Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Gaussian Smoothing of a Finitely Supported Probability Mass Function: Density Bounds and Control of the Directional Score Integral by the Move Information
lemmalem:discrete-smoothing-score-bound-2026aAnalysisProbabilityStatisticsLet be a natural number, let be a real number with , and let , , , and (for ) be as in…Move Score of a Discrete Probability Mass Function
definitiondef:discrete-move-score-2026aProbabilityStatisticsLet and be natural numbers, let be a discrete probability mass function on Euclidean space , and put . Let be a family of points of (the moves) and let …Discrete Probability Mass Function on Euclidean Space
definitiondef:discrete-pmf-euclidean-2026aProbabilityLet be a natural number and let be Euclidean space. A discrete probability mass function on is a function whose sum over isThe Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder
lemmalem:gaussian-smoothing-weight-2026aAnalysisProbabilityLet be a natural number, let be Euclidean space with the Euclidean norm , the dot product , and the Euclidean distance , and let be Lebesgue measure on the Borel -algebra . Let…The Empirical State Measure Deviates from the Realized Mean-Field Flow by at Most the Noise Majorant
lemmalem:realized-flow-deviation-majorant-2026aProbabilityAdopt the setting, hypotheses and notation of the pre-stopping envelope lemma: in particular the number of agents and the number of states (both natural numbers), the horizon , the solution of the controlled -agent dynamics with regular event and empiri…Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates
theoremthm:n-agent-cost-lqg-lower-bound-2026cAnalysisProbabilityAdopt the common data, the family of solutions, the standing hypotheses, hypotheses (I), (I) and (CB), the notion of an admissible parameter vector, and the real number of the asymptotic lower bound theorem: in particular the stationary mean-field triple , the…Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate
lemmalem:filtering-certificate-bound-2026cProbabilityStatisticsLet be a probability space, let be a sub--algebra of , let be a natural number, and let be a tuple of square-integrable random variables on it. For each f…Directional Form of the Multivariate van Trees Inequality
corollarycor:directional-van-trees-2026aProbabilityStatisticsAdopt the setting, notation and hypotheses (i)--(iv) of the multivariate van Trees inequality: the probability space , the natural number , the measurable space with its -finite measure , the square-integrable random v…Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses
theoremthm:n-agent-cost-lower-bound-2026aAnalysisProbabilityCommon data. Fix an affine-controlled transition-rate family on states with compact convex control set and control bound , its transition-rate family with rate bound , an observation-rate family…Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound
lemmalem:n-agent-cascade-ledger-2026aAnalysisProbabilityAdopt the setting, notation, parameters and standing hypotheses of the tracked energy bound lemma, formed for one and the same data as there. In particular: the affine-controlled transition-rate family on states with compact convex control set…Observation-Measurability of the Cascade Good Sets and the Restricted Filtering Bound
lemmalem:fluctuation-cascade-filtering-2026aProbabilityAdopt the setting and notation of the block cascade lemma — the solution of the controlled -agent dynamics with regular event , empirical state measure , control , observation filtration and system filtration…Stopped Completion of Squares on a Cascade Block
lemmalem:fluctuation-block-completion-squares-2026aProbabilityAdopt the setting, notation and hypotheses of the completion-of-squares theorem for the fluctuation cost: the fluctuation processes , of a solution with regular event , empirical state measure , control and system…The Tracked Energy Bound for the Block Cascade under Local Joint Coercivity
lemmalem:fluctuation-tracked-energy-2026aProbabilityAdopt simultaneously the settings and notation of the block cascade lemma, of the anchored pre-stopping envelope lemma, of the localized joint coercivity lemma and of the post-exit comparison lemma, all formed for one and the same data: the…The Block Cascade of Anchored Good-Set Clocks: Adapted Good Sets, Matched Escape Bounds, and the Energy Ledger
lemmalem:fluctuation-block-cascade-2026aProbabilityAdopt the setting, notation and conventions of the extended good-set stopping-time lemma: the affine-controlled transition-rate family on states with compact convex control set , its transition-rate family wi…Post-Exit Comparison at a Stopping Time for the Recentred N-Agent Cost under To-Go Value Regularity
lemmalem:n-agent-post-exit-comparison-2026aProbabilityAdopt the setting, hypotheses and notation of the pathwise tracking lemma: the affine-controlled transition-rate family on states with control set , its transition-rate family with rate bound ,…Anchored Pre-Stopping Envelope and Restricted Moment Bounds for the State Fluctuation Process
lemmalem:fluctuation-anchored-envelope-2026aProbabilityAdopt the setting, notation, hypotheses and definitions of the pre-stopping envelope lemma: the affine-controlled transition-rate family on states with compact convex control set , its transition-rate family with r…Anchored Good-Set Clocks on a Subinterval and the Block Escape Bound
lemmalem:anchored-good-set-clocks-2026aProbabilityAdopt the setting, notation and definitions of the extended good-set stopping-time lemma: the affine-controlled transition-rate family on states with compact convex control set , its transition-rate family with rat…First-Order Expansion of the Recentred N-Agent Cost about a Stationary Mean-Field Triple and Its Coercive Lower Bound
lemmalem:n-agent-cost-first-order-identity-2026aProbabilityAdopt the setting of the fluctuation processes of the controlled -agent dynamics: a transition-rate family on states with control set , a nonempty subset of Euclidean space , and rate bound ; an observation-rate family ;…Pre-Stopping-Time Envelope and Restricted Moment Bounds for the State Fluctuation Process
lemmalem:fluctuation-pre-stopping-envelope-2026aProbabilityAdopt the setting, notation and definitions of the extended good-set stopping-time lemma: the affine-controlled transition-rate family on states with compact convex control set , its transition-rate family with rat…The Extended Good-Set Stopping Time of the Realized Control: Clipped-Out Time and Energy Hitting Bounds
lemmalem:extended-good-set-stopping-time-2026aProbabilityAdopt the setting and notation of the good-set stopping-time lemma, together with those of the progressive measurability lemma for the realized control and of the causality and adaptedness lemma for the realized mean-field flow on which it rests: the…