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 18: A stronger invariant candidate from two complex structures.

Provenance

Preserved from interaction notebook section 18, 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.

18. A stronger invariant candidate from two complex structures

Take two supplied orientation matrices with

JiT=−Ji,Ji2=−I6. J_i^T=-J_i,\qquad J_i^2=-I_6.

Their simultaneous-rotation invariant is

CJ=−16tr⁡(J1J2),AJ=112∥J1−J2∥F2=1−CJ. C_J=-\frac16\operatorname{tr}(J_1J_2),\qquad A_J=\frac1{12}\|J_1-J_2\|_F^2=1-C_J.

Thus AJ≥0A_J\ge0, with equality at identical matrices. This is an algebraic mismatch measure for these supplied orientations, not a physical distance between pattern centres. A term λJAJ\lambda_J A_J with positive coefficient would assign a lower energy to equal matrices in that specified trial functional. It does not itself derive that functional, establish relaxation toward its minimum, or make any particle coordinates mirror one another.

Actual field patterns can contain multiple orientations and spatially varying structure. A pair of these matrices describes only the chosen orientation data, not the complete interaction.