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 20: Generalization to unequal particle depths.

Provenance

Preserved from interaction notebook section 20, 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.

20. Generalization to unequal particle depths

For a conditional uniform-extension control, write ordinary spatial coordinates as (x,y)(x,y) and suppose one field-density profile is independent of yy:

ρa(x,y)=ρa(x). \rho_a(x,y)=\rho_a(x).

For an integrable second density and a specifically chosen scalar overlap, iterated integration gives

Iab=∫dx ρa(x)∫dy ρb(x,y). I_{ab}=\int dx\,\rho_a(x)\int dy\,\rho_b(x,y).

This identity requires the stated profiles, measure and integrability. It concerns that scalar overlap. Uniformity in a direction means that profile has no independently localized centre offset in that direction; it does not make the complete interaction independent of the other pattern's polarization or shape.

The actual tails and their bound/free orientation determine which coordinates affect a pair response. No encounter-suppression exponent or strength hierarchy follows from the depth difference alone. In particular, an assumed scalar marginal does not derive an electromagnetic or weak interaction. Those identifications require the complete same-field dynamics and a causal observable response.