# Dilute Bose gas reference regime planner

Family 267: Positive-temperature Bose–Einstein condensation and exact quantum depletion. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Many-body researchers can use condensation and depletion limits to organize admissible hard-sphere or radial-potential simulation regimes.

## Applicability and commercial boundary

The thermodynamic limit precedes the dilute limit; the positive-temperature existence statement has sufficiently small fixed density. A finite trapped experimental gas needs separate corrections and effective constants.

## Initial business decision

Conditional research. Buyer budget, commercial novelty and profitability are unvalidated.

## Next verification action

Extract finite-volume or density error estimates and compare with a public simulation model.

## Evidence scope

The catalog statement was individually reviewed. This record does not imply a manuscript proof review or formal-scope comparison. No independent Lean check was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Main source

[Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf).