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Math-2879 · mathematics · Probability

A nonlinear-capacity criterion for four-state reconstruction on arbitrary trees

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every deterministic infinite rooted tree with uniformly bounded number of children and $0<\lambda <1$, is four-state reconstruction equivalent to positivity of $sup_\theta $ |theta| over nonnegative root-to-infinity flows satisfying $sup_ray$ $sum_(e$ on $ray)(\lambda ^{-2|e|}\theta (e))^2\leq 1$? Reconstruction means a positive limiting expected total-variation distance of the root posterior from the uniform four-state law.

Short English statement

For every deterministic infinite rooted tree with uniformly bounded number of children and $0<\lambda <1$, is four-state reconstruction equivalent to positivity of $sup_\theta $ |theta| over nonnegative root-to-infinity flows satisfying $sup_ray$ $sum_(e$ on $ray)(\lambda ^{-2|e|}\theta (e))^2\leq 1$? Reconstruction means a positive limiting expected total-variation distance of the root posterior from the uniform four-state law.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2879-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2879

On any bounded-degree deterministic tree, four-state ferromagnetic reconstruction is claimed equivalent to positive $L^3$ capacity for resistances $\lambda^{-2|e|}$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A nonlinear-capacity criterion for four-state reconstruction on arbitrary trees

OpenAI's claim

OpenAI presents a proof in the selected manuscript. On any bounded-degree deterministic tree, four-state ferromagnetic reconstruction is claimed equivalent to positive $L^3$ capacity for resistances $\lambda^{-2|e|}$.

Mathematical statement of the claimed result

Result 1.
Let $T$ be an infinite tree with a distinguished root; each vertex has at most $B<\infty$ children. On this tree, broadcast four uniform states through $P_\lambda(j|i)=\lambda\mathbf1_{i=j}+(1-\lambda)/4$, $0<\lambda<1$. Let $a_\infty$ be the limiting expected total-variation advantage of the root posterior from level observations. For a nonnegative conserved flow $\theta$, with root mass $|\theta|$, define \[ \operatorname{Cap}_{3,\lambda}(T)=\sup\left\{|\theta|: \sup_{\xi\in\partial T}\sum_{e\in\xi}(\lambda^{-2|e|}\theta(e))^2\le1\right\}. \] The source claims \[ a_\infty(T,\lambda)>0\quad\Longleftrightarrow\quad\operatorname{Cap}_{3,\lambda}(T)>0. \] The edge depth is the depth of its child endpoint.

Outline of the argument

A cubic posterior-information saving is converted to a nonlinear flow inequality by changing variables. Reconstruction then telescopes into a positive admissible flow; conversely a scaled flow specifies noisy terminal observations whose accumulated information loss remains controlled.

Formal verification

For family 229, the three-state Comparator entries only establish reconstruction when $d\lambda^2>1$. They omit threshold nonreconstruction, the graph recovery corollary, and the four-state companions. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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