Polarization Normal Form and Whole-Field Circulation Rank

A field can have a simple motion at each point and a richer orientation pattern across space. This matters when comparing a spinor's local plane matrix with the polarization of a complete particle pattern. All ten directions below are ordinary spatial directions in the matter reduction within the single infinite-dimensional physical spacetime; time is separate.

1. A Normal Form for One Chiral Spinor

Choose five orthogonal spatial planes and their usual Clifford creation and annihilation basis. A positive-chirality Weyl spinor has even occupation grades. After a suitable ordinary spatial Spin rotation, every nonzero such spinor can be written

$$\chi=\alpha|0\rangle+\beta|1234\rangle, \qquad |\alpha|\geq|\beta|.$$

These are mathematical coefficients of the field. The normal form does not select the physical orientation of a particle. To see why it exists, maximize the vacuum overlap over spatial rotations. The orbit spans the even module, so this maximum is nonzero. Varying the two real pair generators for every coordinate pair sets both real components of every degree-two coefficient to zero. The remaining vacuum stabilizer rotates the degree-four coefficient vector to one direction, giving the displayed form.

Let $A=|\alpha|^2$, $B=|\beta|^2$, $N=A+B$, and normalize the antisymmetric Clifford plane matrix by $N$. In suitable orthogonal spatial coordinates, this matrix has five two-by-two rotation blocks. Their nonnegative strengths are called its skew singular values. Here they are

$$(1,c,c,c,c),\qquad c=\frac{A-B}{N},\quad 0\leq c\leq1.$$

Bivectors change at most two occupation numbers, so they cannot connect the two displayed terms. The first four plane weights are therefore their population difference, while the fifth is their sum. In the stated charge-conjugation convention the nonconjugated vector bilinear satisfies

$$|V|^2=8AB,\qquad c^2=1-\frac{|V|^2}{2N^2}.$$

This quantifies the distinction between a pure seed with five full rotation planes and a more general chiral seed. It is not a new mass law, a physical trajectory or a result for arbitrary Dirac fields with independent chiral parts. The spatial current must still be calculated from the complete field.

2. One Local Circle Does Not Limit a Whole Pattern to One Plane

Take five disjoint smooth real spatial bumps. On the support of bump $f_a(x)$, let a vector rotate at the same fundamental frequency in spatial plane $(e_{2a-1},e_{2a})$:

$$E(x,t)=\sum_{a=1}^{5}f_a(x) \bigl[e_{2a-1}\cos(\omega t)+e_{2a}\sin(\omega t)\bigr].$$

At any one point, only one bump is nonzero, so the vector traces a circle. The time-averaged spatial integral of its circulation is nevertheless a block-diagonal antisymmetric matrix with five nonzero plane blocks, proportional to $\omega\int f_a^2$. Its rank is ten. Equal bump norms make the blocks equal, but unequal nonzero norms keep the full rank.

This is a smooth kinematic construction, not a solved interacting field. It establishes why a pointwise monochromatic ellipse argument cannot limit the orientation structure of the complete spatial pattern. Every term uses the same fundamental; no separately resonating higher physical mode is needed.

3. Connection to the Existing Sector Work

The adjacent-depth identity relates odd-depth axes to even-depth planes, while the common-fundamental texture account separates portion count from directional response. The present results add an exact ten-direction local classification and a control showing why the whole field can carry more planes than one local vector. These are compatible geometric tools; they do not select a sector, count or coupling by themselves.

The radial current remains $j=-2FGJ\widehat x$. In a specified gapped-tail comparison, its asymptotic speed ratio includes the angular factor:

$$\frac{|j|}{\rho}\longrightarrow\frac{\kappa}{m_0}|J\widehat x|.$$

Replacing that last factor by one requires an orthogonal complex structure on the relevant directions, as in the stated pure even preparation. Full rank alone is insufficient. This conditional tail statement supplies neither a calibrated amplitude nor an independent physical clock.

Evidence and Scope

The original normal-form and whole-field examples were additions in a separate polarization review, SHA-256 d7c5132fa75224dcc0f04c9b79f6bf72d9545dee34ba46c79d40d67a43744405. A later independent analytical assessment checks those additions and the corrected compatibility packet: SHA-256 5e58e69062731f12c280b9e2c1eacb04694d63b39c513c698bdae84467a983be. The proofs are analytical; recorded author matrix tests were inspected, not independently replayed. No physical species or new experimental prediction is identified by this note.