A two-dimensional complex with embedding-dimension gap two

This research note proves the existence of a finite two-dimensional simplicial complex with minimum piecewise-linear embedding dimension three and minimum original-simplex affine embedding dimension five. It answers affirmatively Brehm's 2006 question whether these minimum embedding dimensions can differ by more than one, already in dimension two. The first barycentric subdivision embeds linearly in three dimensions. The argument makes a known balanced-Radon and random-complex affine obstruction uniform under all small facet deletions, then removes triangles sharing an edge. An elementary local-fan construction supplies the PL embedding. Existing Brehm–Sarkaria, Newman and Lee–Nevo methods are credited. Two independently fixed bounded literature audits found no checked theorem subsuming this exact result; categorical first priority is not asserted. Exact finite parameters and independent offline checks are supplied, but no explicit facet list for the enormous witness is generated. Finite checks supplement the universal proof and are not formal verification. Generative AI tools were used extensively in problem solving, proof checking, literature review, manuscript preparation and independent adversarial review. This is an unrefereed preprint; no human peer review or formally machine-verified proof is claimed. Research prose is licensed CC BY4.0 and verification code MIT. The source-and-verification archive contains the manuscript source, pinned standard-library checks, provenance and source/priority evidence. The repository publication program records the separate fresh full preprint reviews.