Beam-adapted $3+3$ frame links recover the pair invariant exactly

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 49: Beam-adapted $3+3$ frame links recover the pair invariant exactly.

Provenance

Preserved from interaction notebook section 49, 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.

49. Beam-adapted $3+3$ frame links recover the pair invariant exactly

This is a matrix identity for supplied orientation data. The matrices are not particle positions, and the chosen grouping does not impose a relation between spatial trajectories.

For $Q_1,Q_2\in SO(3)$, define

$$J(Q)=\begin{pmatrix}0&Q\\-Q^T&0\end{pmatrix},\qquad C_J=-\frac16\operatorname{tr}(J(Q_1)J(Q_2)).$$

If $Q_1Q_2^T$ has principal rotation angle $\phi$, then

$$C_J=\frac13\operatorname{tr}(Q_1Q_2^T)=\frac{1+2\cos\phi}{3}.$$

Defining the coefficient invariant $\nu=(1+\cos\phi)/2$ gives

$$\nu=\cos^2\frac\phi2,\qquad 1-C_J=\frac16\|Q_1-Q_2\|_F^2.$$

These are conditional algebraic descriptions of relative frames. They do not equate a frame angle with physical centre separation or establish any position-locking force.

Related

• Research status — current proof boundaries and correction guidance.
• One- and two-vector outer-product mixed ansatzes — preceding section in the research sequence.
• Conditional relative-separation theorem under two individual beam locks — following section in the research sequence.
• Correction to the rank-three requirement: a generic rank-two orbital bivector uniquely selects an $SO(3)$ frame by orientation-preserving Procrustes completion — later audit limits or corrects this route.