Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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…- Let be a probability space and let be either a real number with or the symbol ; for a real number write for the smaller of and , with . A Poisson clock with horizon on…
Fresh-Start Property for Independent Poisson Clocks Read at Levels Satisfying a Clock-Reading Bound
lemmalem:poisson-clocks-fresh-start-2026aProbabilityLet be a probability space, let be a nonempty finite set (the clock labels), let be a -algebra (the initial data), let be either a real number with or , and for each…Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property
lemmalem:poisson-clock-jump-system-forward-equation-2026aProbabilityLet be a probability space, let and be real numbers, let be a nonempty finite set (the state space), and let be a nonempty finite set (the clock labels) with elements. Write for the…Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set
lemmalem:finite-state-forward-equation-uniqueness-2026aProbabilityLet be the real numbers, let be a nonempty finite set, let and be real numbers, and let . For and with let be real numbers such that, for all such , the map is…Probability of the Good Event on the Synthetic Copy: Poisson Tail for the Cell Counts, the Window Discrepancy Bound for the Clock-Good Event, and the Bound on the Complement of the Good Event
lemmalem:copy-good-event-probability-2026aProbabilityAdopt the setting, hypotheses (OC), (X), (W) and notation of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (and hence of…The Copy Clocks Have Independent Poisson Increments on the Clock Interval: Law Identity with the Uniform Poisson Path and Applicability of the Window Discrepancy Bound
lemmalem:copy-clocks-poisson-increments-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: the natural numbers , , , , the real numbers…Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set
lemmalem:copy-tracked-records-almost-sure-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 (and hence of…Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity
lemmalem:aggregate-recursion-ties-deletion-2026aProbabilityAdopt the setting and notation of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution (and hence of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks): natural numbers , , , the control set ,…Information Bound on the Synthetic Copy with Mean-Field Data: Prior Energy of the Profile, Observation Information Along the Profile Response, and the Non-Close Record Mass
theoremthm:copy-information-mean-field-bound-2026aProbabilityAdopt the setting, hypotheses (OC), (X), (W), (G), (P) and notation of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass (and hence of…Relative Perturbations of a Causal Intensity: Pair-Exponent Bound, the Pair Intensity as a Relative Perturbation, Replacement of the Pair Likelihood by the Base Likelihood, and the Weighted Pair-Covariance Sum
lemmalem:relative-perturbation-pair-covariance-2026aProbabilityLet be a natural number, let be a real number, and let be the observation record space with horizon and channels, with channel set . Causal intensities on , their total intensities and their…Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data
lemmalem:copy-close-records-profile-comparison-2026aProbabilityAdopt the setting, hypotheses and notation of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound (and hence of…Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound
lemmalem:copy-weighted-response-quadratic-form-2026aProbabilityAdopt the setting, hypotheses (OC), (X), (W) and notation of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (and hence of…Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection
lemmalem:injection-weights-mean-field-clock-2026aAnalysisProbabilityLet , , and be natural numbers ( is unrelated to ), and let , and be real numbers with . Let be the set of transition labels on states, which has elem…Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path
lemmalem:insertion-response-removal-form-2026aAnalysisProbabilityAdopt the setting, notation and data of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, with the sole exception of its discrepancy hypothesis (D), which is not assumed here: the natural numbers…Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass
theoremthm:copy-information-assembly-2026aProbabilityAdopt the setting, hypotheses (OC), (X), (W) and notation of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (and hence of…Chernoff Bound for the Under-Likelihood Set of a Relatively Perturbed Causal Intensity: Elementary Exponential Inequalities, the Tilted Power-Product Exponent, and the Markov Step
lemmalem:likelihood-ratio-under-likelihood-chernoff-2026aProbabilityLet be a natural number, let be a real number, and let be the observation record space with horizon and channels, with channel set . Causal intensities on , their total intensities and their…