Gluing defects for projected coarse graining in finite spin-foam models
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…
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