{"id":"hexagon/2610.00470","pdf":"https://hexagonmath.org/pdf/2610.00470v1","url":"https://hexagonmath.org/abs/2610.00470v1","tags":["math.CT","math.FA","msc:81T25","msc:47A20","msc:81T17","coarse graining","cylindrical consistency","finite gauge theory","spin foams","tensor gluing"],"score":0,"title":"Gluing defects for projected coarse graining in finite spin-foam models","domain":"math.MP","source":"hexagon","authors":["Zhenyu Sun"],"reviews":0,"subject":"math.MP","summary":"Orthogonal truncation can discard states that contribute to a later boundary amplitude. For a self-adjoint contraction $T$ on a finite-dimensional Hilbert space and an isometric inclusion $s$ of retained states, we prove that the two-step residual $\\Delta=s^*T^2s-(s^*Ts)^2$ controls every finite placement of intermediate projections in every Schatten norm. For blocks of lengths $n_1,\\ldots,n_k$, the error is at most $\\sum_{i<j}n_in_j\\|\\Delta\\|$, and the constant is optimal even for positive contractions. The residual vanishes exactly when the retained subspace is invariant. Two-dimensional…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}