# Expectation-threshold and decomposition reference

Family 175: Talagrand’s expectation thresholds, discrete convexity, and graph decompositions. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Random-graph researchers can organize integral/fractional threshold comparisons and decomposition claims.

## Applicability and commercial boundary

Unspecified universal factors and independently embedded pieces do not automatically yield a consistent network partition or constructive threshold algorithm.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Inspect effective pieces and threshold-computation cost.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Graph Decompositions at the Integral Expectation Threshold](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf).