Chiral Ray Geometry and Normal Rotation Have the Same Pure-State Metric

Projective rays here describe field coefficients. Particle positions remain ordinary spatial coordinates in the infinite-dimensional containing space.

The normal charge and alignment construction maps a normalized pure spinor ray to an ordinary normal rotation structure. This map also preserves the full local polarization metric, up to one exact common factor. It therefore connects the chiral carrier and the normal orientation energy quantitatively, beyond a count of components.

At complex normal rank three, every chiral ray is pure, so no extra polarization restriction is needed. At rank four the full chiral ray has additional directions that the normal rotation differential misses. This is a conditional distinction between geometric carriers, not a derivation of physical particle assignments or an identification of orientation coordinates with additional spatial dimensions.

Definitions and the Exact Metric Identity

Let $W\cong\mathbb C^q$ describe $2q$ ordinary normal spatial directions. Use Hermitian Euclidean Clifford matrices satisfying $\{\gamma_i,\gamma_j\}=2\delta_{ij}I$ on $\Lambda^*W$. For a normalized spinor $\psi$ define

$$(J_\psi)_{ij}=i\langle\psi,\gamma_i\gamma_j\psi\rangle\quad(i\ne j), \qquad (J_\psi)_{ii}=0.$$

It is real skew. On a pure ray it obeys $J_\psi^2=-I$. Changing the phase of $\psi$ leaves $J_\psi$ unchanged. Fix the Clifford signs so the vacuum gives $q$ blocks $\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$.

For a ray variation, the Fubini–Study metric in the unit-vector convention is

$$ds_{\rm FS}^2=\langle d\psi,d\psi\rangle -|\langle\psi,d\psi\rangle|^2.$$

The metric on normal skew matrices is the ordinary Frobenius norm. Restricted to the pure-ray orbit,

$$\boxed{\|dJ_\psi\|_F^2=16\,ds_{\rm FS}^2.}$$

The factor is sensitive to the displayed Clifford and metric normalizations. It is common to the tested ranks two, three and four, rather than an independently chosen coefficient for each sector.

Derivation at the Vacuum and Transport to the Full Orbit

At the vacuum, the horizontal pure-ray tangents are the real and imaginary two-creation states

$$e_i\wedge e_j,\quad i e_i\wedge e_j,\qquad i<j.$$

They are orthonormal for the real Fubini–Study tangent metric and number $q(q-1)$. Differentiating the bilinear gives

$$(\delta J)_{ij}=i\left( \delta\psi^\dagger\gamma_i\gamma_j\psi +\psi^\dagger\gamma_i\gamma_j\delta\psi\right).$$

Direct Clifford multiplication gives the Gram matrix $16I$ on those tangents. The checker retains every real and imaginary tangent, not one selected polarization.

A single plane makes the normalization transparent. Let $X$ be the real normal skew generator with entries $X_{02}=1$, $X_{13}=-1$ and their skew partners. Its Spin lift is

$$\rho(X)=\frac12(\gamma_0\gamma_2-\gamma_1\gamma_3).$$

It produces the unit-speed ray $\psi(\theta)=\cos\theta\,|0\rangle+\sin\theta\,|12\rangle$, up to an irrelevant sign convention. The ray metric is $d\theta^2$, while $\|[X,J_0]\|_F^2=16$.

The normal rotation group acts transitively on the pure rays in one orientation component. Its Spin action preserves the ray metric; its vector action sends $J$ to $RJR^{-1}$ and preserves the Frobenius metric. Transporting the complete vacuum Gram identity therefore proves the identity everywhere on that orbit. The code additionally checks a nontrivial rational Spin rotation with sine $4/5$ and cosine $3/5$ at every tested rank.

This proves a metric statement. It does not replace a first-order Dirac action by a second-order sigma-model action.

Why Rank Three Is the Complete Chiral-Ray Case

The pure orbit has real dimension $q(q-1)$, whereas the full positive-chiral projective carrier has dimension $2(2^{q-1}-1)$. At ranks two and three these agree, and every chiral ray is pure. At rank three the orbit is $SO(6)/U(3)\cong\mathbb{CP}^3$ with real dimension six and a four-component positive-chiral module. Its unitary stabilizer supplies the familiar traceless eight-generator algebra and the central circle.

At rank four, the positive-chiral ray space has real dimension fourteen but the pure orbit has dimension twelve. The extra two real tangents at the vacuum are

$$|1234\rangle,\qquad i|1234\rangle.$$

Both have Fubini–Study norm one, yet their first bilinear variation is zero: a product of two Clifford matrices cannot connect vacuum and exterior degree four. Thus $\|dJ\|^2$ is not a complete kinetic metric on the unrestricted rank-four chiral ray.

The exact finite-amplitude example is

$$\psi(\theta)=\cos\theta|0\rangle+\sin\theta|1234\rangle, \qquad J_\psi=\cos(2\theta)J_0.$$

At $\theta=\pi/4$, the normal bilinear vanishes although the spinor ray is nonzero and normalized. The full polarization cannot be interpreted as an orthogonal normal rotation. Restricting the rank-four carrier to its pure orbit restores the factor-sixteen metric, but that restriction is an additional premise.

The odd entries use the explicit even-carrier pullback. An orthogonal complex structure is not imposed on their entire odd-rank actual normal bundles. The table also differs from the fixed-six-normal carrier at depth two, where $q$ remains three. A carrier change must be stated before treating this distinction as depth selection.

What an Orientation Kinetic Term Can and Cannot Replace

For a pure ray, the retained term $f^2\|\partial J\|_F^2/2$ equals $8f^2 ds_{\rm FS}^2$ on horizontal polarization variations. It supplies the full local projective metric at rank three. The absolute coefficient $f$ remains a constitutive input until derived from the same field.

Amplitude and common phase are absent from the ray metric. If a separate second-order complex-field comparison uses $\Psi=\rho e^{i\phi}\psi$, its complete norm splits as

$$\|d\Psi\|^2=d\rho^2+\rho^2\left[ (d\phi+a)^2+ds_{\rm FS}^2\right], \qquad a=-i\psi^\dagger d\psi.$$

Replacing only the projective contribution by the normal-rotation metric leaves both the radial and phase contributions. With an overall coefficient $Z/2$ on this comparison norm, the matching relation is $f^2=Z\rho^2/16$. This is a conditional normalization identity; no physical amplitude has been fitted, and a Dirac field needs its own action and symplectic reduction.

The canonical normal-structure connection relates the pure vacuum line's phase connection to the determinant of the ordinary normal complex carrier. That central Spin relation is separate from choosing the observed electric generator or determining particle occupation. The same normal geometry may organize both transport and phase without settling physical charge.

A Purity Defect Has Different Quadratic and Quartic Tests

For any normalized spinor, $\|J\|_{\rm op}\le1$. Indeed $i\gamma(v)\gamma(w)$ has operator norm one for orthonormal real vectors $v,w$, so its expectation bounds $|v^TJw|$ by one. Since $J$ is skew, choosing $w$ along $Jv$ gives the operator bound. Hence

$$\mathcal D=\operatorname{Tr}(I+J^2)=2q-\|J\|_F^2\ge0.$$

Vanishing means every skew singular value equals one and $J^2=-I$. In an adapted Clifford frame the associated occupation numbers then have extremal expectation zero or one; their positive complements annihilate the state, which is pure. Thus $\mathcal D$ is an amplitude-independent purity diagnostic on normalized rays.

On the rank-four extra-direction curve,

$$\mathcal D=8\sin^2(2\theta)=32\theta^2+O(\theta^4),$$

whereas

$$\|I+J^2\|_F^2=8\sin^4(2\theta)=128\theta^4+O(\theta^6).$$

A squared-defect penalty has no quadratic restoring term along this direction. The unsquared nonnegative scalar does. Either proposed energy remains an additional action choice, defined here on normalized rays; its extension through zero field amplitude is unresolved. At rank three both defects vanish identically on the chiral ray, because purity is automatic.

Contribution and Verification

This adds an exact kinetic interpretation to the existing rank-three purity and charge-alignment result. The same ordinary six-normal geometry supports a complete chiral projective polarization metric and its unitary carrier without an extra purity restriction. Higher-rank controls expose the lost directions rather than counting them as physical colors. The result reuses the earlier geometric premises and is not statistically independent sector evidence.

The checker and report preserve 340 exact conditions, complete tangent Gram matrices and rational Spin transports at ranks two, three and four, with every occupied-sector assignment displayed. The canonical-connection review independently confirms the factor-sixteen local normalization. No particle simulation, amplitude calibration or observed electric-charge selection enters this calculation.