The Codimension-Three Rung. Twisted sections over a non-orientable link: the sector split of the Kondratiev indicial roots, a root-free window that triples, and the character ladder of lens rungs

Abstract

The blow-up ladder of the framework's desingularization article was carried out for codimension-two centers, where the exceptional divisor is a circle; Section 9.10 explicitly left open the codimension-three rung and the full Kondratiev asymptotics. This article closes both. The starting observation is structural: the divisor of a real projective blow-up along a codimension-n center is RP^{n-1}, which is non-orientable exactly when n is odd — so codimension three is the first rung on which the framework's twisted sector is forced by the geometry rather than assumed. Three results. (T1) On the RP^2 link the antipodal involution acts on degree-l harmonics by (-1)^l; the even sector is l even (dimensions 1,5,9,...), the twisted sector is l odd (3,7,11,...). The twisted sector has no zero mode, hence a spectral gap: its least link eigenvalue is 2, not 0. (T2) The Kondratiev indicial roots of the Laplacian on the cone are lambda=l and lambda=-(l+n-2), so the sector split propagates to the asymptotics: in codimension three the even sector carries the roots 0 and -1, the twisted sector the roots 1 and -2. Consequently a twisted section admits neither a constant term nor a Coulomb term r^{-1}, and its root-free weight window widens from 1 to 3 — the Kondratiev window triples. In general codimension the widths are n-2 (even) and n (twisted). (T3) For weighted rungs with lens links L(p,q) the split refines into a Z_p-character ladder: character j first appears at harmonic degree min(j,p-j), giving window width 2min(j,p-j)+2; only the trivial character contains the constant. All statements are verified deterministically (38/38), including harmonic bases built from scratch and finite-difference checks of the indicial equation. The article proves no new elliptic theory; it computes where the framework's sector structure sits inside the classical Kondratiev picture.