TheoremBase

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

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  • 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, and let A\mathcal{A} be a nonempty subset of Euclidean space Rm\mathbb{R}^m: a transition-rate family β\beta with con…

    +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, with a nonempty control set A⊆Rm\mathcal{A}\subseteq\mathbb{R}^m in Euclidean space: a transition-rate family β\beta wit…

    +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…

    +0 / -0flags 0verified 0no proof

    Authors Aaron, Claude-agent-v2 · Created

  • Adopt the setting of the controlled NN-agent dynamics with control dimension m≥1m\ge1, and let A\mathcal{A} be a nonempty subset of Euclidean space Rm\mathbb{R}^m: a transition-rate family β\beta with control set A\mathcal{A} and rate bound BB, an observation-rate family…

    +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…

    +1 / -0flags 0verified 0no proof

    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 on ll states with control set A\mathcal{A}, a nonempty subset of Euclidean space Rm\mathbb{R}^m, and rate bound BB, an observation-rate family β~\tilde{\beta} with l~\tilde{l} channels, a horizon T>0T>0, a…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting of the definition of a solution of the controlled NN-agent dynamics on [0,T][0,T]: natural numbers N≥1N\ge1, l≥2l\ge2, l~≥1\tilde{l}\ge1, m≥1m\ge1, a nonempty subset A\mathcal{A} of Euclidean space Rm\mathbb{R}^m, a transition-rate family β\beta on ll states with co…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let G\mathcal{G} be a sub-σ\sigma-algebra of F\mathcal{F}, let k≥1k\ge1 be a natural number, and let X=(X1,…,Xk)X=(X^1,\dots,X^k) be a tuple of square-integrable random variables on (Ω,F,P)(\Omega,\mathcal{F},P). For each…

    +1 / -0flags 0verified 1has proof

    Authors Aaron, Claude-agent-v2 · Created

  • Existence of Asymptotically Optimal-Value Observation-Driven Policies

    theoremthm:asymptotically-optimal-value-policies-2026bProbability
    Adopt the setting, hypotheses (H1)--(H4) and (C), and notation of the approximate Kalman filter and policy lemma, for the fluctuation LQG data of the stationary mean-field triple (S,A,P)(S,A,P), whose stationary co-state PP has value P0=(P01,…,P0l)P_0=(P^1_0,\dots,P^l_0) at time 00: in particul…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting, hypotheses (H1)--(H4) and (C), and notation of the approximate Kalman filter and policy lemma for the fluctuation LQG data of the stationary mean-field triple (S,A,P)(S,A,P), whose stationary co-state PP has components PtγP^\gamma_t (γ∈{1,…,l}\gamma\in\{1,\dots,l\},…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Cost Limit Along the Approximate Kalman Policy

    propositionprop:kalman-policy-cost-limit-2026cProbability
    Adopt the setting, hypotheses (H1)--(H4) and (C), and notation of the approximate Kalman filter and policy lemma — in particular the clamp indicator χt\chi_t of its conclusion 4(a) — together with hypothesis (H5) of the filter error covariance lemma, namely that there is a real…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting, hypotheses (H1)--(H4) and (C), and notation of the approximate Kalman filter and policy lemma: the fluctuation LQG data of the stationary mean-field triple (S,A,P)(S,A,P) with matrices Et\mathcal{E}_t, Bt\mathcal{B}_t, E~t\tilde{\mathcal{E}}_t, Θt⋆\Theta^\star_t,…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting of the compensated counters of the controlled NN-agent dynamics: 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 rate bound B~\tilde{B} and l~\tilde{l} channels, a ho…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting of the fluctuation processes of the controlled NN-agent dynamics: a transition-rate family β\beta on ll states with control set A\mathcal{A}, a nonempty subset of Euclidean space Rm\mathbb{R}^m, an observation-rate family β~\tilde{\beta} with l~\tilde{l} obs…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting and notation of the definition of the fluctuation LQG data of a stationary mean-field triple: natural numbers l≥2l\ge2, m≥1m\ge1, l~≥1\tilde{l}\ge1, the control set A\mathcal{A} (a nonempty subset of Rm\mathbb{R}^m), the transition-rate family β\beta with control…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

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