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 28: Historical quadratic kernel as a candidate pre-contact force.

Provenance

Preserved from interaction notebook section 28, 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.

28. Historical quadratic kernel as a candidate pre-contact force

Historical V2 contained

Sinthist=12∑d,d′gdd′∫(ξd⋅ξd′)2 [ΨˉPdΨ] [ΨˉPd′Ψ]. S_{\rm int}^{\rm hist} = \frac12\sum_{d,d'}g_{dd'} \int (\xi_d\cdot\xi_{d'})^2 \,[\bar\Psi P_d\Psi] \,[\bar\Psi P_{d'}\Psi].

For d=d′=6d=d'=6, this gives a depth-six density-density kernel proportional to

(ξ⋅ξ′)2. (\xi\cdot\xi')^2.

After averaging one isotropic source distribution, historical V2 obtained

Vmf(ξ)∝g666⟨∣ξ′∣2⟩ ∣ξ∣2. V_{\rm mf}(\xi) \propto \frac{g_{66}}6\langle |\xi'|^2\rangle\,|\xi|^2.

This is a genuine extended coordinate-space mean field and therefore could influence trajectories before exact point contact.

This is a specified trial kernel. Its coefficients, spatial profiles and relation to the same-field action must be supplied before it predicts a physical force. A local field overlap can also act through extended tails; spatial locality does not require coincident pattern centres.

28.1 Sign and geometric behavior of the historical d=6d=6 pair kernel

The raw pair invariant

(ξ1⋅ξ2)2 (\xi_1\cdot\xi_2)^2

is nonnegative and is minimized when the two vectors are orthogonal, not when their spatial centers coincide. It is therefore not a central pair potential of the form V(∣ξ1−ξ2∣)V(|\xi_1-\xi_2|).

Its gradient with respect to ξ1\xi_1 is

∇ξ1(ξ1⋅ξ2)2=2(ξ1⋅ξ2)ξ2. \nabla_{\xi_1}(\xi_1\cdot\xi_2)^2 =2(\xi_1\cdot\xi_2)\xi_2.

Thus by itself it tends to change relative orientation/amplitude, not universally reduce the separation

∣ξ1−ξ2∣. |\xi_1-\xi_2|.

The harmonic well obtained historically arose only after averaging over an isotropic source density centered on the same mode, not from a literal attractive two-particle distance law.

This is another important negative result:

the historical (ξ⋅ξ′)2 kernel is not, by itself, the missing six-dimensional electron-capture force. \boxed{ \text{the historical }(\xi\cdot\xi')^2\text{ kernel is not, by itself, the missing six-dimensional electron-capture force.} }

To produce correlation between the observed and unobserved separation components dynamically, the relevant two-body operator must depend on full relative separation, a covariant geodesic analogue, or a connection whose equations reduce that separation.