# Graph community recovery feasibility audit

Edition 9 October 2026 v2. **Bounded model/benchmark audit; low commercial confidence**.

## Research finding

The exact three-state graph consequence uses iid uniform labels, positive fixed a,b, graph-only observations and asymptotic weak recovery. Four-state ferromagnetic/tree-capacity companions have distinct domains. Selected formal scope covers only three-state supercritical reconstruction.

## Problem and buyer

A graph analytics team needs to distinguish an algorithm’s failure from insufficient information in its fitted network model. A synthetic-data or community-detection vendor may need calibrated test regimes and an honest explanation of what latent labels can be recovered.

## What the finding could enable

A declared-model audit, exact parameter-rectangle classification, finite null scoring and preserved existing-algorithm comparisons can expose scope mistakes, label leakage and algorithm-specific failures. Automatic model fitting is not implemented.

## Technical and commercial limits

The graph consequence is for a symmetric sparse three-community model with positive connectivity parameters and mean degree greater than one. Weak recovery is above-chance correlation, not perfect classification. Estimated model parameters and community imbalance create uncertainty. Observed Poisson-tree advantage is averaged without conditioning on survival. The below-threshold and graph consequences do not inherit the selected formal badge.

## Minimal architecture

Declared model and rational parameter bounds -> scope/envelope audit -> iid-label synthetic graphs -> strip planted metadata -> fixed conventional recovery methods -> permutation-aligned score and finite baselines -> evidence report. Real-network fit remains a separate obligation.

## Existing alternatives and differentiation

NetworkX already generates stochastic block models with configurable block sizes and probability matrices. Existing community algorithms can be evaluated on those graphs. The proposed differentiation is exact-model scope, fit uncertainty, finite-size regime coverage and reproducible comparison. [NetworkX SBM generator](https://networkx.org/documentation/stable/reference/generated/networkx.generators.community.stochastic_block_model.html).

## Monetization hypothesis

Earlier AUD 4,000-12,000 assessment and AUD 400-1,500/month figures remain speculative. Existing libraries supply generators and numerical methods. Only pursue recurring benchmark/evidence integration after a buyer demonstrates costly failures and willingness to pay.

## Validation experiment

1,048 passing controls, 729 exact parameter rectangles and 54 preserved graphs compared with NetworkX modularity and a conventional SciPy Bethe-Hessian baseline. Actual model fit, held-out checks, full misspecification experiments and customer value remain open.

## Conditions to reject or defer

Reject if the buyer’s objective is supervised prediction, if their network is far from the model, or if current benchmark tooling already covers the need. Defer real-world impossibility claims and four-state graph-recovery claims unsupported by the inspected source.

## Next concrete action

Keep this as a low-confidence evidence-platform module; next technical work would address held-out real-model fit and suitable misspecification benchmarks. No real-data impossibility verdict or source-optimal algorithm claim is justified.

## Detailed evidence

[Exact scope, source companions, conventional comparisons and preserved failures](../../prototypes/community-recovery-2026-10-09-v2/report.md). A threshold calculation is source-conditional, a model declaration is not model validation, and finite overlap above one-third is not by itself evidence of recovery. The [Facility Plan Auditor](https://mathideas.stera.ventures/facility/) remains the first product experiment.