Spinor Polarization and Spatial Circulation

The spatial spinor profile already used in the sector case supplies a concrete connection between polarization and internal field circulation. Its seed orientation determines an actual current of that profile. The current can be nonzero locally while its integral is exactly zero. These are conditional results for the specified field equation; identifying the current with electric current, energy flow or the particle's fundamental clock remains open.

All coordinates refer to the same ten ordinary spatial directions plus time. A depth-d calculation describes dependence on d of those directions. It does not introduce another physical manifold. Field components and their squared norm are classical field data, not a probability distribution for the electron's position. The electron's centre trajectory is a separate physical variable.

1. Start with the Existing Spatial Solution

Let the Hermitian matrices $\gamma_i$ obey

$$\gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij}I, \qquad D=-i\sum_{i=1}^d\gamma_i\partial_i.$$

These matrices encode the transformation of spinor components under rotations of the physical coordinates. Choose a constant seed $\chi$, with squared magnitude $s=\chi^\dagger\chi>0$. For $r=|x|$, $\widehat x=x/r$, a positive length a and a real amplitude A, write

$$\psi(x)=F(r)\chi+iG(r)(\widehat x\cdot\gamma)\chi, \qquad F=A(1+r^2/a^2)^{-d/2},\qquad G=(r/a)F. \tag{C1}$$

The apparent unit-vector singularity at the origin is removable because $G\widehat x=Fx/a$. The squared field magnitude is

$$\rho=\psi^\dagger\psi=s(F^2+G^2) =sA^2(1+r^2/a^2)^{1-d}. \tag{C2}$$

Direct differentiation gives

$$D\psi=V(r)\psi,\qquad V(r)=\frac{d}{a(1+r^2/a^2)} =g\rho^{1/(d-1)},\qquad g(A^2s)^{1/(d-1)}=\frac da. \tag{C3}$$

For a unit seed this is exactly the case's equation (6.3). Keeping s explicit distinguishes the convention for seed normalization from the uncalibrated physical field amplitude. The nonlinear power depends on d: this is a family of spatial equations, not yet one established depth-independent master action.

2. Expand the Current Instead of Guessing Its Direction

Define the real vector and antisymmetric matrix of the seed by

$$v_i=\chi^\dagger\gamma_i\chi,\qquad B_{ij}=-i\chi^\dagger\gamma_i\gamma_j\chi\quad(i\ne j), \qquad B_{ii}=0.$$

B records oriented spatial planes. With rotation generators $S_{ij}=-i[\gamma_i,\gamma_j]/4$, it is twice the seed's spin bilinear: $B_{ij}=2\chi^\dagger S_{ij}\chi$. That algebraic statement alone does not specify a flow. The flow follows by substituting (C1) into the current definition:

$$j_i=\psi^\dagger\gamma_i\psi =(F^2-G^2)v_i +2G^2\widehat x_i(\widehat x\cdot v) -2FG B_{ij}\widehat x_j. \tag{C4}$$

Repeated spatial indices are summed. The cross term uses $[\gamma_i,\gamma_j]=4iS_{ij}$; the quadratic radial term uses

$$(\widehat x\cdot\gamma)\gamma_i(\widehat x\cdot\gamma) =2\widehat x_i(\widehat x\cdot\gamma)-\gamma_i.$$

Thus (C4) is an identity for the actual nonconstant profile, not an inference from the seed alone. Its last term circles in the planes encoded by B. The first two terms retain the seed vector and vary with position.

For an even-depth chiral seed, $v=0$, since each spatial gamma matrix maps to the opposite chirality. Consequently

$$j=-2FG B\widehat x=-\frac{2F^2}{a}Bx, \qquad x\cdot j=0. \tag{C5}$$

The current is tangent to spheres about the profile centre. The complete profile has both seed and opposite-chirality radial components, so it can have this nonzero current even though the seed's vector bilinear vanishes.

3. Which Depths Give Which Circulation?

At depth four every nonzero chiral seed gives $B^2=-s^2I_4$. In the positive-chirality convention of the connections note, B is self-dual. It has two rotation planes and form norm squared $\|B\|^2=2s^2$. It is not the rank-two area tensor of one vector ellipse.

At six every chiral seed is Cartan-pure, and $B^2=-s^2I_6$ gives three planes. At ten that conclusion requires the additional Cartan-purity condition. General chiral seeds need not give an orthogonal complex structure or full rank. The already known seeds $(|0\rangle\pm|1234\rangle)/\sqrt2$ have the same rank-two B, whereas the pure reference seed has rank ten. Here the ket notation names a component basis, not another spatial arena or a quantum position state.

At minimal depths three and five, $|v|=s$ and $Bv=0$. The expression $\widehat x(\widehat x\cdot v)$ in (C4) nevertheless prevents calling the whole vector-dependent current an everywhere axial stream. These results do not select depth five as uniquely weakly interacting.

For even chiral seeds, division of (C5) by (C2) gives

$$\frac j\rho=-\frac{2a}{a^2+r^2}\frac Bs x. \tag{C6}$$

On the pure branch, $(B/s)^2=-I$, so $|j|/\rho=2ar/(a^2+r^2)$. This ratio describes norm flow in the chosen evolution units. The angular rate varies with radius. It is not the common oscillator fundamental and must not be substituted for it. Even the interpretation of these streamlines as material trajectories requires a physical identification of the current.

4. Local Circulation with Zero Integrated Current

For $d>2$, the radial integrals over the bound coordinates converge:

$$I_F=\int F^2\,d^dx =A^2a^d\pi^{d/2}\frac{\Gamma(d/2)}{\Gamma(d)}, \qquad I_G=\int G^2\,d^dx=\frac{d}{d-2}I_F. \tag{C7}$$

To reproduce them, substitute $t=r^2/a^2$ in the radial measure and use $\int_0^\infty t^{p-1}(1+t)^{-p-q}dt=\Gamma(p)\Gamma(q)/\Gamma(p+q)$. The angular area is $2\pi^{d/2}/\Gamma(d/2)$. The G integral requires $q=d/2-1>0$, explaining the threshold.

The angular averages are $\langle\widehat x\rangle=0$ and $\langle\widehat x_i\widehat x_j\rangle=\delta_{ij}/d$. Applying them to (C4) yields

$$\int j\,d^dx =v\left[I_F+\left(\frac2d-1\right)I_G\right]=0. \tag{C8}$$

This cancellation includes odd depths with nonzero seed vector. It is exact throughout the finite-norm family, not a numerical near-cancellation. The same integrals give

$$N_d=\int\rho\,d^dx =sA^2a^d\pi^{d/2} \frac{\Gamma(d/2-1)}{\Gamma(d-1)},$$

matching the case's bound-direction norm. Uniform extension in the unused spatial directions would still make the full ten-dimensional integral divergent; these integrals do not solve that support-completion problem.

There is also a direct spatial area relation. Define $(x\wedge j)_{ij}=x_i j_j-x_j j_i$. Angular averaging at fixed radius gives

$$\langle x\wedge j\rangle_{S^{d-1}}=\frac{4FGr}{d}B. \tag{C9}$$

The vector-dependent terms average to zero, while the B term uses the same second angular moment. This aligns the current's mean oriented area with the seed's spin bilinear. It does not assign an absolute angular momentum without the correct kinetic and current normalization.

5. What Makes This a Conserved Current?

A specified candidate time equation is

$$i\partial_t\Psi=D\Psi-g(\Psi^\dagger\Psi)^{1/(d-1)}\Psi. \tag{C10}$$

Hermiticity and the real scalar coefficient give $\partial_t\rho+\nabla\cdot j=0$. For the spatial profile, the sign convention can be checked directly:

$$\nabla\cdot j=i\left[\psi^\dagger D\psi-(D\psi)^\dagger\psi\right]=0.$$

The current therefore transports conserved field norm under this candidate equation. It has not been shown to transport the electric charge or energy that an experiment measures. The profile (C1) is the zero-temporal-frequency stationary member of (C10). Adding a constant frequency offset could attach a supplied phase clock, but would not derive the kind-specific fundamental or the multiplier C. The internal current calculation and the common-fundamental waveform construction remain two ingredients to join under one law.

5.1 The Critical Equation Cannot Supply a Nonzero Stationary Clock

For a stationary field $\Psi=e^{-i\omega t}\phi$, define $N=\int\phi^\dagger\phi$, $T=\int\operatorname{Re}(\phi^\dagger D\phi)$, $U=\int(\phi^\dagger\phi)^{p_*}$, and $p_*=d/(d-1)$. The energy of (C10) is $E=T-gU/p_*$. Assume a nonzero finite N, convergent integrals and admissible differentiable amplitude and spatial-dilation variations, or a justified cutoff calculation with vanishing boundary remainders. No positivity of T or minimum assumption is required.

The amplitude variation $\phi\mapsto\sqrt\eta\phi$ and norm-preserving spatial variation $\phi(x)\mapsto\lambda^{d/2}\phi(\lambda x)$ give, respectively,

$$T-gU=\omega N,\qquad T-gU=0,\qquad \omega N=0.$$

Thus (C10) forces zero stationary temporal frequency on this domain, without a radial or polarization restriction. The known bubble at d=2 falls outside it: both its norm and squared dilation derivative have radial tail proportional to $r^{1-d}$, which diverges logarithmically at two. The result does not exclude driven configurations, nondecaying backgrounds or changed field laws.

For an explicitly amended energy $E+\sum_jK_j$, let $a_j$ be the exponent of $K_j$ under the norm-preserving spatial dilation and $b_j$ its exponent under $\phi\mapsto\sqrt\eta\phi$. Subtracting the two variations gives the constructive necessary condition

$$\omega N=\sum_j(b_j-a_j)K_j.$$

A supplied shift $\Omega N$ attaches the phase $e^{-i\Omega t}$ without deriving its physical clock. The positive quartic contact $K=\lambda_c\int\rho^2/2$ instead gives $\omega N=(2-d)K$. It cannot leave the old bubble unchanged, since that would demand the constant $\omega=\lambda_c\rho(x)$ at every point. A new self-consistent profile is required. These are restrictions on candidate equations, not a count selector or the C-times mass-frequency derivation.

For comparison, the scalar-coefficient spatial equation has its own derivative decomposition, valid where $V\ne0$:

$$j_i=\frac1{V(x)}\left[ \operatorname{Im}(\psi^\dagger\partial_i\psi) +\partial_j(\psi^\dagger S_{ij}\psi)\right]. \tag{C11}$$

It follows by inserting $D\psi=V\psi$ into $2Vj_i=\psi^\dagger\gamma_iD\psi+(D\psi)^\dagger\gamma_i\psi$ and splitting the Clifford product into symmetric and antisymmetric parts. The position-dependent factor $1/V$ stays outside the derivative. It cannot be moved inside without an extra gradient term. This derivation does not replace a massive matrix coefficient by a scalar in a different equation.

6. Consequences for Polarization and Coupling

The depth-four B already entering the Hodge-channel comparison now has an explicit realization in the current of the existing spatial profile. At six the same calculation supports three circulation planes; at ten the distinction between pure and general chiral polarization matters. This strengthens the connection between the shared spinor construction and spatial polarization without deriving independent color or a measured interaction strength.

The reciprocal contact benchmark supplies a separate, specified energy term whose spatial and orientation derivatives respond to both patterns. It also shows why the full component overlap can matter when the two-form current agrees. Its Gaussian controls and quartic contact are not solutions of (C3). Combining that contact with this bubble changes the equation and requires checking the resulting profiles and complete kinetic action. The two calculations cannot simply be added and called the finished mass operator.

The next physical question is now more concrete: which common same-field dynamics preserves a suitable localized profile, its fundamental clock and conserved quantities while producing reciprocal polarization-dependent response? The present calculation supplies the explicit current to use in that test. It does not yet fix physical amplitudes, count selection, photon reconstruction, electron/muon charge equality or neutrino interaction rates.

Reproduction and Review

The profile and conventions were read from the sector case at snapshot 3701, SHA-256 81def44492b22640ab31ffb33f51ea0658df7c380026532a114f9f1ac464001e. The existing complete-polarization source has SHA-256 8fcbc63dea7d23a39531fda2f568f483145538ff4ed8587d54e5f9d5c9698daa.

The principal's reproducible sources are calculation-record/polarization-reciprocal-response-20261102/bubble-current-checks.py and bubble-current-results.json in MariaDB. All 231 checks passed: the spatial equation, density, current, divergence and depth controls, exact radial integrals, and controls exposing the invalid single-chirality radial substitution. They are finite verification evidence alongside the displayed derivations, not proofs of physical identification.

Claude independently calculated the profile current and reviewed the integrated cancellation in agents/claude/polarization-20261102/. The governing submission is polarization-bridge-packet-corrected.md read with packet-correction-2.md. Its scripts profile_current_check.py, profile_current_check2.py and profile_current_check3.py preserve independent numerical evidence. The principal reviewed their equations and corrected the original carrier mismatch, scalar-coefficient substitution, divergence sign and depth-ten scope. This cross-review covers the displayed current identities and conditional interpretations; it is not blanket acceptance of the original packets or a full-theory validation.

The clock restriction and amended-action condition were independently reviewed in agents/claude/clock-virial-review-20261102/review-clock-virial.md, read with calculation-record/claude-polarization-integration-20261103/clock-review-amendment.md. The corrected dilation-domain statement above is essential. The original principal derivation and 49-check program are in calculation-record/polarization-clock-virial-20261102/; the subsequent exact correction checks are in calculation-record/claude-polarization-integration-20261103/. This review is scoped to these conditional identities.