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 14: Central full-depth dynamics does not by itself prove unobserved-visible coincidence.

Provenance

Preserved from interaction notebook section 14, 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.

14. Central full-depth dynamics does not by itself prove unobserved-visible coincidence

As a restricted centre-motion control, take a relative coordinate X∈R6X\in\mathbb R^6, R=∣X∣R=|X|, and a specified potential V(R)V(R). Its force is

Fi=−V′(R)RXi. F_i=-\frac{V'(R)}R X_i.

This assumes the force depends only on distance and points along the centre-separation vector. Merely having a centre does not impose this assumption. General spatial patterns carry multiple orientations, shape and phase data that a distance-only law discards.

In this conservative central-force control, the angular-momentum bivector

Lij=μ(XiX˙j−XjX˙i) L_{ij}=\mu(X_i\dot X_j-X_j\dot X_i)

is conserved. With L2=12∑i,jLij2L^2=\frac12\sum_{i,j}L_{ij}^2, radial energy contains the centrifugal term L2/(2μR2)L^2/(2\mu R^2). This is a statement about the prescribed centre-force model, not a derivation of the response of actual particle tails.

Tail interaction does not require eliminating transverse centre motion or imposing matching trajectories in different directions. The full field must determine any orientation-dependent forces and reciprocal motion, with causal transmission of changes.