Positive Spatial Kinetics Constrain the Color-Current Polarization

In the four-plus-six spatial reference split, the base has four ordinary spatial dimensions and its perpendicular complement has six. Time is a separate coordinate in the infinite-dimensional physical spacetime. This four-plus-six split concerns ten spatial directions in the matter domain.

The current self-contact has been tested on positive branches of the actual spatial Clifford symbol. This removes the need to choose a purely chiral source polarization for that control. The positive branch balances its two containing spatial source chiralities automatically, and the independent normal triplet remains intact at three/four.

The calculation also sharpens the force limitation: its unpolarized pair operator is $(4/3)I-\sum_AT_A\otimes T_A$. The coefficient of the color Casimir is negative, and its positive contact eigenvalue is larger in antisymmetric color than symmetric color. Recognizing the familiar Casimir algebra is not by itself a strong-force sign prediction.

These are plane-wave principal-symbol controls, not complete localized particle solutions. The supplied observed mass-frequency targets are neither replaced by $|p|$ nor rederived here.

1. Kinetic Balance of Spatial Source Chirality

For unit momentum direction $\widehat p$ in an occupied depth $d$, use the full symbol and its positive projector,

$$h_d(\widehat p)=\sum_{i=1}^d\gamma_i\widehat p_i,\qquad \Pi_{\widehat p}=\frac{I+h_d(\widehat p)}2.$$

The neighboring even source carrier has $2m=d$ or $d+1$, and its oriented Clifford volume $\Gamma_{2m}$ anticommutes with every active $\gamma_i$. Thus

$$\boxed{\Pi_{\widehat p}\Gamma_{2m}\Pi_{\widehat p}=0.}$$

Every unit spinor on this positive symbol branch therefore has source weights $\|P_+\psi\|^2=\|P_-\psi\|^2=1/2$. This is a kinetic identity, independent of physical amplitude or a fitted mixing ratio. The source chirality refers to the containing ordinary spatial Clifford volume, not automatically to spacetime weak chirality or antiparticles.

The fixed normal-triplet projector commutes with $h_d$ at depths two, three and four. It fails once normal tangent directions become active at five, six and ten, as in the full kinetic comparison. The common Hodge-current self-contact remains zero at two, five, six and ten even after imposing the positive-symbol control. The occupied-sector selection survives that additional constraint.

2. The Actual Positive-Branch Fierz Norm

At three/four, choose an orthonormal four-dimensional spatial base positive spin frame $U_{\widehat p}$ with two columns. The complete normal triplet then gives

$$\psi=(U_{\widehat p}\otimes I_3)\zeta,\qquad \zeta\in\mathbb C^2\otimes\mathbb C^3.$$

This retains the full positive tangent-symbol spin multiplicity. Compressing the six middle Clifford blades to this kinetic branch gives one complete Pauli contraction, whereas a purely chiral source gave two copies. For arbitrary $\zeta$ represented as a two-by-three matrix, put

$$n=\|\zeta\|^2,\qquad p_s=\operatorname{Tr}(\zeta\zeta^\dagger)^2.$$

The actual current norm is

$$\boxed{\sum_{A,I}J_{A,I}^2=\frac{7n^2-5p_s}{6}, \qquad\frac13n^2\le\sum_{A,I}J_{A,I}^2\le\frac34n^2.}$$

The factors follow from Clifford completeness and $\operatorname{Tr}(T_AT_B)=\delta_{AB}/2$, not from dividing by mass or choosing a physical field amplitude. Relative to the earlier purely chiral source, the contact strength is halved for this specified kinetic preparation. Independent spin/color symmetry survives, but source preparation affects its normalization.

3. Independent Momentum Directions Give the Same Unpolarized Color Operator

Let two factorized color patterns have unit color vectors $c,d$, independent unit momentum directions $\widehat p,\widehat q$, and their own positive-symbol spin frames. Their relative phase is averaged uniformly; that is a preparation or averaging assumption, not synchronization of different particle frequencies.

For each source chirality, the general phase-averaged mixed quartic can be written using its norm, spin purity and color purity. In particular, if the two four-spinor factors are $s,t$, put

$$p_\chi=\|P_\chi s\|^2,\quad q_\chi=\|P_\chi t\|^2, \quad h_\chi=|(P_\chi s)^\dagger P_\chi t|^2, \quad c_o=|c^\dagger d|^2.$$

Subtracting both isolated self energies gives

$$\frac{\Delta V}{\gamma} =\frac13\sum_\chi\left[(14-10c_o)p_\chi q_\chi+(14c_o-10)h_\chi\right].$$

The positive symbol identity sets $p_\chi=q_\chi=1/2$. For independent unpolarized spin preparations $\rho_s=\Pi_{\widehat p}/2$, $\rho_t=\Pi_{\widehat q}/2$,

$$\sum_\chi\langle h_\chi\rangle =\frac14\sum_\chi\operatorname{Tr}(P_\chi\Pi_{\widehat p}P_\chi\Pi_{\widehat q}) =\frac14.$$

Therefore

$$\boxed{\left\langle\frac{\Delta V}\gamma\right\rangle =\frac{3-c_o}{2},\qquad W_{\rm color}=\frac{3I-\mathsf S}{2} =\frac43I-\sum_AT_A\otimes T_A.}$$

The result is independent of the two momentum directions. This is a concrete compatibility of spatial spin, kinetic polarization and the independent normal color algebra. It does not require selecting one spatial orientation for both particles.

The operator has eigenvalue $2$ in antisymmetric color and $1$ in symmetric color. The scalar term and negative Casimir coefficient remain. Actual forces additionally depend on spatial profiles, the complete same-field response, coherence and gradients; no attraction or confinement follows from this positive local contact alone. A color-carrier tensor operator and a pair of classical factorized color amplitudes also remain distinct descriptions.

4. Scope and Reproduction

The positive symbol is a local high-frequency control. A localized spatial eigenmode has a full position-dependent symbol, connection, contact operator and complementary bands. Those can mix momenta or change polarization. The identity above must not be transferred to a complete massive particle without checking that mode and its canonical normalization.

The checker and report record 234 conditions across 30 cases. They test every occupied depth with full ambient Clifford matrices, compare the fixed normal projection, verify exact source balance and the sharper Fierz norm, and independently integrate phase and finite spin-basis averages for unrelated momentum directions at three/four. Source code hashes identify the executed current and normal-kinetic dependencies. No particle simulation, individual coefficient fit or calculation at unoccupied depths is used.