A Structural Proof of an Exact Threshold-Four Result for a 14-Vertex Rooted Graceful-Labeling Extension with Lean Formal Verification and Fresh-Clone Reproducibility

We study a rooted one-gap extension problem associated with graceful labelings of trees. The underlying 14-vertex tree considered here belongs to a previously known graceful class; thus its gracefulness is not the contribution of this work. For this explicitly defined rooted tree, we prove instead that the minimum rooted gap threshold is exactly four.

The lower bound is obtained by a structural reduction rather than exhaustive enumeration over all graceful labelings. Complement symmetry and a weighted cut-sum identity exclude thresholds one and two. For threshold three, the 78 possible pairs of exceptional edges reduce under rooted automorphisms to seven orbit types. Five are eliminated by first-moment constraints, while the remaining two reduce to fifteen finite residual configurations. These residual cases are ruled out by the second-moment identity ∑_{d=1}^{13} d² = 819. An explicit graceful labeling provides a threshold-four certificate and hence the matching upper bound.

The local theorem was formalized in Lean 4.34.0 with mathlib 4.34.0, with an explicit trust audit. The final development contains no sorry, native decide, or bv decide; the final declaration depends only on the standard axioms propext, Classical.choice, and Quot.sound.

The project was rebuilt and rechecked from a fresh GitHub checkout on a clean GitHub-hosted Ubuntu runner. These verification statements refer to the publication baseline identified in Section 8. The result concerns only the stated rooted tree and does not constitute a proof of the Graceful Tree Conjecture.