One- and two-vector outer-product mixed ansatzes

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 48: One- and two-vector outer-product mixed ansatzes.

Provenance

Preserved from interaction notebook section 48, 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.

48. One- and two-vector outer-product mixed ansatzes

The simultaneous commutant of $(J_1,J_2)$ admits no mixed map between its canonical

$$E_0^{(2)}\oplus E_4^{(4)}$$

pieces. A collision, however, supplies additional covariant data, beginning with the relative momentum

$$p=p_{\rm rel}.$$

Decompose it with the pair-defined projectors:

$$p_0=P_0p, \qquad p_4=P_4p.$$

The simplest real symmetric mixed tensor constructed from the pair and the collision momentum is

$$\boxed{ M_p=p_0p_4^T+p_4p_0^T. }$$

This is no longer forbidden by the static-pair commutant theorem because $p$ breaks the pair stabilizer.

Its off-diagonal block is

$$B_p=p_0p_4^T:E_4\to E_0.$$

Being an outer product,

$$\boxed{\operatorname{rank}B_p\le1.}$$

All $2\times2$ minors vanish identically, and a generic numerical specialization has rank exactly one.

This particular outer-product ansatz opens one mixed direction. The rank bound does not classify all tensors constructed using a vector and the complex structures.

48.1 Adding impact parameter or a second current direction

A scattering state also contains an independent transverse/impact vector $b$. The next mixed tensor can contain

$$B_{p,b} =p_0p_4^T+b_0b_4^T.$$

Generically this has rank two, the maximum possible rank of a map

$$E_4^{(4)}\to E_0^{(2)}.$$

Therefore the complete pair-native $2+4$ decomposition can at most correlate two independent active combinations directly with its fixed two-plane.

This confirms that the free-pair $2+4$ geometry is not the same structure as the laboratory-visible $3+3$ collision split. The latter requires the observer/beam geometry as additional physical data.

48.2 Consequence for the tested ansatzes

For the restricted commuting class and the displayed outer-product constructions:

$$(J_1,J_2) \quad\Longrightarrow\quad \text{no mixed map in the simultaneous commutant},$$

while

$$(J_1,J_2,p_{\rm rel}) \quad\Longrightarrow\quad \text{rank-one mixed map},$$

and

$$(J_1,J_2,p_{\rm rel},b) \quad\Longrightarrow\quad \text{generically rank-two mixed map}.$$

For the original beam question, the natural next step is therefore to include the laboratory-selected observed three-plane $O_3$ and its orthogonal complement $H_3$. A beam experiment physically supplies this split; it need not be generated by the unordered pair alone.

Related

• Research status — current proof boundaries and correction guidance.
• Simultaneous commutant and its limited no-mixing theorem — preceding section in the research sequence.
• Beam-adapted $3+3$ frame links recover the pair invariant exactly — following section in the research sequence.
• Correction to the rank-three requirement: a generic rank-two orbital bivector uniquely selects an $SO(3)$ frame by orientation-preserving Procrustes completion — later audit limits or corrects this route.