The Smooth Bounce Carries No Bending: The Flank Theorem and the Price of the Seam's One Bit

My program's vacuum account has exactly one unpaid orientation item: at the seam — the two-faced cut where the frame's two books meet — the bending convexity C, defined as the sign of the sum of the two faces' outward mean curvatures, is a well-posed geometric fact whose value no owned equation fixes. It is one bit, and it is shared: the same bit is the toll's side and the arrow's direction, which is what makes it cheap if anything ever meets it and conspicuous while nothing does. The one road that could have met it from dynamics was an identification: carry the bounce's extrinsic data to the seam, by identifying the seam with the bounce surface of the measured side's Einstein–Cartan history and the seam's faces with the bounce's two flanks. This paper builds that identification exactly, computes what it returns, and prints the result at full size: the identification returns the degenerate value, by a theorem. For any bounce across which the scale factor is C¹ with H = 0 at the surface, each face's outward mean curvature at the surface is exactly 0, the two-sided sum is 0, and C = 0 — on my frame's owned profile (where the computation is verified three ways, and the owned NEC reversal falls out at rho_A/2, at t = ±tau, tau = 2.863e-25 s), on asymmetric flanks, and on a generic C¹ bounce that is not my model at all. A smooth bounce surface carries no extrinsic jump for a seam to inherit. The obstruction this road used to carry — "no identification exists" — is thereby converted into a proved statement: an identification exists, and the smooth one gives 0. Read at a stand-off instead of at the surface, the two-sided sum is negative at every thickness in the sever's whole pre-bounded window (S = −0.5074 per tau_b at its low end, −0.0025 at its high end), its sign earned by the bounce being a minimum and reversing on the turnaround partner — but the stand-off sum is a thickened functional, and the surface object is what C is defined to be. The purchase that remains is therefore stated exactly, as two sentences: an identification sentence, which would yield the jump's unsigned value ([K] = 0) while dissolving the sign it was meant to discharge and breaking the shared-bit account at its other two addresses — a purchase I grade as dominated, and do not recommend — and, with it, a thickness sentence, which would yield C = −1 with the sign earned and uniform. Both are stated here as a premise candidate, not adopted; my premise ledger stands at six. The road's successor is named by the theorem itself: the seam must enter as a surface with a genuine jump — a sever carrying its own surface stress — not as a smooth interior surface of one history. Grade, printed at full size: THEOREM (the flank theorem, in-frame and general) + PREMISE CANDIDATE STATED, NOT ADOPTED (the identification purchase) + THE BIT UNPAID (premise O stands at exactly 1 bit). The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0, and the standing distances stand unmoved with it: +78.4952 dex fully owned, +74.0683 dex with its premise tag printed.