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 37: Next proof obligation.

Provenance

Preserved from interaction notebook section 37, 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.

37. Next proof obligation

The pair-response problem is to derive a causal interaction from the one-field dynamics of both actual spatial patterns, including their extended tails, multiple orientations and reciprocal response. A projector/connection reduction is one possible tool when obtained from that dynamics; it is not a mandatory physical route.

Any proposed reduction must retain its kinetic normalization, energy and momentum balance, and distinguish supplied profiles and coefficients from derived particle solutions. An electromagnetic identification remains open until calculated. Neither exact centre coincidence, an imposed coordinate lock nor an independently chosen statistical encounter penalty is a prerequisite for tail interaction.