Incidence Geometry of Projected Operator Composition

This paper studies when two nonzero projected operator factors have a zero product. The exact criterion is image-kernel incidence: the product vanishes precisely when the image of the right factor is contained in the kernel of the left factor. In a sectorized operator setting, this is the local obstruction to promoting a Boolean support path through an intermediate sector to a nonzero routed composition. The theorem layer studies the closed zero-product locus and its constructible nonzero-factor part. It derives the dimensions and codimensions of fixed-rank and fixed-double-rank incidence strata. On a stratum where the two factors have ranks r and s, the incidence condition has relative codimension r times s. Full column rank of the left factor or full row rank of the right factor excludes nonzero zero-products and provides quantitative singular-value lower bounds. The exact examples realize five routed-product classes: both-protected, left-only, right-only, unprotected-nonzero, and unprotected-zero. In the declared complex128 Rubik realization, 2592 nonzero-factor routed records split into 2016 unprotected nonzero products and 576 machine-zero image-kernel alignments. The 576 exceptional records are concentrated in four sector triples. These are finite computational observations rather than exact symbolic incidence identities. The conclusions stop at one routed matrix product. They do not identify a support path with a routed product, a routed product with a full word, a full word with a commutator, or low-order commutator support with Lie depth. Ambient codimension in the free matrix-pair space also does not determine the pullback incidence geometry of a representation-derived family. Version 2.1 adds an exact fixed-frame carrier certificate for the four canonical Rubik zero-route concentrations and a source-addressed registration of the 19-orbit canonical-frame census. Exact arithmetic certifies every zero in the registered numerator. Nonvanishing of all complementary supported routes is numerical, so the observed 2/9 rate remains a bounded Computational Observation rather than a universal law. Source repository: https://github.com/dooven-prime/rime-lite Paper VII of the Representation-Induced Mechanics and Evolution (RIME) program. Independent researcher; no institutional affiliation.