Two individual locks reduce to one exact relative restoring mode

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 51: Two individual locks reduce to one exact relative restoring mode.

Provenance

Preserved from interaction notebook section 51, 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.

51. Two individual locks reduce to one exact relative restoring mode

Take a supplied positive quadratic functional on abstract three-component vectors and orientation matrices, with $\kappa>0$:

$$V=\frac\kappa2\left(\|h_1-Q_1^Tu_1\|^2+\|h_2-Q_2^Tu_2\|^2\right).$$

Set $U=(u_1+u_2)/2$, $u=u_1-u_2$, $H=(h_1+h_2)/2$ and $h=h_1-h_2$. Minimizing over $H$ at $U=0$ gives the exact algebraic reduction

$$\boxed{ V_{\rm rel}=\frac\kappa4\left\| h-\frac{Q_1^T+Q_2^T}{2}u \right\|^2. }$$

This identity concerns the stated quadratic functional. Identifying its abstract vectors with electron positions would impose an additional coordinate-lock hypothesis, which is not part of the interaction model. No scattering force, trajectory correlation or physical restoring coefficient follows from completing this square.

Related

• Research status — current proof boundaries and correction guidance.
• Conditional relative-separation theorem under two individual beam locks — preceding 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.