Independent Normal Color and Two-Sided Hodge Contact Share One Current Action

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 and symmetric-Hodge constructions can be combined without identifying their respective three-component spaces. Independent color acts on the ordinary normal spinor factor. Hodge contact acts on a spatial form index of that same field's current. One explicit positive quartic energy then has an active independent-color channel at depths three/four and an exact zero at the other maintained depths.

This is a proposed common interaction, not a derivation of the master law or the measured strong force. Its value is that the full source, field order, spin action and geometric parity are now calculated together. The original holomorphic middle-form source and its twelfth-order three-source energy are different candidates; their component counts need not be reinterpreted as physical color.

1. Two Different Indices of One Field Current

Use the selected ten-dimensional ordinary parent and its polarization-defined four-dimensional spatial base/six-normal split. Let $\Psi$ be the full rank-32 spatial Dirac field. The actual normal rotation construction gives eight Hermitian generators $T_A$, normalized on a normal triplet by $\operatorname{Tr}(T_AT_B)=\delta_{AB}/2$.

At three/four these generators commute with every active tangent Clifford vector and observer-spin generator. Their action is independent color on a separate factor of the full spatial field. They need not have that independent interpretation once normal directions enter the active tangent operator.

For an even carrier dimension $2m$, and an increasing middle multi-index $I$, define the Hermitian Clifford blade

$$K_I=i^{m(m-1)/2}\gamma_{i_1}\cdots\gamma_{i_m}.$$

Its color-current component is

$$\boxed{J_{A,I}(\Psi)=\Psi^\dagger B_{A,I}\Psi, \qquad B_{A,I}=\frac12\{K_I,T_A\}.}$$

Every $B_{A,I}$ is Hermitian, so $J_{A,I}$ is real. The anticommutator defines the same source even where a normal generator fails to commute with a particular blade. At three/four it reduces to $K_IT_A$. Index $A$ is the normal color adjoint; index $I$ is an ordinary spatial middle form. The latter's selfdual triple is not the color triplet.

Odd depths use the existing neighboring even-carrier shadow. This remains an explicit extension premise, rather than an intrinsic odd-dimensional middle-degree star. The full kinetic field is retained at every depth. A source can inspect chiral or normal components without replacing its spatial kinetic operator by a pointwise chiral compression.

2. One Symmetric Contact Gives the Maintained-Sector Rule

Let $H=(*+*^\dagger)/2$ on the form index. The common positive energy is

$$\boxed{E_{\rm cur}=\gamma\sum_{A=1}^{8}\|HJ_A(\Psi)\|^2, \qquad \gamma>0.}$$

This is fourth order in field amplitude. No independently chosen zero or coefficient is inserted at any depth. Since $H=*$ at even $m$ and $H=0$ at odd $m$, the source is retained at three/four and cancelled at two, five/six and nine/ten. It vanishes for every field value, phase and polarization in the inactive cases.

The independent-color kinetic test must also be included:

Thus three/four are the only active independent normal-color channels in this fixed-parent comparison. Seven/eight retains a positive geometric current interaction, but a fixed normal $SU(3)$ does not preserve its full tangent kinetic symbol. Its higher spatial response components do not automatically give thirty-five independent colors or an $SU(35)$ force. This does not prove absence of all higher-sector forces, especially in a different containing parent or with additional equivalent-mode carriers.

The energy is invariant under normal $SU(3)$ at three/four and under independent spatial spin rotations there. It is covariant under a simultaneous rotation of the field, its actual spatial tangent frame and the normal geometric data. The checker verifies both statements directly. Rotating only the field by normal $SU(3)$ at seven/eight generally changes its energy, as expected from the distinct kinetic symmetry.

3. Variation and a Positive Canonical Mode Completion

Writing $j$ for the real vector of all source components, the variation with respect to $\overline\Psi$ is

$$\frac{\partial E_{\rm cur}}{\partial\overline\Psi} =2\gamma\sum_{A,I}(H^\dagger Hj_A)_I B_{A,I}\Psi.$$

Both occurrences of the field in the quadratic source are retained. A finite same-field mode completion is

$$L=\frac i2(\Psi^\dagger\dot\Psi-\dot\Psi^\dagger\Psi) -\omega\|\Psi\|^2-E_{\rm cur},\qquad\omega>0.$$

It has positive conserved Hamiltonian $\omega\|\Psi\|^2+E_{\rm cur}$ and conserved total mode norm. At three/four its independent normal color charges are conserved as well. The exact source gradient and these conservation identities are verified, including a separate finite-difference variation check.

This is a canonical retained-mode construction. It does not replace the previously checked spatial Clifford symbol or derive the reduction of that spatial action. The coefficient $\gamma$ carries the required amplitude and time units and is not fitted independently for particles. Nonlinear self-interaction can shift an isolated pattern's frequency; the supplied observed mass-frequency targets have not been rederived from this action. Positive source energy alone does not fix the observed force sign.

4. Exact Spin and Color Pair Algebra

For a chiral source at four, retain two spin components and a normal triplet. The kinetic field itself remains full. The six Hermitian middle blades restrict to two copies of the three Pauli matrices, while the normal color matrices remain independent.

Write the source amplitude as a two-by-three matrix $\psi$. Let

$$n=\|\psi\|^2,\qquad \rho_s=\psi\psi^\dagger, \qquad p=\operatorname{Tr}\rho_s^2.$$

Direct Clifford and color Fierz contraction gives

$$\boxed{\sum_{A,I}J_{A,I}^2=\frac{7n^2-5p}{3}, \qquad \frac23n^2\le\sum_{A,I}J_{A,I}^2\le\frac32n^2.}$$

A factorized spin/color state attains the lower bound. A maximally entangled two-spin/two-color state attains the upper bound. These bounds use arbitrary field norm and do not calibrate physical amplitude.

For two factorized unit modes $a=s\otimes c$, $b=t\otimes d$, put $s_o=|s^\dagger t|^2$ and $c_o=|c^\dagger d|^2$. Average the energy of $a+e^{i\phi}b$ over relative phase and subtract the two self energies. The result is

$$\boxed{\Delta E/\gamma= \frac{14-10s_o-10c_o+14s_oc_o}{3}.}$$

Spin dependence remains explicit. Independent spin and color symmetries do not require every interaction to be spin independent. For an unpolarized spin preparation, $\langle s_o\rangle=1/2$, so the color pair contraction becomes

$$3I-\mathsf S_{\rm color} =\frac83I-2\sum_AT_A\otimes T_A.$$

The standard color-Casimir operator is therefore present, with an additional scalar term. The source's algebra is stronger evidence of a meaningful connection than the number three alone. It is not a complete strong-force prediction: a positive local contact energy and a spatial overlap kernel must be varied before attraction or confinement can be inferred. Scalar offsets, spin preparation, source coherence and response signs cannot be discarded just to retain the desired Casimir term.

5. Physical Scope of the Join

The shared mathematics now has a concrete division of work. The ordinary normal spinor factor supplies independent color. The spatial Clifford blades produce its same-field current. Symmetric Hodge contact determines which instantaneous channel survives. The current Hamiltonian supplies an explicit reciprocal variation and conserved mode energy. Each ingredient enters one stated action, rather than borrowing a conclusion from a different field order or carrier.

The selection of this Hodge kernel remains a physical premise. The complementary Hermitian Hodge channel, velocity interactions and generic norm contacts are still allowed by symmetry. A normal connection already present in spatial covariant kinetic terms is not removed merely because this current energy vanishes. Its dynamics or color-singlet occupation at depth two must also be addressed. Likewise a finite-delay replacement changes the cancellation, as the passive-path calculation establishes.

The Hermitian current has a useful link to the harmonic source question. For the conditional real preparation $\Psi(t)=f_K(\omega_0t)u$, its source is exactly $J(t)=f_K^2J(u)$. A positive-frequency complex preparation instead gives $J(t)=|z_K|^2J(u)$, with a different higher-harmonic spectrum. The current construction derives the bilinear source type; it does not make those wave preparations dynamically interchangeable. Their responses must use their actual Fourier coefficients and conserved full spatial modes.

Verification

The single checker and report preserve 6,559 passing conditions across ten cases with exact executed dependencies. They construct every source matrix through parent depth ten, compare all maintained and neighboring sectors, verify full geometry covariance and normal-color symmetry, independently vary the quartic, test canonical conservation identities, and derive the sharp Fierz and pair-phase formulas. No particle simulation, separate particle amplitude fit or claim of complete strong-force dynamics is used.

The Complex Harmonic Source Has Its Own Response

The occupied-sector complex-density comparison now computes the positive-frequency preparation with the same receiver used for the real sawtooth. Its bilinear source retains the same mean and fundamental but different higher harmonics. The geometry zeros remain exact, while the signed lepton derivative response changes and a single-frequency down preparation has no dynamic density beat. This makes the temporal occupation and readout substantive parts of a proposed physical current law.

A Local Self-Contact Formulation

The chirality-weighted self-wedge construction now reproduces this action and its full variation at all occupied depths. Exterior order reversal cancels the odd-middle contacts; the Clifford chirality identity makes the three/four contact positive. Both source chiralities preserve parity, and normal contraction explains the three-shadow density without pulling back a four-form incorrectly. The actual algebraic current is generally nonclosed, so this proposed interaction is not merely an exact-curvature boundary term. Its selection and complete physical force still need derivation.

Kinetic Polarization Changes the Pair Normalization

The positive spatial symbol comparison tests the actual current on the full kinetic carrier. It forces equal containing source-chirality norms, retains the occupied-sector zeros, and gives the spin-averaged pair operator $(3I-\mathsf S)/2=(4/3)I-\sum_A T_A\otimes T_A$ independently of the two momentum directions. The negative Casimir coefficient makes the antisymmetric contact eigenvalue larger, so the recognizable color algebra must not be presented as a derived attractive strong-force channel.

A Connection Coefficient Can Impose the Missing Source Constraint

The Hodge-gated Gauss construction puts this geometric factor in the actual connection dielectric. Its inactive zero then constrains the Gauss and spatial kinetic sources, rather than merely removing the quartic contact. Explicit singlet branches of the surviving normal algebra preserve every active Clifford direction at two and five/six. This is a changed common action with a singular zero-dielectric premise; an ordinary positive floor removes the exact constraint, and active-sector confinement remains a separate calculation.