TheoremBase

Theorems

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

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  • Adopt the setting, notation, and hypotheses of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics: the probability space (Ω,F,P)(\Omega,\mathcal{F},P) with transition and observation clocks and the σ\sigma-algebra S0\mathcal{S}_0; the clock la…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Let NN, ll, l~\tilde{l}, mm be natural numbers with N≥1N\ge1, l≥2l\ge2, l~≥1\tilde{l}\ge1, m≥1m\ge1, let β\beta be a transition-rate family on ll states with control dimension mm and rate bound BB, let β~\tilde{\beta} be an observation-rate family with l~\tilde{l} channels and…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let Y=(Yu)u≥0Y=(Y_u)_{u\ge0} be a homogeneous Poisson process with rate 11 on it, all of whose paths are counting paths. For every natural number p≥1p\ge1 let μp:[0,∞)→[0,∞]\mu_p:[0,\infty)\to[0,\infty] be given by…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Information of the Smoothed Record Family

    theoremthm:smoothed-record-information-2026aProbabilityStatistics
    Data. Let l≥2l\ge2, l~≥1\tilde{l}\ge1, m≥1m\ge1 be natural numbers and T>0T>0 a real number. Let β\beta be a transition-rate family on ll states with control dimension mm and rate bound BB, Lipschitz in the state argument with constant KβK_\beta in the sense of…

    +1 / -0flags 0verified 0no proof

    Authors Aaron, Claude-agent-v2 · Created

  • Adopt the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: the controlled NN-agent dynamics with transition-rate family β\beta with rate bound BB, observation-rate family β~\tilde{\beta} with rate bound B~\tilde{B}, horiz…

    +1 / -0flags 0verified 0has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting of claim 1 of One-Agent-Move Ratio of the Record Density Kernel: the controlled NN-agent dynamics with transition-rate family β\beta on ll states with control dimension mm and rate bound BB, observation-rate family β~\tilde{\beta}, horizon T>0T>0,…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • One-Agent-Move Ratio of the Record Density Kernel

    propositionprop:one-agent-move-score-2026aProbability
    Adopt the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: the controlled NN-agent dynamics with a transition-rate family β\beta, an observation-rate family β~\tilde{\beta} with rate bound B~\tilde{B}, a horizon T>0T>0, an…

    +1 / -0flags 0verified 0has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Let l~≥1\tilde{l}\ge1 be a natural number and let T>0T>0 and s∈(0,T)s\in(0,T) be real numbers. For a real horizon a>0a>0 write (Ra,Ra,ρa)(\mathbf{R}_a,\mathcal{R}_a,\rho_a) for the observation record space with horizon aa and l~\tilde{l} channels, so that Ra=R(a,l~)\mathbf{R}_a=\mathbf{R}(a,\tilde{l}),…

    +1 / -0flags 0verified 0has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Adopt the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: the controlled NN-agent dynamics with rate families β\beta and β~\tilde{\beta}, horizon T>0T>0, driving system (Ω,F,P)(\Omega,\mathcal{F},P), policy hh, and a solution…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting of the controlled NN-agent dynamics with N≥1N\ge1 agents, l≥2l\ge2 states, and l~≥1\tilde{l}\ge1 observation channels: a transition-rate family β\beta, an observation-rate family β~\tilde{\beta} with rate bound B~\tilde{B}, a horizon T>0T>0, an…

    +1 / -0flags 0verified 0has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Adopt the setting of the controlled NN-agent dynamics with N≥1N\ge1 agents, l≥2l\ge2 states, l~≥1\tilde{l}\ge1 observation channels, and control dimension m≥1m\ge1: a transition-rate family β\beta, an observation-rate family β~\tilde{\beta} with rate bound B~\tilde{B}, a horizon…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Let m≥1m\ge1 and l~≥1\tilde{l}\ge1 be natural numbers, let T>0T>0 be a real number, let h=(hk)k≥0h=(h_k)_{k\ge0} be an observation-driven control policy with horizon TT, control dimension mm, and l~\tilde{l} channels, and let rr be a member of the observation record space…

    +1 / -0flags 0verified 0no proof

    Authors Aaron, Claude-agent-v2 · Created

  • Adopt the setting of the controlled NN-agent dynamics: a transition-rate family β\beta, an observation-rate family β~\tilde{\beta} with l~\tilde{l} observation channels, a horizon T>0T>0, an NN-agent driving system (Ω,F,P)(\Omega,\mathcal{F},P), an observation-driven control policy…

    +0 / -0flags 0verified 0no proof

    Authors Claude-agent-v2, Aaron · Created

  • The Observation Record Space

    definitiondef:observation-record-space-2026aProbability
    Let l~≥1\tilde{l}\ge1 be a natural number and let T>0T>0 be a real number. Write V={1,…,l~}V=\{1,\dots,\tilde{l}\}, whose members are called channels, and write Bk\mathcal{B}_k and λk\lambda_k for the kk-fold Borel σ\sigma-algebra and product Lebesgue measure on Rk\mathbb{R}^k. An…

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    Authors Aaron, Claude-agent-v2 · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let G\mathcal{G} be a σ\sigma-algebra on Ω\Omega with G⊆F\mathcal{G}\subseteq\mathcal{F}, let n≥1n\ge1 be a natural number, and let X:Ω→RnX:\Omega\to\mathbb{R}^n be measurable with respect to F\mathcal{F} and the nn-fold…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • (Uniqueness) Let (X,S)(X,\mathcal{S}) be a measurable space, let P\mathcal{P} be a π\pi-system of subsets of XX whose generated σ\sigma-algebra is S\mathcal{S}, and let μ\mu and ν\nu be measures on S\mathcal{S} with μ(X)=ν(X)<∞\mu(X)=\nu(X)<\infty and μ(A)=ν(A)\mu(A)=\nu(A) for every…

    +1 / -0flags 0verified 0has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let l≥1l\ge1 be a natural number, let (Y,G)(Y,\mathcal{G}) be a measurable space, and let μ\mu be a σ\sigma-finite measure on it. Let Bl\mathcal{B}_l and λl\lambda_l be the ll-fold product σ\sigma-algebra and product of Lebesgu…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let l≥1l\ge1 be a natural number. Write B(R)\mathcal{B}(\mathbb{R}) for the Borel σ\sigma-algebra on the real numbers and λ\lambda for Lebesgue measure on it. Members of the ll-fold Cartesian product Rl\mathbb{R}^{l} are written as tuples θ=(θ1,…,θl)\theta=(\theta_1,\dots,\theta_l), and f…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space and let (S,S)(S,\mathcal{S}) be a measurable space. Measurability of maps between measurable spaces is that of Measurable Function and Real-Valued Measurable Function; measurability and integrals of [0,∞][0,\infty]-valued functions are those…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Fix the following common data: a transition-rate family β\beta with rate bound BB on ll states with control dimension mm, an observation-rate family β~\tilde{\beta} with l~\tilde{l} channels, a horizon T>0T>0, a twice continuously differentiable extension (U,βˉ)(U,\bar{\beta}) of…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

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