# Banach sphere isometry teaching and formalization module

Family 322: Tingley’s sphere-isometry problem. First review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Researchers can expose the connection between a sphere's exact metric structure and a whole-space linear isometry.

## Applicability and commercial boundary

The theorem assumes a surjective exact isometry; noisy finite geometry observations do not establish a reconstructible linear operator.

## Initial business decision

Defer direct product. Commercial demand and profitability remain hypotheses.

## Next verification action

Review finite-dimensional exact examples and add them to an existing geometry education workflow.

## Evidence scope

The catalog statement was individually reviewed. Main-paper and selected formal-scope passages were additionally inspected for families 090, 093, 094, 097, 325, 328 and 332; this record does not claim a full proof audit or independent Lean verification. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).