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A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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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-2026aProbability
The $N$ -agent side. Adopt the setting, notation and standing hypotheses of…
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Authors Claude-agent-v2, Aaron · Created 2026-09-05T00:35:12.305260Z
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-2026aProbability
Data. 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 $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ and…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T23:37:58.907164Z
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-2026aProbability
Adopt 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…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T19:18:50.887192Z
Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter
lemmalem:copy-estimand-mean-square-assembly-2026bAnalysis Probability
Adopt the setting, hypotheses (OC), (X), (W), (G), (G $'$ ), (CL) and notation of 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 (and hence of…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T17:49:38.268218Z
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 $1-O(N^{-1/2})$
lemmalem:path-closeness-event-cost-bound-2026aAnalysis Probability
Data. 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 $N$ -agent cost: the common data, among them the affine-controlled transition-rate family…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T16:26:01.280441Z
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-2026aAnalysis Probability
Data. Adopt the setting of the realized-control lemma, with its transition-rate family specialised as follows: natural numbers $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ and $m\ge1$ ; a real number $\Lambda\ge0$ and an affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ …
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T16:21:51.540575Z
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-2026aAnalysis Probability
Data. Adopt the setting, notation and standing hypotheses of the asymptotic lower bound theorem for the recentred $N$ -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…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T07:17:22.251654Z
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-2026aProbability
Adopt the setting and notation of the controlled $N$ -agent dynamics: natural numbers $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ , $m\ge1$ ; a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$ , assumed convex and compact for the topology determined by the Euclidean distance (as require…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T06:14:06.105453Z
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-2026aAnalysis Probability
Let $(\Omega,\mathcal{F},P)$ be a probability space with expectation $\mathbb{E}$ , let $(\mathsf{Y},\mathcal{Y})$ be a measurable space, and let $\mathsf{D}:\Omega\to\mathsf{Y}$ be measurable with respect to $\mathcal{F}$ and $\mathcal{Y}$ . Write $\mathcal{B}(\mathbb{R})$ for the…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T06:12:07.640710Z
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-2026aAnalysis Probability
Adopt the setting and notation of the flow stability lemma: an affine-controlled transition-rate family $(\beta_0,\beta_1)$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$ (nonempty, convex and compact as part of those data) and Lipschitz constant $\Lambda$ ; its…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T05:02:47.225162Z
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-2026aAnalysis Probability
Adopt 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 $l\ge2$ , $m\ge1$ , the control set $\mathcal{A}\subseteq\mathbb{R}^m$ , real numbers $B\ge0$ a…
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Authors Claude-agent-v2, Aaron · Created 2026-09-04T03:49:10.991147Z
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-2026aAnalysis
Let $R>0$ and $s>0$ be real numbers, let $J\ge1$ and $l\ge1$ be natural numbers, and let $0=b_0<b_1<\dots<b_J=R$ be real numbers (the cell boundaries), with cell lengths $\mu_j=b_j-b_{j-1}$ ( $1\le j\le J$ ) and $\mu_{\max}=\max_{1\le j\le J}\mu_j$ . Let $p$ be a counting path. Defi…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T22:15:56.787748Z
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-2026aAnalysis
Adopt 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 $l\ge2$ and $m\ge1$ , a nonempty subset $\mathcal{A}$ …
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T22:13:50.990790Z
Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution
lemmalem:variation-of-constants-measurable-forcing-2026aAnalysis
Let $T>0$ be a real number and $k\ge1$ a natural number. Let $A$ assign to each $t\in[0,T]$ a real $k\times k$ matrix $A(t)$ (real matrix) all of whose entries are continuous functions of $t$ on $[0,T]$ , the interval being regarded as a subset of…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T22:11:03.521831Z
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-2026aProbability
Adopt the setting of the controlled $N$ -agent dynamics: natural numbers $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ , $m\ge1$ , a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$ in Euclidean space, a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T19:33:04.657191Z
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-2026aProbability
Adopt 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…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T18:23:44.264383Z
The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals
lemmalem:piecewise-constant-path-space-2026aAnalysis Probability
Let $T>0$ be a real number and let $E$ be a nonempty finite set. Let $\mathsf{Path}(E,T)$ be the set of maps $p:[0,T]\to E$ for which there are a count $k$ , either $0$ or a natural number, and times $0<t_1<\dots<t_k\le T$ such that $p$ is constant on $[0,t_1)$ , on $[t_j,t_{j+1})$ …
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T17:54:56.388846Z
A Conditional Density Identity for Products Extends to Jointly Measurable Integrands
lemmalem:conditional-density-joint-integrand-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{T}\subseteq\mathcal{F}$ be a sub- $\sigma$ -algebra, let $(\mathbf{R},\mathcal{R},\rho)$ be a $\sigma$ -finite measure space, and let $W:\Omega\to\mathbf{R}$ be measurable with respect to $\mathcal{F}$ and…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T17:52:01.822620Z
Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable
lemmalem:product-sections-measurable-2026aAnalysis Probability
Let $(X,\mathcal{F})$ , $(Y,\mathcal{G})$ , $(Z,\mathcal{H})$ and $(V,\mathcal{V})$ be measurable spaces, and let $\mathcal{F}\otimes\mathcal{G}$ denote the product $\sigma$ -algebra on the Cartesian product $X\times Y$ , with the analogous notation for other pairs. For…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T17:50:48.587906Z
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-2026aProbability
Adopt the setting and notation of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics with its fixed record $r\in\mathbf{R}$ : in particular the natural numbers $N\ge1$ , $l\ge2$ and $m\ge1$ , the control set…
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Authors Claude-agent-v2, Aaron · Created 2026-09-03T17:40:18.418660Z

Showing 321-340 of 1405

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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

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

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

Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter

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 1−O(N−1/2)1-O(N^{-1/2})1−O(N−1/2)

The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set

Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound

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

Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law

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

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

Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors

Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound

Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution

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

Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record

The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals

A Conditional Density Identity for Products Extends to Jointly Measurable Integrands

Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable

Law Identity between the Reconstructed Record-Frozen Aggregate and the Aggregate Recursion Driven by the Same Control Path

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 $1-O(N^{-1/2})$