Canonical free-pair operator from two electron complex structures

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 45: Canonical free-pair operator from two electron complex structures.

Provenance

Preserved from interaction notebook section 45, 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.

45. Canonical free-pair operator from two electron complex structures

To test whether a free electron pair can generate the locking map $Q$ without importing a nucleus-selected $3+3$ split, construct the most immediate pair operator from the two orthogonal complex structures,

$$\boxed{ R_{12}=-J_1J_2. }$$

Since

$$J_i^T=-J_i, \qquad J_i^2=-I_6,$$

one has

$$R_{12}^TR_{12}=I_6,$$

so $R_{12}\in SO(6)$ for the explicit electron-sector realization.

Using the diagonal-$SU(4)$ canonical pair

$$Z_1=e_1, \qquad Z_2=\cos\theta\,e_1+\sin\theta\,e_2,$$

the exact gamma-matrix calculation gives

$$\boxed{ R_{12}=I_2\oplus R(2\theta)\oplus R(-2\theta), }$$

up to ordering of the real basis, where $R(\varphi)$ is a real planar rotation.

Equivalently,

$$\boxed{ \operatorname{tr}R_{12}=2+4\cos(2\theta), }$$

and the characteristic polynomial is

$$\boxed{ \chi_{R_{12}}(\lambda) =(\lambda-1)^2 \left(\lambda^2-2\cos(2\theta)\lambda+1\right)^2. }$$

Thus, for generic pair separation, the real six-dimensional representation has a canonical decomposition

$$\boxed{ \mathbb R^6 =E_{\rm fix}^{(2)}\oplus E_{\rm rot}^{(4)}. }$$

In a convenient canonical basis the four-dimensional rotating block can be displayed as two planar rotations $R(2\theta)\oplus R(-2\theta)$, but that planar $2+2$ refinement is not itself uniquely selected by the pair invariants. The invariant content is the fixed two-plane plus the active four-space.

45.1 Consequence for the free-pair locking problem

This is a genuine pair-defined frame structure: it depends only on $(J_1,J_2)$ and therefore requires no external nucleus or observer-selected $3+3$ decomposition.

However, it is not the previously used

$$\mathbb R^3\oplus\mathbb R^3$$

split. The generic free pair naturally gives the invariant split

$$\boxed{2+4,}$$

not $3+3$. A displayed $2+2+2$ block form is basis-dependent inside the active four-space.

Therefore the exact $3+3$ Schur-lock model cannot yet be promoted to an $SU(4)$-invariant free-electron theorem merely by replacing the nucleus with the second electron.

This is an important negative result: the pair geometry itself selects canonical planes, but their dimensions differ from the nucleus-relative decomposition.

45.2 Relation to the single pair invariant

The spectrum of $R_{12}$ is completely determined by the Fubini--Study separation $\theta$, hence by

$$\nu=\cos^2\theta.$$

The canonical relative rotation angle is $2\theta$, and

$$\operatorname{tr}R_{12} =2+4(2\nu-1) =8\nu-2.$$

Using

$$C_J=\frac{4\nu-1}{3},$$

one gets

$$\boxed{ \operatorname{tr}R_{12}=6C_J. }$$

This is consistent with

$$C_J=-\frac16\operatorname{tr}(J_1J_2) =\frac16\operatorname{tr}R_{12}.$$

45.3 Revised free-electron question

The appropriate free-pair problem is therefore not immediately

$$h=Q^Tu$$

for two three-planes. Instead it is whether the master interaction dynamically correlates the three pair-defined two-planes supplied by $R_{12}$ in such a way that the observer's resolved three-dimensional motion becomes correlated with the unresolved directions.

The nucleus-induced $3+3$ structure remains valid for bound electron--nucleus systems. The free electron pair has a different canonical symmetry reduction and must be treated separately.

Related

• Research status — current proof boundaries and correction guidance.
• Minimal self-adjoint unobserved-visible lock with exactly zero relaxed backreaction — preceding section in the research sequence.
• Quaternionic structure of the free-pair active four-space — following section in the research sequence.