Interaction price

In plain terms

Here is the problem. Free field dynamics keeps observables confined to their modes forever. Interactions spread them. The qualitative statement is folklore; the quantitative question is the interesting one: at what RATE does interaction buy delocalisation, and is the rate you can prove also the rate that actually occurs?

The corpus answers with a sandwich. Upper bound: after one period of an interacting drive, every local observable stays within twice coupling-times-time of some exactly local operator -- locality violation priced linearly. Lower bound, on an explicit witness: a computable commutator forces every exactly local operator to be at least a comparable linear distance away.

Between the two, "costs at most" becomes "costs exactly", up to a constant between one-over-pi and two.

In CSD

The price ladder is the field track's quantitative locality backbone: it feeds the propagator's interacting corrections, the cutoff-uniform renormalised couplings, and the channel-level coarse-graining budget, where the whole error of a renormalisation step is the interaction's price because the free part intertwines exactly.

The attainment closed a declared boundary: the upper bound had stood alone with its sharpness explicitly unclaimed, and the witness construction retired that caveat.

Mathematically

price_linear_attained: on the two-mode witness pair, and for tau lambda in the window from zero to pi (both bounds are hypotheses of the theorem, not decoration -- the sine is only monotone there), the distance from the interacting Heisenberg observable to the subalgebra supported on the free modes satisfies tau lambda / pi <= dist <= 2 tau lambda -- the lower bound via the commutator functional (any supported operator commutes with a disjoint probe, so one computed commutator bounds the distance to the whole subalgebra) with the commutator entry of modulus 2 sin(tau lambda / 2) exactly, and the Jordan inequality converting the sine to the linear form.