Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- A random vector in R^d is a measurable map from a probability space to R^d, with coordinates; its law is the image of the probability measure under it, a Borel probability measure on R^d.
Probability Measures on Euclidean Space and Random Vectors: Standing Notation
settingset:probability-measures-euclidean-2026aAnalysisProbabilityStanding notation for probability measures on Euclidean space and random vectors: Borel maps, the set of probability measures and their integrals, push-forwards, pairs of points with coordinate projections and product measures, and a probability space with expectations.Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets
lemmalem:euclidean-pairs-borel-toolkit-2026aAnalysisProbabilityToolkit for pairs of Euclidean points: the coordinate projections of a concatenated space are Borel, pairings of measurable maps are measurable, the product of two Borel probability measures transports to a probability measure on the concatenated space with the given marginals, t…What the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise Models, and What Is Not Proved
remarkrem:white-noise-heat-equation-torus-2026bAnalysisProbabilityPDEReads the well-posedness corollary for the white-noise heat equation as the dynamic programming equation of a controlled stochastic heat equation on the torus, explains why the noise is square-integrable white noise and why the equation lives on a negative-order Sobolev space, an…- 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…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…The Van Trees Data of the Trimmed Synthetic Copy Supplied by an N-Agent Solution: Transport to the Record Support, Regularity of the Smoothed Joint Density, the Information Bound in the Injection Direction, and the Pairing of the Cell Coefficients with the Profile Response
lemmalem:copy-van-trees-data-from-n-agent-solution-2026aProbabilitySetting. Adopt the setting, notation, hypothesis (L) and conventions of…The Closeness, Fourth-Moment and Discrepancy Hypotheses of the Estimand Assembly Lemma Supplied by an N-Agent Solution under the Cost Bound: Control-Lipschitz Bound on the Record Discrepancy, Close Records from the Path-Closeness Event, and the Explicit Mean-Square Bound
lemmalem:copy-estimand-assembly-hypotheses-from-n-agent-solution-2026aProbabilityThe -agent side. Adopt the setting, notation and standing hypotheses of…The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy
lemmalem:observation-centred-fluctuation-copy-transport-2026aProbabilityData. Adopt the Data paragraph of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set (its comparison data and its path data are not used here): natural numbers , , and…Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum
lemmalem:copy-clock-discrepancy-cell-count-moments-2026aProbabilityAdopt the setting and notation of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (as also adopted by…