Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- The epsilon-delta definition of the limit of a real-valued function at a point of an interval, and the notation for it. Uniqueness of the limiting value is established separately.
- Fixes the standing objects and vocabulary of single-variable calculus on an interval: continuity, the derivative at an interior point, the Riemann integral, and the standard theorems about them, all carried by reference.
- Fixes the notation and background facts about the real numbers, intervals, continuity and sequences that the calculus items adopting it use throughout. It introduces no new concept and carries its background results by reference.
- Defines, from eight parameters, the data of a two-state controlled population of Ising type: the compact box of admissible transition rates, the rates themselves, the entropic running cost with its splitting interaction, the two observation channels, and the smooth extensions of…
Regimes of the Ising Population Model Outside the Fluctuation Optimality Theorem
remarkrem:ising-excluded-regimes-2026bProbabilityRecords which regimes of the Ising population model the fluctuation optimality theorem excludes and why: the ferromagnetic sign of the interaction breaks joint coercivity, a vanishing background observation rate breaks uniform observation positivity, and the convexity weight is a…The Ising Population Model: A Worked Instance of the Fluctuation Theory
exampleex:ising-population-model-2026aProbabilityA self-contained account of a two-state controlled population whose controls are the transition rates and whose observations are noisy per-state counts: its mean-field equilibrium, the scalar fluctuation problem it reduces to, and the closed-form asymptotically optimal value of i…The Asymptotically Optimal Value of the Recentred N-Agent Cost of the Ising Population Model
corollarycor:ising-asymptotic-value-2026aAnalysisProbabilityThe Ising population model satisfies every standing hypothesis of the fluctuation optimality theorem at its even-split equilibrium, so the asymptotically optimal value of its recentred N-agent cost is the explicit integral of the two scalar Riccati solutions.The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form
lemmalem:ising-riccati-families-2026aAnalysisProbabilityBoth Riccati families of the Ising equilibrium reduce to scalar equations along the tangential direction and are solved in closed form; this verifies the Riccati hypothesis of the completion-of-squares theorem and identifies the Kalman covariance and the two integrands of the flu…The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity
lemmalem:ising-fluctuation-lqg-data-2026aProbabilityLinear AlgebraComputes every matrix of the fluctuation linear-quadratic-Gaussian data at the even-split equilibrium of the Ising model in closed form, and shows that the fluctuation Hessian is positive definite with an explicit coercivity constant.The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal
lemmalem:ising-stationary-triple-2026aAnalysisProbabilityThe constant triple with an even population split, unit transition rates and vanishing co-state is a stationary mean-field triple whose control is the unique Hamiltonian minimiser and is globally optimal for the mean-field problem, and the first-order to-go regularity hypothesis…The Ising Population Model Instantiates the Data of the Fluctuation Theory
lemmalem:ising-data-well-posed-2026aAnalysisProbabilityThe Ising rates, observation rates and costs form an affine-controlled transition-rate family, an observation-rate family and population cost data with the stated smooth extensions, and satisfy the compactness, uniform observation positivity and Lipschitz cost hypotheses of the f…- Constructs a globally , convex, nonnegative function on the real line that coincides with the entropic rate cost near the rest rate 1, vanishes only there, and has bounded, Lipschitz first and second derivatives.
Asymptotically Optimal Value of a Set of Real Sequences
definitiondef:asymptotically-optimal-value-sequences-2026aAnalysisLet be a nonempty set of sequences of real numbers, indexed by the natural numbers , and let be a real number. We say that is an asymptotically optimal value of if (i) for every sequence in…Joint Measurability of the Tracked Events and of the Tracked Energy Density over the Block Cascade, and Measurability in Time of Their Expectations
lemmalem:tracked-energy-density-measurable-2026aAnalysisProbabilitySetting. Adopt the setting, notation, parameters and standing hypotheses of Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound, formed for one and the same data as there. In particular: the natural numbers ,…The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States
theoremthm:lqg-value-is-asymptotically-optimal-2026aAnalysisProbabilityCommon data. Adopt the common data and the standing hypotheses of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses, and the notation of…Injection Certificates on the Trimmed Synthetic Copy for the Family of N-Agent Solutions: the Law-Transported Van Trees Certificate Hypothesis Holds under Deterministic Initial States, Uniform Observation Positivity and a Observation Extension
lemmalem:van-trees-certificates-from-n-agent-solutions-2026aProbabilitySetting. Adopt the setting, notation and standing hypotheses of…Instantiation of the Trimmed-Copy Van Trees Data from an N-Agent Solution at the Power Scales: Construction of the Copy Side, Verification of the Copy Hypotheses, and Domination of Its Constants by the Scale-Set Majorants
lemmalem:van-trees-assembly-copy-instantiation-2026aProbabilityThe -agent side and the intermediate time. Adopt the two paragraphs The -agent side and The intermediate time of the setting of…Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data
lemmalem:van-trees-assembly-mean-field-data-2026aAnalysisData. Let , and be natural numbers, let be a nonempty subset of Euclidean space, let be a transition-rate family on states with control set and rate bound , let …The Scale Set for the Van Trees Assembly: Eventual Validity of the Copy-Side Constraints, Lower Bounds for the Chernoff Exponents, and Vanishing of the Error Majorants
lemmalem:van-trees-assembly-scale-set-2026aAnalysisData. Let , and be natural numbers, and let , , , , , , , , , , , , ,…- Let be the real numbers, let be the exponential function and let be the natural logarithm. For a real number and a real number , the real power of with exponent is