A Parity-Even Self-Contact Produces the Occupied-Sector Hodge Rule

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 normal-color Hodge current has a more direct local-action formulation. Pair each real spatial chiral current with itself using exterior multiplication, and weight its two chiralities by their oriented Clifford signs. The resulting scalar energy is positive at three/four and identically zero at the other occupied depths. It equals the earlier positive current action there, including its complete field variation.

This is a concrete candidate interaction with a common geometric rule. Choosing this self-contact as the field's interaction remains a physical premise. The calculation neither derives the complete strong force nor excludes other allowed interactions.

1. The Same Full Field and Its Chiral Source Currents

Retain the full spatial Dirac field $\Psi$ in the selected ordinary ten-dimensional parent. Normal rotation gives Hermitian color generators $T_A$ independent of tangent spin at depths three/four. The four-dimensional spatial base and normal construction defines that geometry; no inherent three-plus-three electron split is assumed.

On an even source carrier of dimension $2m$, let

$$\Gamma_{2m}=i^m\gamma_1\cdots\gamma_{2m},\qquad P_\chi=\frac{I+\chi\Gamma_{2m}}2,\quad\chi=\pm1.$$

Here $\Gamma_{2m}$ is Hermitian and squares to one. Its sign refers to ordinary spatial Clifford chirality, not automatically to observed spacetime weak chirality. Define

$$K_I=i^{m(m-1)/2}\gamma_{i_1}\cdots\gamma_{i_m},\quad B_{A,I}=\frac12\{K_I,T_A\},\quad J_{\chi,A,I}=\Psi^\dagger P_\chi B_{A,I}P_\chi\Psi.$$

These are real commuting components of an $m$-form $J_{\chi,A}$. Both source chiralities are retained. The kinetic field is not replaced by $P_\chi\Psi$: that replacement would erase its Clifford derivatives. A source projection is a different operation from compressing the kinetic operator.

2. One Local Polynomial Contact

With the oriented carrier volume form $\operatorname{vol}_{2m}$, propose

$$\boxed{V_{\rm self}\operatorname{vol}_{2m} =\gamma\sum_{A,\chi}\chi\, J_{\chi,A}\wedge J_{\chi,A},\qquad\gamma>0.}$$

The interaction is fourth order in field amplitude. It is the invariant diagonal color contraction, with separate spatial form and color indices. It does not identify a spatial spin-one triple with three colors.

For odd $m$, graded commutativity gives

$$J_{\chi,A}\wedge J_{\chi,A} =(-1)^mJ_{\chi,A}\wedge J_{\chi,A}=0.$$

This is the precise two-sided cancellation: interchanging identical odd-degree contact factors reverses the sign. It is not the assertion that a tensor product of two identical color triplets contains a singlet.

For even $m$, the oriented Clifford-volume identity gives

$$*\bigl(P_\chi K_I P_\chi\bigr) =\chi P_\chi K_I P_\chi.$$

Since $K_I$ preserves chirality for even $m$, only the corresponding diagonal block of $T_A$ contributes. Consequently the actual projected current satisfies $*J_{\chi,A}=\chi J_{\chi,A}$, even if a different carrier's full color generator has off-diagonal chiral blocks. Therefore

$$\chi J_{\chi,A}\wedge J_{\chi,A} =\|J_{\chi,A}\|^2\operatorname{vol}_{2m},\qquad \boxed{V_{\rm self}=\gamma\sum_{A,\chi}\|J_{\chi,A}\|^2\ge0.}$$

At the occupied active depths three/four, $T_A$ commutes with the entire base Clifford algebra. The unprojected current is $J_A=J_{+,A}+J_{-,A}$, with orthogonal selfdual and antiselfdual parts. Thus

$$V_{\rm self}=\gamma\sum_A\|J_A\|^2 =\gamma\sum_A\|HJ_A\|^2, \qquad H=\frac{*+*^\dagger}{2}.$$

At the other occupied depths both sides vanish. The source action itself reproduces the earlier Hodge-filter rule, rather than introducing a zero coefficient for each particle family.

Odd depths still use their stated containing even-carrier shadow. The table is a comparison of occupied assignments under that common premise, not a derivation of particle occupation or of the entire field law.

3. Both Chiralities Make the Contact Parity Even

Changing orientation reverses $*$ and $\Gamma_{2m}$, and interchanges $P_+$ and $P_-$. Relabeling $\chi$ leaves the weighted energy and its variation unchanged. An actual reflection of a base direction also preserves the three/four energy when both source chiralities are present.

A single selected chirality would have a different parity interpretation. The proposed action includes both and retains the full tangent kinetic field. Its independence of the orientation convention and its physical base-parity test are separately checked.

Writing $j_{\chi,A}$ for component vectors and $B_{\chi,A,I}=P_\chi B_{A,I}P_\chi$, its full variation is

$$\frac{\partial V_{\rm self}}{\partial\overline\Psi} =\gamma\sum_{A,\chi,I}\chi \bigl[(*+*^T)j_{\chi,A}\bigr]_I B_{\chi,A,I}\Psi.$$

It agrees with the earlier quartic gradient at every occupied depth, not only with the energy value. The canonical retained-mode action $L=i(\Psi^\dagger\dot\Psi-\dot\Psi^\dagger\Psi)/2-\omega\|\Psi\|^2-V_{\rm self}$ retains positive conserved mode energy and conserved total norm. Its reduction from a complete spatial action remains open; nonlinear frequency shifts must still be checked against the supplied mass-frequency targets.

4. The Fourth Direction Gives a Genuine Three-Dimensional Shadow

A four-form pulled back directly to three dimensions is zero. The depth-three construction instead has the containing four-dimensional spatial base and its ordinary unit normal $n$. With $\operatorname{vol}_4=\operatorname{vol}_3\wedge n^\flat$, normal contraction gives the three-dimensional density

$$-\jmath^*\iota_n\bigl(J_{\chi,A}\wedge J_{\chi,A}\bigr).$$

For an oriented frame with $n=e_4$, write

$$B_A=(J_{23},-J_{13},J_{12}),\qquad E_A=(J_{14},J_{24},J_{34}).$$

The parent top coefficient is $2E_A\cdot B_A$. Spatial chirality imposes $E_A=\chi B_A$, so

$$\chi\,2E_A\cdot B_A=2\|B_A\|^2 =\|J_{\chi,A}\|^2.$$

The three-shadow energy therefore retains precisely the positive four-carrier contact. It uses an additional ordinary rotation direction supplied by the containing sector. The three components of $B_A$ rotate as spatial spin response; the independent normal color index remains $A$. Neither component count should be substituted for the other.

5. Sharp Bounds with the Full Base Dirac Factor

On the fixed normal triplet, the full base Dirac amplitude has two chiral two-spin-by-three-color blocks $\psi_\chi$. Define $n_\chi=\|\psi_\chi\|^2$ and $p_\chi=\operatorname{Tr}(\psi_\chi\psi_\chi^\dagger)^2$. The established chiral Fierz relation gives

$$\frac{V_{\rm self}}\gamma =\frac{7(n_+^2+n_-^2)-5(p_++p_-)}3.$$

For total triplet norm $n=n_++n_-$,

$$\boxed{\frac13n^2\le\frac{V_{\rm self}}\gamma\le\frac32n^2.}$$

Balanced factorized chiral blocks attain the lower bound. A maximally entangled spin/color source in one chirality attains the upper bound. These bounds concern the complete base-Dirac normal triplet; they do not apply to a full ambient field containing normal-color singlets. Physical amplitude remains uncalibrated.

6. This Current Is Not Automatically a Topological Curvature

If an $m$-form were an exact closed differential curvature $F=dA$, then

$$F\wedge F=d(A\wedge F).$$

Its integrated self-contact would be a boundary term under the appropriate domain conditions. A bulk force cannot be inferred from that term alone. The present $J$ is an algebraic spinor current and generally does not obey $dJ=0$.

For an explicit analytic control, take an actual unit factorized chiral spin/color amplitude $u$ with current norm $2/3$, and the decaying trial field $\Psi(x)=a(x)u$, $a(x)=\exp(-|x|^2/2)$ on the four-dimensional spatial base. Then $J(x)=a(x)^2J(u)$ and $dJ=2a\,da\wedge J(u)$ is nonzero. Its integrated contact is

$$\int_{\mathbb R^4}V_{\rm self}\,d^4x =\gamma\frac23\int e^{-2|x|^2}\,d^4x =\gamma\frac{\pi^2}{6}.$$

Under $\Psi\mapsto\alpha\Psi$, its derivative at $\alpha=1$ is $2\gamma\pi^2/3$, despite vanishing tails at infinity. This is a nonzero bulk variation. The Gaussian is a test profile, not a claimed particle solution or amplitude calibration. The checker independently verifies the exact-curvature boundary identity on the occupied even carriers.

7. Physical Scope

One local polynomial contact now supplies the maintained-sector cancellation, positive source energy, both spatial chiralities, independent color/spin at three/four and a legitimate three-shadow contraction. These properties reuse the same spatial Clifford and exterior algebra.

The choice of self-wedge contact is still a hypothesis. Generic metric-norm interactions, complementary Hodge response and a separately prescribed normal kinetic connection are not removed. Noncommuting matrix-valued forms also need their own ordering analysis; the cancellation proved here uses real commuting current components and the invariant diagonal color contraction.

The action's existing scalar and spin-dependent pair terms remain. Recovering a color-Casimir term does not fix its physical force sign, remove scalar offsets or reproduce confinement. The current preparation must also belong to the full spatial kinetics. These are specific remaining dynamical questions, not defects in the exact cancellation identity.

Verification

The checker and report preserve all executed source hashes. They compare energy and gradient at all six occupied depths, check parity and orientation, retain independent normal color, verify the full-triplet Fierz bounds and sharp examples, compute the three-shadow contraction, and distinguish a nonclosed actual spinor current from an exact curvature boundary term. No particle simulation or calculation at unoccupied depths is used.

Positive Spatial Kinetic Branches Have Been Checked

The positive-symbol current test balances the two source chiralities without fitted amplitudes. At three/four its independent normal triplet retains a two-spin positive branch and has half the purely chiral Fierz norm. Independent unpolarized momentum directions give the same color pair operator, with scalar term and negative Casimir coefficient retained. This is a local kinetic control, not a complete massive spatial solution or a force-sign derivation.