# Variational segmentation structure diagnostics

Initial decision: **Conditional diagnostic**. First dossier, 9 October 2026 Australia/Brisbane. Buyer demand, commercial novelty and profitability are unvalidated.

## Research finding

Family 366 classifies interior discontinuity structures of reduced planar absolute Mumford–Shah minimizers as smooth arcs, crack tips or three-way junctions meeting at 120 degrees, with local finiteness properties. Family 367 separately states critical dimension seven for one-phase Bernoulli minimizers. They concern different functionals; the application must not merge their assumptions or singularity classifications.

## Problem and buyer

A vision or numerical-analysis team developing variational segmentation needs useful regression examples and a way to understand boundary structures across resolution and initialization. A first buyer hypothesis is a specialist industrial image-analysis group that repeatedly reviews geometrically implausible segmentations, although the initial theorem-derived use is research oriented.

## What the finding could enable

A solver-development plugin can supply theorem-linked structure fixtures, extract local boundary graphs and compare junction angles, tip shapes and mesh sensitivity. A new app could help an engineer distinguish a noisy digital boundary from a claimed continuum minimizer and decide which additional optimization checks to run. Geometry diagnostics are not new by themselves; the proposed contribution is a precise bridge between the new classifications and regression-test evidence.

## Technical and commercial limits

The Mumford–Shah statement assumes an absolute minimizer and bounded fidelity on a planar Lipschitz domain, and covers interior behavior. A local numerical minimum, piecewise-constant variant, 3D segmentation or physical crack network is a different object. Digital junction extraction depends on scale and tolerance. A 120-degree junction is not proof of global optimality, and a deviating junction does not by itself establish a real object was misclassified.

## Minimal architecture

Functional and model manifest -> solver-run import -> boundary graph at multiple scales -> junction and tip measurements -> energy/initialization history -> analytic fixture comparisons -> scope-aware report. Keep Mumford–Shah and Bernoulli test suites distinct. Start with an open library and an offline viewer rather than a new production image platform.

## Existing alternatives and differentiation

Scikit-image already provides iterative Chan–Vese and morphological segmentation implementations and exposes useful solver outputs. These methods do not become certified absolute Mumford–Shah minimizers because their contours look regular. The proposed plugin needs to add actionable diagnostics beyond current energy histories, multi-start comparisons and human inspection. [Scikit-image segmentation API](https://scikit-image.org/docs/stable/api/skimage.segmentation.html).

## Monetization hypothesis

Test an AUD 3,000–8,000 evaluation integration for a team maintaining a variational solver; consider an AUD 300–1,000 monthly supported library only if regression review recurs. At an illustrative AUD 5,000 fee, 25 specialist hours costed at AUD 150/hour consume AUD 3,750, leaving AUD 1,250 before support and overhead. Research labs may have no commercial budget, and a broad library may be better offered as an open component of a paid review service.

## Validation experiment

Use analytic straight-arc, tip and equal-angle triple-junction fixtures, plus pixelated and perturbed versions. Measure the stability of extracted geometry across scale. Run at least two initializations on an existing solver and determine whether the report explains a real discrepancy beyond a standard energy plot. Record when the functional differs from the source theorem.

## Conditions to reject or defer

Reject if the diagnostic has no effect on debugging decisions, if users work exclusively with unrelated deep segmentation models, or if global-minimizer assumptions make theorem-derived checks uninformative for their workflow. Defer a commercial segmentation-accuracy promise until task-specific evaluation supports it.

## Next concrete action

Extract an exact triple-junction reference and design an honest approximate-geometry tolerance model. Do not build a full segmenter before testing whether the diagnostic changes a review decision.

## Pinned research sources

- Family 366: [Interior regularity of planar Mumford–Shah minimizers](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf).
- Family 367: [The critical dimension for one-phase Bernoulli minimizers](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf).
- Family 367: [selected formal scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/367.md); not independently checked here.

The repository statements are treated as source claims. No full proof review, buyer interview or clinical/engineering certification has been completed.