This preprint develops the local extraction mechanism for the canonical odd-branch crossing datum in hierarchy-compressed transport. In the B-line of the \kappa-theory series, odd-order turning points carry a canonical local crossing datum \mathcal C_m=(\Phi_m,A_m), once the distinguished decaying branch is fixed on the evanescent side. The purpose of the present note is to explain how this datum is read from normalized odd local structure. For odd local order m=2k+1, the local turning-point model has sign-changing geometry: one side is evanescent and the other oscillatory. After normalization to the odd local normal form, one fixes the distinguished decaying input on the forbidden side and reads off the leading real oscillatory continuation on the allowed side. This continuation determines the canonical crossing phase \Phi_m and amplitude coefficient A_m. The first two odd hierarchy levels, m=1,\qquad m=3, provide the first realizations of this extraction scheme, yielding the first extracted data \mathcal C_1 and \mathcal C_3. This note is the machinery companion to the odd-branch datum note. It does not restate the hierarchy law, does not treat the even branch, and does not develop semi-global exponent laws, codimension theory, or physical matching problems.