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 4: General depth-mediated interaction rule.

Provenance

Preserved from interaction notebook section 4, 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.

4. General depth-mediated interaction rule

Suppose a stated calculation supplies maps from two field-current descriptions to a common resolved domain. Write those maps as TbT_b and TcT_c. A reduced bilinear expression can then be written schematically as

Ibc=⟨TbJb,KTcJc⟩. I_{bc}=\langle T_bJ_b,K T_cJ_c\rangle.

This is conditional operator notation. The maps, kernel, integration measure, normalization and retained current components must be specified. For a time-dependent physical response, the reduction must also retain the causal history required by the same-field equations. A frozen spatial bilinear is not by itself that causal law.

If a map is defined as complete marginalization of a normalized integrable density, it preserves total weight. That mathematical identity does not establish that the interaction uses that marginal, discards the remaining spatial structure, or has a known coupling strength. A nonzero invariant contraction supplies an allowed algebraic structure; the actual occupied patterns and dynamics determine its response.

No mediator particle is introduced. Identifying any such reduction with electromagnetism is an additional physical derivation, not a consequence of the photon's depth or the existence of a projection map.