Conditional relative-separation theorem under two individual beam locks

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 50: Conditional relative-separation theorem under two individual beam locks.

Provenance

Preserved from interaction notebook section 50, 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.

50. Conditional relative-separation theorem under two individual beam locks

Consider abstract three-component vectors subject to the supplied linear constraints $h_i=Q_i^Tu_i$, with $Q_i\in SO(3)$. This is a constrained-algebra comparison, not an electron-trajectory premise.

At $u_1=u/2$, $u_2=-u/2$, the difference is

$$h=h_1-h_2=Au,\qquad A=\frac12(Q_1^T+Q_2^T).$$

For relative rotation angle $\phi$ and $\nu=\cos^2(\phi/2)$,

$$\sigma(A)=\{1,\sqrt\nu,\sqrt\nu\},\qquad |h|^2=|u_{\parallel}|^2+\nu|u_{\perp}|^2.$$

Consequently $\nu|u|^2\le|h|^2\le|u|^2$. If $\xi=(u,h)$, then

$$(1+\nu)|u|^2\le|\xi|^2\le2|u|^2.$$

These bounds follow from the imposed vector constraints. They supply no physical correlation between observed and unobserved positions, and no particle encounter probability.

Related

• Research status — current proof boundaries and correction guidance.
• Beam-adapted $3+3$ frame links recover the pair invariant exactly — preceding section in the research sequence.
• Two individual locks reduce to one exact relative restoring mode — 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.