Quaternionic structure of the free-pair active four-space

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 46: Quaternionic structure of the free-pair active four-space.

Provenance

Preserved from interaction notebook section 46, 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.

46. Quaternionic structure of the free-pair active four-space

The apparent $2+2+2$ form of $R_{12}$ in a convenient basis does not mean that the free pair canonically selects two real two-planes inside its rotating four-space. A stronger invariant structure is present.

Let

$$R=-J_1J_2,$$

and decompose its symmetric and antisymmetric parts,

$$C=\frac12(R+R^T), \qquad A=\frac12(R-R^T).$$

For generic separation $0<\theta<\pi/2$, define the pair-fixed projector

$$\boxed{ P_0 =\frac{C-\cos(2\theta)I}{1-\cos(2\theta)} }$$

and

$$P_4=I-P_0.$$

The explicit chiral-spinor calculation gives

$$\boxed{ \operatorname{rank}P_0=2, \qquad \operatorname{rank}P_4=4. }$$

The normalized antisymmetric part

$$\boxed{ J_{\rm pair} =\frac{A}{\sin(2\theta)} }$$

vanishes on the fixed plane and satisfies

$$J_{\rm pair}^2=-P_4.$$

Thus it is an orthogonal complex structure on the active four-dimensional real subspace.

Now restrict the first electron complex structure to that active subspace,

$$I_4=P_4J_1P_4.$$

The exact identities are

$$I_4^2=-P_4, \qquad J_{\rm pair}^2=-P_4,$$

and

$$\boxed{ I_4J_{\rm pair}+J_{\rm pair}I_4=0. }$$

Define

$$K_4=I_4J_{\rm pair}.$$

Then

$$K_4^2=-P_4,$$

and the quaternion relations hold exactly on the active four-space:

$$\boxed{ I_4J_{\rm pair}=K_4, \qquad J_{\rm pair}K_4=I_4, \qquad K_4I_4=J_{\rm pair}. }$$

All three operators are skew-adjoint orthogonal complex structures.

Therefore

$$\boxed{ E_{\rm rot}^{(4)}\text{ carries a canonical quaternionic structure generated by the pair }(J_1,J_2). }$$

The commutant of two independent quaternion generators on this real four-space has real dimension four, as expected for the opposite quaternionic action. This confirms that the active space is naturally one quaternionic line rather than a canonically ordered pair of real two-planes.

46.1 Correction to the earlier free-pair split

The basis expression

$$R=I_2\oplus R(2\theta)\oplus R(-2\theta)$$

remains correct, but its two rotating planes are not invariantly distinguished by the pair alone. The canonical decomposition is

$$\boxed{ \mathbb R^6 =\mathbb R^2_{\rm fixed} \oplus \mathbb H_{\rm pair}, }$$

where $\mathbb H_{\rm pair}$ denotes the active real four-space equipped with the quaternionic structure above.

This is stronger than the earlier $2+2+2$ description and removes an artificial choice of two-plane basis within the active four-plane.

46.2 Consequence for a free-electron locking interaction

A free-pair interaction built only from pair invariants should respect this quaternionic symmetry unless another physical structure breaks it.

Therefore a candidate that singles out one particular real $2+2$ decomposition inside $E_{\rm rot}^{(4)}$ is not pair-canonical.

The minimal pair-native quadratic interaction must instead be constructed from the invariant projectors

$$P_0, \qquad P_4,$$

and quaternionic-invariant tensors on $E_{\rm rot}^{(4)}$.

This sharply restricts the next search.

Related

• Research status — current proof boundaries and correction guidance.
• Canonical free-pair operator from two electron complex structures — preceding section in the research sequence.
• Simultaneous commutant and its limited no-mixing theorem — following section in the research sequence.