Orthonormalized scattering-frame candidate and exact Schur compatibility

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 56: Orthonormalized scattering-frame candidate and exact Schur compatibility.

Provenance

Preserved from interaction notebook section 56, 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.

56. Orthonormalized scattering-frame candidate and exact Schur compatibility

The strongest candidate from section 55 can be made exact by separating directional information from metric distortion before constructing the frame link.

From the observed and unobserved scattering data form

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

For nondegenerate impact geometry these matrices are invertible. Take their orientation-preserving polar factors

$$\widehat O=\operatorname{polar}_{SO(3)}(O_s), \qquad \widehat H=\operatorname{polar}_{SO(3)}(H_s).$$

Define

$$\boxed{ Q_{\rm scat}=\widehat O\widehat H^T. }$$

Then automatically

$$Q_{\rm scat}^TQ_{\rm scat}=I, \qquad \det Q_{\rm scat}=1,$$

and under independent rotations of the two three-frames,

$$O_s\mapsto R_OO_s, \qquad H_s\mapsto R_HH_s,$$

the candidate transforms covariantly,

$$\boxed{ Q_{\rm scat}\mapsto R_OQ_{\rm scat}R_H^T. }$$

Thus $Q_{\rm scat}$ has exactly the transformation law required of the electron mixed frame link, while being built from scattering-state data rather than inserted as an independent matrix.

56.1 Exact Schur compatibility

Insert $Q_{\rm scat}$ into the previously verified lock block

$$K_{\rm lock} =\kappa \begin{pmatrix} I&-Q_{\rm scat}\\ -Q_{\rm scat}^T&I \end{pmatrix}.$$

For an arbitrary observed operator $K_{\rm obs}$, take

$$K_6= \begin{pmatrix} K_{\rm obs}+\kappa I&-\kappa Q_{\rm scat}\\ -\kappa Q_{\rm scat}^T&\kappa I \end{pmatrix}.$$

Because $Q_{\rm scat}$ is orthogonal,

$$BD^{-1}C =(-\kappa Q_{\rm scat})(\kappa^{-1}I)(-\kappa Q_{\rm scat}^T) =\kappa I,$$

so the relaxed observed operator is exactly

$$\boxed{ K_{\rm eff} =K_{\rm obs}. }$$

The algebraic identity establishes this static cancellation. It does not establish unchanged frequency-dependent dynamics: adding unobserved inertia replaces $D$ by $D-\omega^2M_H$ and generally changes the Schur response.

Therefore the scattering-frame candidate passes an important consistency test:

$$\boxed{ \text{it can produce a full }3+3\text{ lock without changing the relaxed observed interaction.} }$$

56.2 What this does and does not derive

This construction removes the arbitrariness in the orientation of $Q$ once a nondegenerate scattering state supplies observed and unobserved momentum/impact directions.

It does not yet derive:

• why the V2 master operator should use the polar-frame map $Q_{\rm scat}$;
• the positive coefficient $\kappa$;
• how the unobserved scattering vectors $p_H,b_H$ are determined from directly prepared beam data;
• or whether the corresponding mixed/extrinsic block is generated by the current master connection.

Thus the candidate is stronger than a hand-chosen $Q$ but remains a proposed completion rather than a theorem of the current V2 action.

56.3 Candidate status

Among the tested constructions, the current ranking is now:

1. Orthonormalized scattering-frame map $Q_{\rm scat}$: full rank, exactly orthogonal, covariant, statically Schur-compatible; a sufficient candidate.
2. Raw scattering triad $E_{\rm tri}$: full rank and covariant, but metric-weighted before polar normalization.
3. Two generic bivectors: capable of full rank, but no preferred V2 combination.
4. Single orbital bivector: generically rank two; already sufficient for unique $SO(3)$ orientation and more economical than a full triad.
5. Pair commutator $[J_1,J_2]$: the tested mixed block is rank one and vanishes at coincidence; this is not a universal rank classification for arbitrary laboratory placements.

The remaining task is to derive a source selecting $Q_{\rm scat}$ or an equivalent orientation from the master operator, and determine the restoring spectrum. A generic rank-two source can already select an $SO(3)$ orientation (Section 99).

Related

• Research status — current proof boundaries and correction guidance.
• Candidate scan for a dynamically selected mixed frame — preceding section in the research sequence.
• Variational derivation of the scattering-selected frame link — 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.