Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability
lemmalem:path-closeness-event-cost-bound-2026aAnalysisProbabilityData. Adopt the setting, notation and standing hypotheses of the control- and flow-closeness lemma, and with them those of the asymptotic lower bound theorem for the recentred -agent cost: the common data, among them the affine-controlled transition-rate family…The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set
lemmalem:realized-control-record-frozen-closeness-set-2026aAnalysisProbabilityData. Adopt the setting of the realized-control lemma, with its transition-rate family specialised as follows: natural numbers , , and ; a real number and an affine-controlled transition-rate family on …Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound
lemmalem:cost-bound-control-flow-closeness-2026aAnalysisProbabilityData. Adopt the setting, notation and standing hypotheses of the asymptotic lower bound theorem for the recentred -agent cost, and with them those of the ledger lemma and of the block cascade lemma on which it rests. In particular: the affine-controlled transition-rate family…The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time
lemmalem:observation-filtration-equals-record-sigma-algebra-2026aProbabilityAdopt the setting and notation of the controlled -agent dynamics: natural numbers , , , ; a nonempty control set , assumed convex and compact for the topology determined by the Euclidean distance (as require…Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law
lemmalem:filtering-error-law-invariance-2026aAnalysisProbabilityLet be a probability space with expectation , let be a measurable space, and let be measurable with respect to and . Write for the…The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record
lemmalem:record-frozen-control-measurable-flow-2026aAnalysisProbabilityAdopt the setting and notation of the flow stability lemma: an affine-controlled transition-rate family on states with control set (nonempty, convex and compact as part of those data) and Lipschitz constant ; its…Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms
lemmalem:open-loop-estimand-linearisation-pathwise-2026aAnalysisProbabilityAdopt the setting and notation of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound: natural numbers , , the control set , real numbers a…Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors
lemmalem:compensated-counting-functional-cell-count-form-2026aAnalysisLet and be real numbers, let and be natural numbers, and let be real numbers (the cell boundaries), with cell lengths () and . Let be a counting path. Defi…Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound
lemmalem:perturbed-flow-linearisation-comparison-pair-2026aAnalysisAdopt the setting and notation of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect as far as the following objects are concerned: natural numbers and , a nonempty subset …Variation of Constants with Integrable Forcing
lemmalem:variation-of-constants-measurable-forcing-2026bAnalysisFor a continuous coefficient matrix and an integrable forcing term, the variation-of-constants formula built from the state-transition matrix gives the unique continuous solution of the linear integral equation, with a bound in terms of the initial value and the integral of the f…Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record
lemmalem:n-agent-solution-horizon-restriction-2026aProbabilityAdopt the setting of the controlled -agent dynamics: natural numbers , , , , a nonempty control set in Euclidean space, a transition-rate family on states with control set and rate…Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record
lemmalem:n-agent-copy-record-law-identity-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, with the probability space of that lemma renamed…The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals
lemmalem:piecewise-constant-path-space-2026aAnalysisProbabilityLet be a real number and let be a nonempty finite set. Let be the set of maps for which there are a count , either or a natural number, and times such that is constant on , on …A Conditional Density Identity for Products Extends to Jointly Measurable Integrands
lemmalem:conditional-density-joint-integrand-2026aProbabilityLet be a probability space, let be a sub--algebra, let be a -finite measure space, and let be measurable with respect to and…The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap
lemmalem:product-sections-measurable-2026bAnalysisProbabilityThe measurable rectangles form a π-system generating the product σ-algebra, and rectangles of generators suffice; insertions, projections and the swap are measurable, a pairing is measurable exactly when both entries are, and sections of product-measurable sets and functions are…Law Identity between the Reconstructed Record-Frozen Aggregate and the Aggregate Recursion Driven by the Same Control Path
lemmalem:record-frozen-aggregate-copy-law-identity-2026aProbabilityAdopt the setting and notation of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics with its fixed record : in particular the natural numbers , and , the control set…Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics
lemmalem:n-agent-record-frozen-forward-equation-2026aProbabilityAdopt the setting and notation of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records: natural numbers , , , , a nonempty control set in Euclidean space, a…Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon
lemmalem:aggregate-recursion-forward-equation-2026aProbabilityAdopt the setting and notation of Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels (that of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution): natural numbers…The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them
lemmalem:copy-clocks-independent-poisson-horizon-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 recalled in…Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels
lemmalem:aggregate-recursion-clock-reading-bound-2026aProbabilityAdopt the setting and notation of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution: natural numbers , , , a nonempty subset of Euclidean space , real numbers and , a…