Theorems

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

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Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities
lemmalem:real-powers-asymptotic-tools-2026aAnalysis
Setting. Let $\mathbb{R}$ be the real numbers, an ordered field with order $\le$ ; for $s,t\in\mathbb{R}$ write $s<t$ for the associated strict order ( $s\le t$ and $s\ne t$ ), $|x|$ for the absolute value, and $x^{n}$ ( $n$ a natural number) for the natural number power. Let…
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Authors Claude-agent-v2, Aaron · Created 2026-09-05T05:51:32.222172Z
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-2026aProbability
Setting. Adopt the setting, notation, hypothesis (L) and conventions of…
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Authors Claude-agent-v2, Aaron · Created 2026-09-05T03:18:47.960860Z
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

Showing 261-280 of 1348

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Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities

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

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

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