# Two companion studies

Ergentics, 2026. New AI-generated images made with `image_gen.imagegen` under
Ergentics art direction and Codex assistance. They are contemporary interpretive
portraits, not authenticated likenesses or historical paintings. Painted paths,
spheres and triangles suggest ideas; they do not supply mathematical evidence.

## A shared shape — Lagrange

[Save the original PNG](assets/lagrange-interpretation.png) · 1122 × 1402 · 2,273,453 bytes.

A reflective figure in a dark study, with three luminous forms above him. The
composition follows the emphasis on mutual distances in Lagrange's
*Avertissement*: a triangle as a relationship between three bodies. The separate
exact diagram and interactive viewing room preserve the measured geometry.

SHA-256: `047c5f95946e94ad652af5d41608fa314d0298fa1ca4dfcb7c599e12b3ce021d`.

## Looking again — Poincaré

[Save the original PNG](assets/poincare-interpretation.png) · 1122 × 1402 · 2,310,341 bytes.

A contemplative portrait with paths returning through the space above. Its
narrative starting point is Poincaré's acknowledgment of Phragmén in his 1890
*Introduction*. The paths are metaphorical. The gallery's comparison of two
phase-shifted states is a separate modern teaching example.

SHA-256: `0b1f566321e6ef9cc6dd0aec974ebb234ad37c1f573f92d21b9da3a0be4e84ce`.

## Making and viewing

Both originals were generated afresh for this project and saved without pixel
edits. Their shared direction specifies a vertical portrait, subtle painted
texture, dark green space, warm parchment light and restrained jade, amber and
violet forms. No historical artwork, museum asset or artist's signature was
used as an image input. Separate HTML labels identify their role; CSS frames
preserve the full image. The originals are about 2.3 MB each and load lazily.

Prompts and artifact identities are retained with the research packet. These
files can accompany the paper as explicitly labeled interpretive art. Their
availability does not establish unrestricted outside reuse authorization.
See the [primary-source trace](primary-trace.md) for the passages behind the
two compositions and the limits of the historical and mathematical review.