Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Research record, section 25: Pair-invariant collapse on the electron sector.

Provenance

Preserved from interaction notebook section 25, recorded 2026-09-05 and corrected 2026-09-06. Original sequence contents. This is an exploratory research record, not an independently established V2 result. Read the related later audits before using an intermediate conclusion.

25. Pair-invariant collapse on the electron sector

For normalized four-component chiral coefficient vectors Z1,Z2Z_1,Z_2 in the stated spinor representation, define

ν=∣Z1†Z2∣2. \nu=|Z_1^\dagger Z_2|^2.

The associated orientation matrices satisfy the exact algebraic identities

CJ=−16tr⁡(J1J2)=4ν−13,AJ=1−CJ=4(1−ν)3. C_J=-\frac16\operatorname{tr}(J_1J_2)=\frac{4\nu-1}{3},\qquad A_J=1-C_J=\frac{4(1-\nu)}{3}.

If the ray angle is defined by ν=cos⁡2θ12\nu=\cos^2\theta_{12}, then AJ=43sin⁡2θ12A_J=\frac43\sin^2\theta_{12}. This is an angle between coefficient rays. It is not a separation between physical particle centres.

At orthogonal rays, CJ=−1/3C_J=-1/3; at identical rays, CJ=1C_J=1. The identities relate two descriptions of the chosen polarization data. They do not force trajectories to approach, remove transverse motion, or supply a position-locking interaction. Physical interaction depends on both actual spatial field patterns and their causal dynamics.