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 32: Can a pulled-back depth-two connection reduce unobserved transverse momentum?.

Provenance

Preserved from interaction notebook section 32, 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.

32. Can a pulled-back depth-two connection reduce unobserved transverse momentum?

Let the local tangent split be

TQ6∣Q2=TQ2⊕N6/2, TQ_6|_{Q_2}=TQ_2\oplus N_{6/2},

with

dim⁡RTQ2=2,dim⁡RN6/2=4. \dim_{\mathbb R}TQ_2=2, \qquad \dim_{\mathbb R}N_{6/2}=4.

For a pure pullback of a one-form A(2)A^{(2)} from Q2Q_2,

A(6)=π62∗A(2), A^{(6)}=\pi_{62}^*A^{(2)},

one has

A(6)(n)=0 A^{(6)}(n)=0

for every vertical vector

n∈N6/2 n\in N_{6/2}

when the splitting is an ordinary bundle submersion and the pullback contains no additional normal connection components.

Likewise the pulled-back curvature

F(6)=π62∗F(2) F^{(6)}=\pi_{62}^*F^{(2)}

has no purely vertical legs:

F(6)(n1,n2)=0. F^{(6)}(n_1,n_2)=0.

Therefore a pure depth-two pullback by itself cannot directly supply a Lorentz force in the vertical four-dimensional normal bundle.

This is an important conditional theorem:

if the electron’s electromagnetic connection is only π62∗A(2), then ordinary EM does not directly damp unobserved normal momentum. \boxed{ \text{if the electron's electromagnetic connection is only }\pi_{62}^*A^{(2)}, \text{ then ordinary EM does not directly damp unobserved normal momentum.} }

To act on the extra electron directions one needs additional connection components or mixed curvature,

F∥⊥≠0 F_{\parallel\perp}\neq0

or

F⊥⊥≠0, F_{\perp\perp}\neq0,

arising from the higher-sector connection rather than from a bare pullback.

The higher-dimensional magnetism module explicitly allows such additional supported curvature planes, but it does not derive them from an electron source or show that they participate in electron--electron Coulomb scattering.

Hence there are two possible electromagnetic regimes:

  1. pure projected EM: correct low-dimensional propagation, no direct unobserved steering;
  2. lifted/mixed-curvature EM: possible unobserved steering, but its source equation and normalization remain to be derived and must preserve the observed Coulomb law.