Boundary Counts Do Not Determine Cover Reconfiguration

Static counting information and bounded-cost reachability record different features of a solution space. We exhibit a four-cycle and a paw, with one distinguished vertex in each, whose vertex-cover counting polynomials agree for every assignment of the distinguished vertex. This equality preserves the complete cover-size distribution under every one-vertex gluing with disjoint interiors. Nevertheless, their two minimum covers have token-addition/removal peak costs four and three, respectively. The unrooted polynomial equality is classical; the point here is the explicit boundary-conditioned reconfiguration comparison. We also study the hypergraph on {2^i, 3·2^i : 0 ≤ i < k} whose edges are distinct solutions of 1/x = 1/y + 1/z. For positive integer vertex costs a ≥ b > 0 on the two respective families, we characterize every edge, obtain optimum cost b floor(2k/3), and prove that every distinct optimum pair has exact peak b floor(2k/3) + b. A weighted example outside that regime has optimum cost six and peak ten. Explicit certificates and finite verification code accompany the proofs. AI assistance: OpenAI's ChatGPT, used in the project under the name SHADOW FOUNDRY, provided substantive assistance with mathematical development and proof checking, literature searches, verification-code generation and execution, manuscript drafting, and LaTeX preparation. This assistance was not limited to language editing. The AI tool is acknowledged as a tool, not listed as an author or treated as an independent referee. Its outputs are not evidence of correctness or priority. Seth Christopher is the sole author and is responsible for the final submitted content. Research status: This is a preprint with self-contained arguments and finite computational checks. It is not independently peer reviewed or proof-assistant certified. No resolution of a named open problem or globally established novelty is claimed. Known antecedents are identified in the manuscript. Licensing: The manuscript, LaTeX source, results compilation, and supporting research documentation use CC BY 4.0. The Python verification software uses the MIT License. These are separate scopes, not cumulative licenses for every file. The grants cover only rights held by the licensor; cited third-party works are not relicensed.