The Dump Is Dust: The Crossing's Deposit and the Earned Sign of the Junction

A companion paper in this program (The Sever as a Shell, production run 18) reduced the junction sign of the vacuum campaign to a single named remainder: with the sever cut read as an Israel thin shell, the sign is \mathcal{C} = \mathrm{sign}(\sigma - 2p), where \sigma is the shell's surface density and p its tangential pressure — and the owned record fixed \sigma \ge 0 but owned nothing at all about the shell's equation of state w_s = p/\sigma. That remainder (the dump-trace bit, R-P10D) is discharged here. The shell at the sever cut is the dump: the accumulated deposit of everything that has crossed the cut. Crossings — the record's owned crossing postulate: sever events are crossings, and crossings carry flux bookkeeping — deposit energy and normal momentum at the cut; nothing in a crossing deposits tangential momentum, and nothing in the accumulated deposit pushes sideways. The dump's tangential pressure is therefore zero by construction, p = 0, w_s = 0: the dump is dust, and \mathcal{C} = +1, earned rather than assigned, at the price of the three sentences itemized in §2, each printed with its cost. The earning is robust in a stated band: an admixed bath flips the sign only if its fraction \chi and equation of state satisfy \chi \cdot w_{\mathrm{bath}} > \tfrac{1}{2}, which a radiation bath can never reach and a normal-incidence crossing flux cannot supply in stiff form. What remains open is printed at the same size: the magnitude E_{\mathrm{dump}}(m) is not computed here, and a second, distinct sign-bit at the toll's emitter address (Bit-E) is not this bit and is not paid by this paper. Campaign headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.