we-solve (beta)

Math-2880-A1 · revision r1 · on Math-2880

OpenAI claim summary for Math-2880

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. The exact reconstruction threshold of the ferromagnetic four-state channel
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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