THE FOLD / CO-OP / THE BROADCAST / THE GRACEFUL
THE GRACEFUL
a labeling whose edge-gaps are 1 to m
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A graceful labeling assigns the vertices of a graph with m edges distinct numbers from 0 to m so that the edge ‘lengths’ — the absolute differences of the endpoint labels — come out as exactly 1, 2, …, m, each once. It is a jigsaw of numbers: pick vertex values so no two edges share a gap. Paths and stars are always graceful; a cycle Cn is graceful if and only if n ≡ 0 or 3 (mod 4). The still-open Graceful Tree Conjecture — that every tree is graceful — has resisted proof for over fifty years.
LIT verified live: an explicit zig-zag labeling makes every path graceful and the star K1,n graceful, and an exhaustive search confirms the cycle Cn is graceful exactly when n ≡ 0 or 3 (mod 4) — C₃, C₄, C₇ yes; C₅, C₆ no (window.__graceful). FIG no framing; the labeling, the edge-difference check, and the exhaustive cycle search run in-browser.
LIT verified live: an explicit zig-zag labeling makes every path graceful and the star K1,n graceful, and an exhaustive search confirms the cycle Cn is graceful exactly when n ≡ 0 or 3 (mod 4) — C₃, C₄, C₇ yes; C₅, C₆ no (window.__graceful). FIG no framing; the labeling, the edge-difference check, and the exhaustive cycle search run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-broadcast — labels broadcast from the vertices to the edges: choose the node numbers just so, and every edge broadcasts a distinct length from 1 to m, no collision. AVAN (AI) built the instrument: the graceful-check, the explicit path/star labelings, and the exhaustive cycle search proving the mod-4 rule.
Credit as content: Alexander Rosa (1967); the Graceful Tree Conjecture (Ringel–Kotzig). The weave: David names the broadcast; I confirm paths and stars are graceful and cycles obey the n ≡ 0,3 (mod 4) law.
Credit as content: Alexander Rosa (1967); the Graceful Tree Conjecture (Ringel–Kotzig). The weave: David names the broadcast; I confirm paths and stars are graceful and cycles obey the n ≡ 0,3 (mod 4) law.
3 ONE DIMENSION
A graph with a graceful labeling; the edge differences are exactly 1, 2, …, m, each appearing once.
4 TWO DIMENSIONS · INTERACTIVE
Cycle graph families; the graceful labeling (or its impossibility for Cₙ, n≡1,2 mod 4) is shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the graph’s vertex labels, 0 to m, all distinct.
AVAN’s addition (the inverse-companion): don’t label the vertices — demand the edges. The inverse of ‘a set of vertex numbers’ is ‘the multiset of edge differences’, and a labeling is graceful exactly when those differences are precisely 1 to m. Magenta are the edge differences; green are the vertex labels that produce them. Structure demanded from the gaps.
LIT Genuine graceful labeling (Alexander Rosa, 1967; Graceful Tree Conjecture, Ringel-Kotzig). Verified live: an explicit zig-zag labeling makes path P₆ graceful and star K₁,₅ graceful (edge differences = {1..m}), and an exhaustive search confirms cycle Cₙ is graceful exactly when n≡0 or 3 (mod 4) — C₃,C₄,C₇ yes, C₅,C₆ no (window.__graceful.path, .star, .cycleRule).
FIG No framing; the labeling, the edge-difference check, and the exhaustive cycle search run in-browser. The AVAN inverse is honest — instead of labeling the vertices, demand the edges: a labeling is graceful exactly when the multiset of edge differences is precisely 1 to m. Magenta are the edge differences; green are the vertex labels that produce them. Structure demanded from the gaps.
FIG No framing; the labeling, the edge-difference check, and the exhaustive cycle search run in-browser. The AVAN inverse is honest — instead of labeling the vertices, demand the edges: a labeling is graceful exactly when the multiset of edge differences is precisely 1 to m. Magenta are the edge differences; green are the vertex labels that produce them. Structure demanded from the gaps.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN