Padding Residual Gram
Abstract
The concrete residual-coordinate overlap is the minimum-head word multiplicity.
Theorem 1.1 (Padding Residual Gram).
Proof. Machine-checked in Lean as D5/S3/Quantum/StationaryPreparation/PaddingResidualGram.padding_inner (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every finite alphabet, arbitrary natural capacities encoded by a, every head, and every pair b and c of submultisets of a, the inner product of padding(b) and padding(c) is determined by their common coordinates. A nonzero tail contributes exactly at its whole-tail coordinate and at head indices bounded by the residual head count. The common coordinates of two residuals stop at their minimum head count.
Equal tails give the corresponding minimum-head word multiplicity, and distinct tails give zero. The last-tail masses telescope, including the sink and tail-free cases. After normalization, comparable residuals have the square-root multiplicity ratio times their tail moment; reversed comparisons use complex conjugation and incomparable pairs vanish.