Atlas

A visual map of the program's territory. The center is the framework. The orbits are the papers. The connecting tissue is the oracle stack that makes the framework's claims computable. The outer ring is the lineage from which the work descends.

How to read it

The central node is the framework: the claim that geometric primitives should be classified by operational invariant rather than symmetry group. The framework's empirical defensibility rests on the methodology paper, drawn here as the foundation below the center. Without the methodology paper, the framework is an assertion; with it, the framework is an apparatus whose failure modes are characterized.

The four orbiting nodes are the papers. Each paper enters the framework through a specific oracle: Paper I through the Contact Distribution Score (CDS), Paper II through the Equilibrium Count Score (ECS), Paper III through the Hertz Stress oracle (SDS) and the equilibrium analysis, Paper IV through the thermal oracle (TDS). The oracles are the connecting tissue. Without them the papers would each be isolated findings; with them, the findings compose into a coherent program.

The outer corners are the lineages from which the work descends. The mono-monostatic arc runs Arnold to Domokos & Várkonyi to Sloan to the catalog work. The developable-roller arc runs Schatz to the Dirnböck & Stachel formalization to Paper I. The dashed lines indicate inheritance, not collaboration; the present work descends from these traditions but the traditions are not consulted on the present work's claims.

What is not on the map

Three things deliberately omitted. First, future work — Paper IV constant-width primitives (Reuleaux and Meissner), Year-2 biomimetics direction, the U₃ characterization decision. These exist as scoped artifacts but have not yet produced empirical claims. Second, adjacent contemporary work — Bogosel 2024 on Meissner polyhedra, the recent U₃ construction paper, Pak–Várkonyi 2024 on centroids and equilibrium. These shape the landscape but are not load-bearing for the present program's findings. Third, the chemistry preprint on hERG QSAR. It was drafted in parallel during the same sprint but does not belong to Substrate Geometry's territory; it is a separate work that shares only the timeline.

For the linear narrative of how the program arrived at this state, see lineage. For the formal claims, see papers. For the discipline that underwrites the claims, see methodology.