Candidate scan for a dynamically selected mixed frame

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 55: Candidate scan for a dynamically selected mixed frame.

Provenance

Preserved from interaction notebook section 55, 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.

55. Candidate scan for a dynamically selected mixed frame

Several V2-native candidates were tested for a mixed orientation-selecting block that could select a beam-relative frame link $Q$ without inserting it by hand.

55.1 Pair commutator

The first candidate is

$$F_{\rm pair}=[J_1,J_2].$$

It is exactly related to the antisymmetric part of

$$R=-J_1J_2$$

by

$$[J_1,J_2]=-2\,\frac{R-R^T}{2}.$$

In the canonical test pair and a fixed laboratory $3+3$ split, its mixed $3\times3$ block has generic rank one and vanishes as the pair approaches projective coincidence.

Therefore

$$\boxed{ \text{the tested rank-one mixed block does not determine a full }SO(3)\text{ orientation.} }$$

This is a result for that test pair and split, not a universal mixed-rank theorem for all placements of the pair relative to the laboratory split. The coordinate-independent pair decomposition is $2+4$.

55.2 Single orbital bivector

A natural scattering object is the relative orbital bivector

$$L=x\wedge p.$$

Its mixed block is

$$E_L=x_Op_H^T-p_Ox_H^T.$$

This is a difference of two rank-one matrices, so

$$\boxed{ \operatorname{rank}E_L\le2. }$$

A single nondegenerate orbital bivector can select a full $SO(3)$ frame relation: its rank-two mixed tensor has a unique orientation-preserving Procrustes completion (Section 99).

55.3 Two generic simple bivectors

A sum of two independent simple bivectors can generically produce

$$\operatorname{rank}E=3.$$

Explicit random tests gave nonzero determinants in generic samples. Therefore rank three is not forbidden once more than one independent scattering two-plane participates.

However, an arbitrary sum of two bivectors has no unique V2-native prescription yet, so this observation establishes possibility rather than a preferred construction.

55.4 Scattering-triad candidate

A sufficient, nonminimal candidate is built from relative momentum and impact data in each half of the beam-adapted split.

Let

$$p_O,b_O\in O, \qquad p_H,b_H\in H,$$

and define the oriented normal directions

$$n_O=p_O\times b_O, \qquad n_H=p_H\times b_H.$$

Form the oriented scattering-frame matrices

$$O_s=[p_O\;b_O\;n_O], \qquad H_s=[p_H\;b_H\;n_H].$$

The mixed tensor

$$\boxed{ E_{\rm tri} =O_sH_s^T =p_Op_H^T +b_Ob_H^T +n_On_H^T }$$

has determinant

$$\boxed{ \det E_{\rm tri} =|p_O\times b_O|^2 |p_H\times b_H|^2. }$$

Therefore it is generically full rank whenever momentum and impact vectors are nonparallel in both three-planes.

This is stronger than a rank-one source, but a rank-two orbital source already suffices for orientation (Section 99).

55.5 Exact oriented-frame limit

If the two scattering triads are orthonormal and oriented, then

$$O_s,H_s\in SO(3),$$

and

$$\boxed{ E_{\rm tri}=O_sH_s^T\in SO(3). }$$

Thus in this limit the mixed tensor is itself exactly the relative rotation carrying the unobserved scattering frame into the observed scattering frame.

This gives a non-circular candidate for

$$\boxed{Q=O_sH_s^T.}$$

Unlike the earlier hand-chosen $F_{\rm mix}$, this $Q$ is constructed from collision data.

55.6 Generic nonorthogonal triads

For raw nonorthogonal vectors, $E_{\rm tri}$ remains full rank and transforms covariantly,

$$E_{\rm tri} \longmapsto R_OE_{\rm tri}R_H^T.$$

Its polar factor therefore transforms as

$$Q_E \longmapsto R_OQ_ER_H^T.$$

Numerical tests confirm covariance to machine precision.

However, for generic nonorthogonal triads,

$$\operatorname{polar}(O_sH_s^T) \ne \operatorname{polar}(O_s) \operatorname{polar}(H_s)^T$$

exactly. The discrepancy in representative random samples was of order $5\times10^{-2}$ to $1.7\times10^{-1}$ in Frobenius norm.

Therefore metric weighting in the unnormalized triads matters.

The clean construction is consequently to form the orthonormalized oriented scattering frames first and then take their relative rotation.

55.7 Degenerate limit

If

$$p_O\parallel b_O$$

or

$$p_H\parallel b_H,$$

then

$$\det E_{\rm tri}=0.$$

The scattering state then fails to define a complete oriented three-frame, so no unique full $Q$ can be extracted from these data alone.

This is physically sensible: a strictly head-on/zero-impact configuration has enhanced rotational symmetry and does not carry enough directional information to select all three axes.

55.8 Current ranking

The tested candidates rank as follows:

The next refinement is to define the orthonormalized scattering frames covariantly and test whether the resulting

$$Q_{\rm scat}$$

can enter a Schur-compensated restoring operator while preserving the observed electromagnetic compression.

Related

• Research status — current proof boundaries and correction guidance.
• Electromagnetic two-form alignment: compatibility, circularity, and rank obstruction — preceding section in the research sequence.
• Orthonormalized scattering-frame candidate and exact Schur compatibility — 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.