Frustration, holonomy and projection constants: the sharp cycle gap for vector-valued sheaf cohomology on graphs

We study the gap between frustration and cycle-holonomy bounds for vector-valued translation sheaves on graphs. For a finite graph G and a finite-dimensional normed stalk X, we identify the cycle gap K(G,X) with the X-valued extension constant of the cycle space Z₁(G) ⊂ ℓ₁(E), equivalently with a projective-tensor distortion and the reciprocal of an injection modulus. The sharp bounds are K(G,X) ≤ λ(X) and K(G,X) ≤ λ(Z₁(G), ℓ₁(E)). Moreover, taking the supremum over all stalks X gives sup K(G,X) = λ(Z₁(G), ℓ₁(E)), while taking the supremum over all graphs G gives sup K(G,X) = λ(X). These results refute the earlier conjecture that the Jung constant J(X) controls the gap on all graphs. Explicit counterexamples are given for Euclidean stalks, including K₄ with X = ℓ₂², as well as exact hypercube examples. The resulting hierarchy is J(X) ≤ E(X) ≤ λ(X), where bananas compute the Jung constant, generalized theta graphs compute Grünbaum’s expansion constant E(X), and arbitrary graphs recover the absolute projection constant λ(X). Further results include characterizations of when cycle bounds are tight, quantitative growth estimates, convergence of p-harmonic minimal representatives as p → ∞, and Hodge-theoretic decompositions of frustration on 2-complexes. The accompanying source and verification package contains exact rational, algebraic, and interval-arithmetic certificates for the computational claims used in the paper.