infinite_crossings_need_infinite_budget
plain-language theorem explainer
Infinite crossings of topological links require infinite budget when each crossing carries positive cost per crossing. Researchers deriving finite-capacity vetoes for rigid rotations in three-dimensional field configurations cite this when bounding link complexity by available energy. The proof is a one-line wrapper that applies the finite crossings from budget lemma.
Claim. Let $c > 0$ be the cost per crossing and $B ≥ 0$ the total budget. Then for every natural number $N$, if $N · c ≤ B$ then $N ≤ B / c$.
background
The TopologicalVeto module derives finite-capacity vetoes in D = 3 from Alexander duality and link penalties. Each topological crossing costs ln φ > 0 by the link_penalty_positive result, and finite energy implies finitely many crossings via the finite_budget_finite_crossings statement in the module. The local setting assumes the Recognition Science forcing chain with D = 3 spatial dimensions and the eight-tick octave.
proof idea
The proof is a one-line wrapper that applies the finite_crossings_from_budget lemma to the given budget and cost hypotheses.
why it matters
This fills the F6.3.3 step in the Foundation paper, establishing the budget obstruction for infinite link crossings. It supports the master veto that rigid rotation cannot arise from finite-energy initial data, since infinite crossings would be needed to reach zero linking over infinite extent. The result connects to the D = 3 forcing step and the positive link penalty in the topological capacity veto chain.
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