Direct projection of the current V2 scalar quartic into a two-electron channel

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 26: Direct projection of the current V2 scalar quartic into a two-electron channel.

Provenance

Preserved from interaction notebook section 26, 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.

26. Direct projection of the current V2 scalar quartic into a two-electron channel

The present V2 mass interaction is not the historical coordinate kernel. The current action-level scalar vertex is

$$\boxed{ \mathcal L_{\rm sc,6} =-\frac{\kappa_6}{2}J_6^2, \qquad J_6=\bar\Psi_6\Psi_6. }$$

Therefore the first honest $K_{66}$ test must begin here.

26.1 Two localized electron-sector contributions

Write the depth-six field locally as two localized contributions,

$$\Psi_6=\psi_1+\psi_2.$$

Then

$$J_6 = \bar\psi_1\psi_1 + \bar\psi_2\psi_2 + \bar\psi_1\psi_2 + \bar\psi_2\psi_1.$$

Define

$$J_1=\bar\psi_1\psi_1, \qquad J_2=\bar\psi_2\psi_2,$$

and the coherent interference scalar

$$X=\bar\psi_1\psi_2+\bar\psi_2\psi_1.$$

Then

$$J_6=J_1+J_2+X.$$

The quartic expands exactly as

$$\mathcal L_{\rm sc,6} =-\frac{\kappa_6}{2} \left( J_1^2+J_2^2+X^2+2J_1J_2+2J_1X+2J_2X \right).$$

Subtracting the two isolated self-energies leaves the two-packet interaction density

$$\boxed{ \mathcal L_{12}^{\rm sc} =-\kappa_6J_1J_2 -\kappa_6(J_1+J_2)X -\frac{\kappa_6}{2}X^2. }$$

This is the first direct $K_{66}$ result from the current V2 action.

26.2 Incoherent/separated limit

When the packets overlap in density but carry negligible coherent scalar interference,

$$X\to0,$$

the cross term reduces to

$$\boxed{ \mathcal L_{12}^{\rm sc} =-\kappa_6J_1J_2. }$$

For a non-derivative interaction, the interaction contribution to the Hamiltonian density is

$$\mathcal H_{12}^{\rm sc} =-\mathcal L_{12}^{\rm sc} = \boxed{\kappa_6J_1J_2}.$$

Thus the sign of the physical two-electron scalar interaction is controlled by the sign of the projected coefficient $\kappa_6$ together with the signs of the scalar bilinears $J_i$.

The current V2 files do not independently fix the sign of $\kappa_6$ as a two-body scattering coefficient. The mass theorem fixes only

$$\mu_6=\kappa_6R_6B_6$$

as the positive observed rest-frequency scale after profile and spinor contractions. Therefore one cannot infer the sign of $\kappa_6$ alone from $m_e>0$ without also fixing the conventions/signs of $R_6$ and $B_6$.

This means the current scalar quartic does not yet predict whether the direct full-depth $e$--$e$ scalar channel is attractive or repulsive.

26.3 Support of the direct scalar channel

The current scalar vertex is algebraic and local. In the coordinate representation its kernel contains

$$\delta_{Q_6}(q,q')$$

and, on the occupied mode ray, the phase-ray delta used by the mass theorem.

Therefore its strict two-particle cross interaction is supported where the two depth-six field contributions overlap on the same $Q_6$ point. At leading order it does not provide a long-range force that gradually pulls an unobserved impact parameter toward zero.

This is decisive for the beam-correlation idea:

$$\boxed{ \text{the current order-zero scalar contact can act once full-depth overlap occurs, but by itself it cannot steer initially separated six-dimensional trajectories into overlap.} }$$

A steering or capture mechanism requires at least one of:

1. a finite-width/nonlocal scalar kernel;
2. a derivative/tensor interaction carrying information away from the diagonal;
3. a gauge/connection-mediated force acting before full contact;
4. the historical coordinate-space quadratic kernel, if it can be re-derived consistently from the current master action.

26.4 Coherent interference terms

The $X$ terms show that the scalar quartic is not merely density-density contact when two coherent pieces of the one master field overlap. The interaction contains

$$-\kappa_6(J_1+J_2)X$$

and

$$-\frac{\kappa_6}{2}X^2.$$

These terms are sensitive to relative spinor phase and scalar polarization. They are therefore a natural place for polarization-dependent full-contact structure.

However, they still have the same local support as the underlying order-zero vertex. They change what happens at overlap, not the pre-contact unobserved-impact trajectory.

Related

• Research status — current proof boundaries and correction guidance.
• Pair-invariant collapse on the electron sector — preceding section in the research sequence.