Spatial source complementarity

Summary: Projecting a normalized odd-dimensional rotation tensor and its null-axis vector onto a lower spatial subspace gives an exact complementarity identity and conditional source-strength bound.

A source determined by the higher geometry

This calculation starts with a higher-dimensional spatial source and asks which of its components survive on a chosen lower subspace. All spaces are ordinary macroscopic spatial directions in one nested manifold. The split is relative to that lower subspace; it is not an inherent division of the higher particle.

Use the normalized skew tensor and unit null vector from the dimensional magnetic kernel:

$$J^T=-J,\qquad J^2=-I+nn^T,\qquad |n|=1 \tag{1}$$

in dimension $d=2m+1$. These imply $Jn=0$ and unit rotational singular values on $n^\perp$. At depth five, the specified pure spatial spinor model supplies both $J$ and $n$. For other odd dimensions, (1) defines a conditional tensor family rather than a claim about all spinors or physical particles.

Let $W$ be an $r$-dimensional lower spatial subspace and let $Z=W^\perp$, of dimension $s=d-r$. In adapted orthonormal frames,

$$J=\begin{pmatrix}A&Q\\-Q^T&C\end{pmatrix}, \qquad n=\begin{pmatrix}n_W\\n_Z\end{pmatrix}. \tag{2}$$

$A$ is the restriction of the rotational two-form to $W$, while $n_W$ is the projected vector. Use $\|A\|^2=-\operatorname{tr}(A^2)/2$ for skew matrices and the ordinary Euclidean norm for vectors.

Exact projection identity

The diagonal blocks of (1) are

$$A^2-QQ^T=-I_r+n_Wn_W^T, \qquad C^2-Q^TQ=-I_s+n_Zn_Z^T.$$

Writing $t=|n_W|^2$ and using $|n_Z|^2=1-t$, their traces give

$$2\|A\|^2+\|Q\|_F^2=r-t, \qquad 2\|C\|^2+\|Q\|_F^2=s-1+t.$$

Subtracting eliminates every mixed component and yields

$$\boxed{\|A\|^2+|n_W|^2-\|C\|^2=r-m.} \tag{3}$$

Consequently

$$\|A\|^2+|n_W|^2\geq r-m. \tag{4}$$

The lower bound is useful when $r>m$. It is a statement about projected source descriptors, before identifying an interaction channel or rate.

Five-dimensional source viewed in three dimensions

For $d=5$ and $r=3$, equation (3) reads

$$\boxed{\|A\|^2+|n_W|^2=1+\|C\|^2\geq1.} \tag{5}$$

Every real three-dimensional skew matrix has one rotational singular value $a\geq0$, with $\|A\|^2=a^2$. Since $A$ is a compression of a norm-one operator, $a\leq1$. These bounds sharpen two endpoint cases:

• If $n_W=0$, then $\|A\|^2=1$ and $C=0$.
• If $A=0$, then $|n_W|=1$ and $C=0$.

Both cases are attainable. In an orthonormal basis of $\mathbb R^5$, take $J=e_1\wedge e_2+e_3\wedge e_4$ and $n=e_5$. The first case is $W=\operatorname{span}(e_1,e_2,e_3)$; the second is $W=\operatorname{span}(e_1,e_3,e_5)$.

Thus orientation alone cannot make both projected descriptors small in this normalized family. If a response reads these as orthogonal, distinguishable channels with positive weights $g_B,g_V$, then

$$g_B\|A\|^2+g_V|n_W|^2\geq\min(g_B,g_V). \tag{6}$$

This bound requires both channels to be present with the stated positive metric. A physical map might omit one channel, assign different derived normalizations, or coherently combine contributions. In those cases the actual map must be evaluated; (6) does not replace it. In particular, the five-dimensional vector supplied by the scalar-exchange benchmark has not been identified as an electromagnetic source by this identity.

Relation to photon observations in a shared three-plane

Let $k\in W$ be a unit propagation direction and take the observer's photon plane $E=W\cap k^\perp$. Because $k^TJk=0$,

$$P_EJk=Ak. \tag{7}$$

Write $Av=b\times v$ in an oriented frame on $W$; this three-dimensional axial-vector notation is only a way of describing the restricted two-form. Then $|b|^2=\|A\|^2$ and

$$C(E,k;J)=|b|^2-(b\cdot k)^2. \tag{8}$$

For a uniform propagation direction within this fixed $W$,

$$\mathbb E_{k\in S(W)}C=\frac23\|A\|^2. \tag{9}$$

This is the sum over two photon polarizations. An unpolarized average has the additional factor $1/2$. If $A=0$, every such photon direction has zero magnetic overlap, even though the higher tensor $J$ is nonzero. For the five-dimensional model, (5) then requires a fully projected vector $n_W$. Its physical coupling must be evaluated separately.

The preparation in (9) fixes the observer subspace and target tensor. It differs from the full isotropic relative-preparation ensemble in isotropic polarization comparison.

Even-dimensional comparison

For a normalized even-dimensional tensor $J^2=-I$ in $d=2m$, the same trace calculation gives

$$\boxed{\|A\|^2-\|C\|^2=r-m.} \tag{10}$$

At $d=6,r=3$, the two restricted norms are equal and both may vanish. For example $J=e_1\wedge e_2+e_3\wedge e_4+e_5\wedge e_6$ and $W=\operatorname{span}(e_1,e_3,e_5)$ give $A=C=0$, while the mixed block carries the complete rotation. This is a relation to a chosen observer three-plane; the electron's six spatial dimensions remain equivalent and have no inherent split.

The odd/even identities provide exact tests of a proposed descending source map. They do not determine a particle's global spatial topology, prepare the required orientations, or assign empirical coupling strengths.

Verification

polarization_coupling_checks.py checks the displayed identities with independent spatial quadrature, energy variations, or full finite-system evolution as appropriate. These checks verify the declared mathematics and benchmarks, not empirical couplings.

Related

• Dimensional magnetic kernel — derives the particular five-dimensional vector and rotational tensor and distinguishes candidate interaction terms.
• Descending photon source — derives the higher-field source before selecting its lower response.
• Photon plane coupling — supplies the overlap used in (7)–(9).
• Isotropic polarization comparison — evaluates a different, explicitly averaged ensemble.