Frustration and holonomy in sheaves of qualia structures: quantitative local-to-global principles for phenomenal unity

Abstract

A recent proposal in the structural approach to consciousness [Lee-Youngzie, Tsuchiya, Robinson, Dietz & Monti, Entropy 28 (2026) 615] models the phenomenal unity of experience sheaf-theoretically: experiential parts form a finite topological space, empirical measures of their qualia form a presheaf of dissimilarity structures, and unity is the gluing of local sections into a global one. The framework rests on Robinson’s theory of sheaves of pseudometric spaces, whose consistency radius measures the disagreement of an assignment of local data, and it leaves open the quantitative questions: how far from unified is a given experience, when does approximate local agreement guarantee approximate global unity, and what is the precise obstruction in contextual cases such as binocular rivalry and the Penrose tribar? We answer these questions. We introduce the frustration f(S) of a sheaf of pseudometric spaces — the infimum of the consistency radius over all assignments — and prove: (i) a holonomy lower bound f(S) ≥ δ(hol)/2n along every closed zigzag through n overlaps with isometric restrictions, where δ is the minimal displacement of the holonomy isometry; (ii) an exact formula f = δ(hol)/2n for cyclic covers with midpoint (e.g. geodesic) stalks, which for the Penrose tribar in the phenomenologically motivated Weber–Fechner metric gives f = |log λ|/6; (iii) a vanishing-and-rectification theorem on acyclic covers — the upper-bound converse to Robinson’s inequality, yielding a two-sided “quantitative sheaf property”; (iv) an exact combinatorial formula for translation systems in one dimension via linear-programming duality (a minimax mean-cycle theorem), together with a proof that it fails in Euclidean dimension ≥ 2, where Chebyshev-type simplex obstructions invisible to any cycle holonomy appear — with the Helly number of the stalk geometry as the conjectured boundary; (v) a dichotomy on two-region covers separating fusion (midpoint stalks: cost g/2, attained by blends) from rivalry (categorical stalks: cost g, two symmetric optima), a formal account of the fusion-to-rivalry transition and of alternation as bistability of frustration minimizers; and (vi) for the category DIS of dissimilarity structures used in the Entropy framework, that all nonempty finite limits exist and force a sharp, previously unnoticed empirical prediction — the unity equation: broad dissimilarity equals the minimum of the corresponding narrow dissimilarities. Every theorem is verified by an accompanying computational suite (25 checks, all passing). We close with the inferential logic these bounds licence for experiments on phenomenal unity, and with open problems.