# Verification and scope report

Title: Maximal Gaps between PL and Linear Embedding Dimensions

- Five-page English manuscript; author name at top, company and contact in a footnote.
- Complete written proof and detailed originating-researcher self-audit.
- 2000 exact diagnostics agree in ordinary and optimized Python.
- Exact rational Radon dependencies on five configurations, unordered bad-pair counts, ordered dependency counts, uniform deletion loss, private-vertex extensions, and exponent arithmetic were checked.
- No diagnostic sample is claimed to certify nonembedding in every placement.
- The original OWR question on printed p. 702, Newman Corollary 10/Lemma 12, Lee--Nevo Lemma 3.1, and Zhao's Janson convention were checked in archived primary sources.
- Desktop compilation and a portable source-only build succeed.
- All five PDF pages were directly visually inspected: no clipping, overlap, missing symbols or unresolved references.

Accepted scope: For every d>=3 there exists a finite d-dimensional simplicial complex with minimum PL embedding dimension d and minimum linear embedding dimension 2d+1, the maximum possible gap. A finite two-dimensional complex has the exact pair (3,5). This answers the full finite-complex existential question in OWR-1194-009, including dimensions two and three. The examples are probabilistic and need not be manifolds.

Whole source question resolved in the written proof: yes.
New self-audited original-source closures: one.
No outside review, proof-assistant formalization, absolute priority
certification, manifold theorem or explicit small facet witness.
AI tools assisted derivation, source search, verification code and exposition.
License: CC BY 4.0.