3511 — Higher Marked-Wall Shape Poles Have Invertible Proper-Face Leading Symbols
Hard-to-vary claim
For the native six-term physical second-shape insertion, the associated-grade lowering symbol is invertible for every marked-wall pole above logarithmic order on its proper face. This excludes an independent generator on the leading pole grade. It does not yet construct the full filtered contraction.
Exact pole census
All six occurrences retain pole order three on (g_1) and (g_2), and pole order one on (g_3) and their selected (B)-wall. The four occurrences whose selected small wall moves under the shape deformation retain order three on that wall. The two occurrences on the invariant wall (s_{12}) reduce exactly to order one.
The Cayley--Menger factor retains the separately typed order (K_0^{-2}) in addition to the square-root twist. It is not included among the marked-wall lowering directions.
Proper-face lowering
For each of the nine labelled linear walls (s), the frozen coordinates admit a normal variable (x_s) satisfying
On the associated grade of pole order (m>1), twisted de Rham differentiation has leading symbol
This is invertible in characteristic zero. Exact division also verifies that no marked wall divides (K_0). Hence no associated grade (m>1) supplies an independent proper-face generator.
All 24 pole-profile, unit-normal, and proper-face checks pass.
Consequence
Treating the cubic poles as three independent residues would overcount the associated-grade object. Their contribution must instead be transported through a source-normalized filtered lowering map. Until that map and its intersection coherence are constructed, the full proper-face coefficient object has not been computed.
Scope and next falsifier
This does not address lower-order terms in the triangular differential, wall intersections, or the Cayley--Menger branch. Those may carry excess or supported classes. The next test is to recursively construct the complete finite lowering homotopy, prove constancy on its full linear fibers, and test pairwise-intersection coherence before forming the simple-residue matrix.
Evidence
research/benincasa/compile_relative_shape_pole_depth.py;research/benincasa/results/relative-shape-pole-depth.json.
Allocator claim: seqclaim-ff354a8ff4fa8e9c154a2984.