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Math-2880 · mathematics · Probability

The exact reconstruction threshold of the ferromagnetic four-state channel

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the symmetric four-state broadcast channel $P(j|i)=\lambda $ $1_{i=j}+(1-\lambda )/4$ with $0<\lambda <1$, is reconstruction on every d-ary tree, integer $d\geq 2$, and on the observed Poisson(d) offspring tree for every real $d>0$ possible exactly when d $\lambda ^2>1$, with nonreconstruction at equality?

Short English statement

For the symmetric four-state broadcast channel $P(j|i)=\lambda $ $1_{i=j}+(1-\lambda )/4$ with $0<\lambda <1$, is reconstruction on every d-ary tree, integer $d\geq 2$, and on the observed Poisson(d) offspring tree for every real $d>0$ possible exactly when d $\lambda ^2>1$, with nonreconstruction at equality?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2880-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2880

Four-state ferromagnetic broadcasting is claimed to have exact threshold $d\lambda^2>1$ on regular and observed Poisson trees, with no reconstruction at equality. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The exact reconstruction threshold of the ferromagnetic four-state channel

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Four-state ferromagnetic broadcasting is claimed to have exact threshold $d\lambda^2>1$ on regular and observed Poisson trees, with no reconstruction at equality.

Mathematical statement of the claimed result

Result 1.
For the ferromagnetic four-state channel $P_\lambda(j|i)=\lambda\mathbf1_{i=j}+(1-\lambda)/4$, $0\le\lambda\le1$, let \[ a_n=\mathbb E\left[\tfrac12\sum_{i=1}^4 |\mathbb P(\sigma_\rho=i\mid T,\sigma_{L_n})-1/4|\right]. \] The source claims $d\lambda^2\le1\Rightarrow a_n\to0$, both on every rooted integer $d$-ary tree with $d\ge2$ and on Poisson-$d$ Galton--Watson trees for every real $d>0$. The random-tree expectation is not conditioned on survival, and the entire tree is observed. Equality is included. Combined with the usual positive-side reconstruction criterion, this identifies the strict threshold.

Outline of the argument

Symmetric posterior experiments satisfy a preserved algebraic constraint. Polynomial inequalities give a strict information saving under branch combination, and reproducible rational-arithmetic certificates verify the inequalities; iteration forces posterior information to zero at and below the boundary.

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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