# Depth-three circuit lower-bound teaching and benchmark module

Family 112: Beyond the square-root exponent for depth-three circuits. First review, 8 October 2026.

## Problem and potential new use

Circuit-complexity researchers and educators could use an explicit language separating a restricted circuit model from polynomial-time decision.

## Applicability and commercial boundary

The theorem does not show hard instances for unrestricted software or create a practical cryptographic primitive.

## Initial business decision

Defer direct product. Buyer demand and profitability remain hypotheses. This first pass does not establish a validated commercial market.

## Next verification action

Inspect the explicit language and identify a bounded demonstrator; prefer adding it to an existing research workbench.

## Evidence scope

Catalog statement reviewed; inspect linked opportunity dossier for any deeper source review. This record alone does not establish full manuscript or proof verification.

Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json) for the exact manuscripts and available scope notes.