Frustration and holonomy in sheaves of qualia structures: quantitative local-to-global principles for phenomenal unity
In [1] we introduced the frustration f(S)f(S) of a sheaf of pseudometric spaces — the least consistency radius (in the sense of Robinson [3, 4]) achievable by any assignment — and proved metric local-to-global principles: holonomy displacement bounds, exact values on cycles, and rectification on acyclic covers. Prompted by correspondence on that paper, we here restrict the stalks to finite-dimensional normed vector spaces and show that the entire theory becomes cohomological. For an affine sheaf — a linear cellular sheaf SS on a multigraph twisted by a 11-cochain zz — we prove the cochain-level reformulation fp(S,z)=21/p−1distp(z,imδ0)=21/p−1∥[z]∥H1,f_p(S,z) = 2^{1/p-1}\operatorname{dist}_p(z,\operatorname{im}\delta^0) = 2^{1/p-1}\|[z]\|_{H^1}, one half (for p=∞p=\infty) the quotient norm of the obstruction class in sheaf cohomology: frustration is a norm on H1H^1, its optimal assignments are minimal-norm cocycle representatives (harmonic for p=2p=2, minimax for p=∞p=\infty), and in finite dimensions the obstruction is complete: f=0f=0 iff [z]=0[z]=0 iff a global section exists. Duality identifies the frustration with a maximization over the dual cycle space kerδT\ker\delta^T; by Rockafellar’s theory of elementary vectors, for rank-one (gain-graph) sheaves at p=∞p=\infty the optimal dual certificates are supported on the circuits of Zaslavsky’s frame matroid — balanced cycles and unbalanced theta/handcuff pairs — yielding a closed combinatorial formula that strictly extends the mean-cycle theorem of [1], and explaining exactly when cycle holonomy fails to determine frustration: the maximizing certificate can be a handcuff. For translation systems we identify the extremal frustration-to-cycle-bound ratio on two-vertex (banana) graphs with the Jung constant J(V)J(V) of the stalk norm — the equilateral-theta gap 2/32/\sqrt3 of [1] is exactly J(ℓ22)J(\ell_2^2) — settling the two-vertex case and the necessity direction of the Helly boundary question posed there, with sufficiency conjectured (and numerically supported): bananas appear to be the worst case in general. For homogeneous linear sheaves the natural invariant is the unit-normalized frustration, which equals σmin(δ)/2\sigma_{\min}(\delta)/\sqrt2 — equivalently, its square is half the spectral gap λmin\lambda_{\min} of the Hansen–Ghrist sheaf Laplacian; on cycles with orthogonal restriction maps λmin=2−2cos(θ∗/n)\lambda_{\min}=2-2\cos(\theta^\ast/n), and we give the exact dictionary between this spectral theory and the metric cycle formula of [1]. Finally we connect all of this to Robinson’s transmission-line sheaves on quantum graphs [J. Differential Equations 260 (2016) 872–896]: in the worked loop-with-tail example his resonance conditions and cohomology-dimension jumps are precisely the zero locus of our quantitative obstruction σmin\sigma_{\min} of the secular map, which we compute in closed form; loss imposes a uniform positive lower bound on it; and his gauge freedom under edge collapse leaves cohomology invariant while transforming frustration by condition-number factors — frustration sees the geometry that cohomology forgets. The main identities and examples are machine-verified (18 further checks, all passing).