test completeness. Named after the Nyquist sampling theorem — sufficient test coverage prevents aliased (missed) requirements. NOT for running tests (see mk:testing); NOT for structural code review (see mk:review).

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Applies game theory to multi-agent coordination protocols with advisory claims, repeated interactions, daemon-mediated correlation, and deviation penalties. Use for Nash equilibrium checks, folk-theorem cooperation analysis, correlated-equilibrium routing, and price-of-anarchy audits in strategic agent systems. NOT for consumer pricing, card-game strategy, or general microeconomics outside agent coordination.

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Expert-level formal methods covering formal specification, model checking, theorem proving, type theory, program verification, and formal languages.

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Expert-level circuit analysis covering KVL, KCL, nodal and mesh analysis, Thevenin and Norton theorems, AC circuits, phasors, and frequency response.

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Expert-level calculus knowledge. Use when working with limits, derivatives, integrals, multivariable calculus, vector calculus, differential equations, series, or optimization. Also use when the user mentions 'derivative', 'integral', 'limit', 'chain rule', 'gradient', 'divergence', 'curl', 'Taylor series', 'Fourier series', 'partial derivative', 'double integral', 'line integral', 'Stokes theorem', or 'Lagrange multipliers'.

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Expert-level probability theory knowledge. Use when working with probability spaces, random variables, distributions, expectation, variance, conditional probability, stochastic processes, Markov chains, martingales, or measure-theoretic probability. Also use when the user mentions 'sample space', 'random variable', 'expected value', 'variance', 'conditional probability', 'independence', 'Markov chain', 'Poisson process', 'moment generating function', 'law of large numbers', 'central limit theorem', or 'stochastic process'.

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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.

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Expert-level condensed matter physics knowledge. Use when working with crystal structure, band theory, semiconductors, superconductivity, magnetism, phase transitions, Fermi liquids, topological materials, or strongly correlated systems. Also use when the user mentions 'band gap', 'Fermi energy', 'semiconductor', 'superconductor', 'phonon', 'crystal lattice', 'Brillouin zone', 'Bloch theorem', 'Hall effect', 'magnetism', 'phase transition', or 'topological insulator'.

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Create a study cheatsheet by extracting and synthesizing key concepts from lecture materials, homework, and past exams. Use this skill whenever a user asks for help making a cheatsheet, study sheet, formula sheet, exam reference sheet, or "one-pager" — especially when they upload lecture slides (PDF/PPTX/DOCX), problem sets, past exams, or scanned/handwritten notes. Trigger this even for casual phrasings like "help me prep for my final," "summarize my course materials into something I can bring to the exam," or "make me a study reference from these slides and homeworks." Tuned for math/science (formulas, proofs, theorems) and CS/engineering (code, algorithms). Processes one file at a time and persists intermediate summaries to disk to avoid blowing past the context window on large courses.

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Automated dimensional analysis — Buckingham Pi theorem, non-dimensionalization, unit validation with pint, and characteristic scale estimation. Use before any physics computation to verify consistency and reduce parameter space.

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Discover conserved quantities and symmetries from trajectory data. Identifies energy, momentum, angular momentum, and custom invariants using neural networks and symbolic methods. Inspired by Noether's theorem.

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Refute-before-prove research discipline: automated counterexample sweeps before any formalization, meter-integrity audits for information-bound experiments, mutation-tested model checkers, pre-registered commit-or-cut gates for speculative directions, wrong-turn reporting, and numeric-claim provenance. Use when validating any new theorem, bound, equilibrium, or security claim; when building or trusting a checker, simulator, or experiment; or when deciding whether a speculative research direction lives or dies. NOT for exploratory brainstorming, product prioritization, writing style, or literature review.

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