Proposition

Lemma 14 ([One free period]). A block of k layers and period p can be joined internally, in place of its outgoing link, for cost at most (k-1)p . Thus this replacement leaves at least p of its total budget available.

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ProveOct 21, 2026

Lemma 9 ([Joining an ordered band]). Consider k ordered layers from one…Proposition 13. After all groups have been processed, every layer is either…Proposition

For an individual block, or any block with k=1 , no internal addition is needed. For a collective block, gives f_H(t)-l_ L_* (t)≤(k-1)p. Apply Lemma 9 ([Joining an ordered band]) with h=(k-1)p .

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