Subjectivity-Intersection Mathematics: All-Orders Intersection Consistency and Compositional Objectivity from Pointed Unitary Braids

Abstract

Subjectivity-Intersection Mathematics asks whether structures treated as objective can be composed as stable consequences of lawful intersections among differentiated standpoint roles rather than supplied as standpoint-free primitives. We study one restricted homogeneous operator representation of that principle. Objective mathematics is used here as a representational instrument downstream of registered standpoint-relative intersections; no formal datum or objective reconstruction is identified with subjectivity or Subjectivity Intersection itself, and no such representation is promoted into an exhaustive objectification of its generative principle. Let H be a complex Hilbert space, e in H a unit vector, and R in U(H tensor H) a checked-braid solution satisfying R(e tensor e) = lambda(e tensor e), with |lambda| = 1. At every finite order, the canonical braid-orbit isometries have constant ordered overlaps equal to the phase-normalized right endpoint slice; leg reversal gives the left family. Within the declared Subjectivity-Intersection representation, we call this neutral canonical overlap property all-orders intersection consistency; the interpretive name adds no mathematical conclusion. The standard constant-overlap Gram criterion forces both phase-normalized slices to be positive contractions, while the pointing condition makes both fix e. No finite-dimensionality, separability, involutivity, self-adjointness, or trace assumption is used. Together with separately proved pointed-unitary covariance and compatibility with the declared shuffled tensor law, this verifies the stipulated technical compositional-objectivity criterion in this representation class. The sole general-mathematics novelty candidate is the complete implication from a pointed unitary braid to canonical all-orders overlaps and bilateral endpoint positivity. Its Gram, localization, compression, covariance, and tensor ingredients are cited rather than claimed; historical priority remains open pending claim-level search and specialist review. Separately, complete population response laws on finite registered sets determine source-relative quotients, and a supplied pair kernel descends exactly under fibre constancy. These facts specify an ideal-population prospective certificate contract, not recovery of its inputs or a finite-sample natural-identification theorem. The atomic, NMR, and FPR1 records are component-level compatibility checks. The cryo-EM check yields a byte-identical canonicalized candidate-matrix artifact over heterogeneous class averages, while its exact-path baseline remains incomplete. No natural certificate is accepted and no result identifies the formal variables with phenomenal subjectivity.