The reciprocal pushforward formulation is more faithful to current V2

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 41: The reciprocal pushforward formulation is more faithful to current V2.

Provenance

Preserved from interaction notebook section 41, 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.

41. The reciprocal pushforward formulation is more faithful to current V2

The bound-within/free-without theorem uses a pushforward rather than simple orthogonal deletion. For a higher-depth current $J_6(x,y)$,

$$M J_6 =(\pi_{62})_*J_6 =\int J_6(x,y)d\mu_F(y).$$

The lower-depth carrier is lifted cylindrically by

$$L=\pi_{62}^*.$$

With normalized fibre pairing, the structural identity is

$$ML=I$$

on the lower-depth test space, in the generalized/box-normalized sense already used by the contact theorem.

Therefore the electromagnetic lifting condition is naturally

$$\boxed{ (\pi_{62})_*\,\mathcal K_6\,\pi_{62}^* =\mathcal K_{\rm EM}^{(2)}. }$$

Any correction $\mathcal N$ satisfying

$$\boxed{ (\pi_{62})_*\,\mathcal N\,\pi_{62}^*=0 }$$

is invisible to the direct depth-two pushforward test.

Such an $\mathcal N$ may nevertheless act nontrivially on variations within the fibre and can therefore correlate higher-depth coordinates.

This is much closer to IDWT than the earlier ordinary six-dimensional Maxwell extrapolation.

Related

• Research status — current proof boundaries and correction guidance.
• Compression is not the same as effective reduced dynamics — preceding section in the research sequence.
• First classification of observationally invisible higher-depth operators — following section in the research sequence.