# Sources and limits

## Primary local source

The Tangled Triangle essay is adapted from Lawrence Rowland's **Tangled Triangle — one-shot project and system architecture** brief and report. The original scenario describes a proposed regional signalling control centre on a triangular site constrained by a canal, road, elevated railway, high-voltage line and old mine passage.

The public essay reuses the scenario as a toy compatibility problem. Its geometry, buffers and candidate locations are illustrative; it is not a site survey, engineering assessment, safety case or design recommendation.

## Formal and applied reading

- Daniel Rosiak, *Sheaf Theory through Examples* (MIT Press, 2022), ISBN 9780262542159.
- R. Ghrist, public teaching material on sheaves, flows and lattices.
- P. Le Masson, B. Weil and A. Hatchuel, “Sheaves as a Framework for Design: an application to architectural design,” *Proceedings of the Design Society* (2023), DOI: [10.1017/pds.2023.317](https://doi.org/10.1017/pds.2023.317).
- Warren B. Powell, writing on sequential decision analytics and state-variable modelling, used as an adjacent prompt to keep state, decision, exogenous information, transition and objective distinct.

The reading corpus also contained exploratory papers applying sheaf language to resource allocation and WBS risk. Those papers are treated here as provocations, not as validation of the site's claims.

## Interpretation used in this site

The useful formal pattern is:

1. define local data on meaningful parts of a project;
2. define the overlaps and the maps that compare data there;
3. test whether local assignments agree on every declared overlap;
4. treat compatible assignments as candidates for a global section;
5. treat incompatibilities as located obstructions worth investigating.

Choosing the cover—the parts, interfaces and levels on which this test is performed—is a substantive modelling decision. A WBS hierarchy alone is not automatically a sheaf, and adding mathematical vocabulary does not make an arbitrary dashboard rigorous.

Essay 001 now makes that boundary executable: six explicit equality/capacity rules compare fictional projected fields, with missing information reported separately. Its compatible tuple is a result of those finite constraints. The stronger sheaf condition requires unique gluing of compatible local sections; see the [Mathlib formulation and proof](https://leanprover-community.github.io/mathlib_docs/topology/sheaves/sheaf_condition/unique_gluing.html). The example does not establish those axioms for project data.

## Images

The two picture essays use generated mnemonic images supplied with the foray. They are aids to thought, not mathematical diagrams, evidence or proofs.