Geometric Color Channels, Pair Response and Charge Consistency

The middle-Hodge construction gives a complete three-component channel only at depths three and four. Combining that channel with its volume tensor gives a pair operator with the standard color decomposition. Combining its rank with the stated chiral weak field content gives charge fractions and an additional parity test.

These calculations use equivalent relative modes of the same spatial field. The frame and spin factors must have the meaning already specified in the Hodge construction. A three-component spatial spin-one representation alone does not give an independent color symmetry.

1. One Geometric Rank Enters Several Calculations

For $d=2m-1$ or $2m$, the complete middle channel has complex rank

$$k_m=\binom{2m-1}{m-1}.$$

The nontrivial rank-three case is exactly $m=2$. With a Hermitian metric and parallel complex volume, the maximal mixing group is $SU(3)$. The complete relative-mode space then has eight traceless generators, fundamental Casimir $4/3$ and a scalar alternating trilinear amplitude. These are consequences of the same carrier, conditional on realizing that unitary mixing as a physical response.

2. Contract the Third Pattern and Obtain the Pair Operator

For three distinguishable mode amplitudes $a,b,c\in\mathbb C^3$, define

$$D(a,b,c)=\epsilon_{ijk}a_i b_j c_k.$$

A term $\gamma|D|^2$ is neutral under the separate overall phase of each pattern. It is therefore compatible with their differing electric charges at the level of this energy term. The amplitudes must be distinguishable field modes or spatial data: $D(a,a,a)=0$.

Let the third pattern have a specified isotropic covariance

$$\langle c_k\bar c_l\rangle=\sigma^2\delta_{kl}.$$

Contracting the two alternating tensors gives

$$\epsilon_{ijk}\epsilon_{i'j'k} =\delta_{ii'}\delta_{jj'}-\delta_{ij'}\delta_{ji'}.$$

Hence the reduced pair energy is

$$\langle\gamma|D|^2\rangle_c =\gamma\sigma^2\left(\|a\|^2\|b\|^2-|a^\dagger b|^2\right).$$

As an operator on $3\otimes3$ this is $\gamma\sigma^2(I-\mathsf S)$, where $\mathsf S$ exchanges the two slots. The normalization convention $\operatorname{tr}(T_AT_B)=\delta_{AB}/2$ gives

$$\mathsf S=\frac13I+2\sum_{A=1}^{8}T_A\otimes T_A,\qquad I-\mathsf S=\frac23I-2\sum_AT_A\otimes T_A.$$

Thus the geometric volume contraction produces a scalar term and a definite color-Casimir term. The Casimir pair eigenvalues are

$$-\frac23\quad\text{on }\overline{\mathbf3}, \qquad \frac13\quad\text{on }\mathbf6.$$

This supplies the relative channel factors to insert into a derived spatial interaction. The sign of $\gamma$, its spatial dependence and its supporting energy determine attraction or repulsion. Positive $\gamma$ penalizes the antisymmetric pair; reversing that preference needs a justified interaction sign and stability of the full energy. The third-pattern covariance is part of the physical reduction, not a consequence of rank three.

For a general positive covariance $C$, write $L(a\otimes b)_k=\epsilon_{ijk}a_i b_j$. The reduced operator is $L^\dagger C^T L$. Its three nonzero eigenvalues are twice the eigenvalues of $C$; the symmetric pair space remains in its kernel. Equal response in the three channels is therefore a testable isotropy condition on the same-field background.

3. Apply Exactly the Same Contraction at Higher Rank

On $\mathbb C^k$, tracing the last $k-2$ slots of the unnormalized volume tensor gives

$$\operatorname{Tr}_{3,\ldots,k}|\epsilon_k\rangle\langle\epsilon_k| =(k-2)!(I-\mathsf S).$$

For the unit-norm volume state, divide by $k!$:

$$\rho_{12}=\frac{I-\mathsf S}{k(k-1)}.$$

A scalar trilinear volume exists at nontrivial full-channel rank $k=3$. Its energy is sixth order; a full rank-$k$ volume energy has degree $2k$.

There is also a uniform sixth-order invariant at every $k\geq3$:

$$\|a\wedge b\wedge c\|^2=\det\operatorname{Gram}(a,b,c).$$

With the same isotropic third-pattern covariance its contraction is

$$\langle\|a\wedge b\wedge c\|^2\rangle_c =(k-2)\sigma^2 \left(\|a\|^2\|b\|^2-|a^\dagger b|^2\right).$$

The coefficient follows by tracing the three-slot antisymmetrizer. It is zero at rank two, one at rank three and grows at larger rank. Thus a mechanism selecting a scalar volume amplitude and a mechanism using the norm of all three-form amplitudes have different dimensional consequences. The latter allows higher-rank pair response.

This identifies the physical selection rule needed for a color explanation: derive which invariant and which occupied modes the common action actually supplies. The rank theorem and volume contraction give a concrete $SU(3)$ channel, while the higher-rank control preserves the unresolved question of force absence.

4. The Same Rank Fixes Conditional Charge Fractions

Assume a four-dimensional observed chiral reduction with one left weak quark doublet in the fundamental of $SU(k)$, the corresponding right singlets, one lepton doublet and right lepton singlets including a neutral neutrino. These are observed-field representation assumptions, not new spatial dimensions.

Use $Q_{\rm electric}=T_3+Y$, fix the charged lepton to $-1$ and require vector-like electromagnetic charges. Neutrino neutrality gives $Y_L=-1/2$. The mixed $SU(2)^2U(1)$ condition gives

$$kY_Q+Y_L=0,\qquad Y_Q=\frac1{2k}.$$

The quark electric charges are consequently

$$q_u=\frac{k+1}{2k},\qquad q_d=\frac{1-k}{2k}.$$

The remaining local gauge and gravitational anomalies cancel with the corresponding right-handed charges. This specializes the anomaly-family calculation without introducing a Yukawa mechanism into IDWT.

At the geometric rank $k=3$, the same carrier yields $q_u=2/3$ and $q_d=-1/3$. This connects the color rank with the charge fractions under the stated neutrality, normalization and chiral-content premises. Those premises still need their own geometric and physical justification.

5. A Further Weak-Anomaly Parity Test

A stand-alone four-dimensional quantum $SU(2)$ gauge reduction containing only fundamental weak doublets and singlets requires an even number of left-handed doublets. This is the familiar global anomaly reviewed in Wang, Wen and Witten, introduction.

The displayed family has $k+1$ such doublets, counting color multiplicity. One generation, or any odd number of identical generations, therefore requires $k$ odd. This test assumes no additional anomaly-cancelling doublets or bulk inflow.

For the middle-Hodge rank,

$$k_m=\frac12\binom{2m}{m},\qquad v_2(k_m)=s_2(m)-1,$$

where $s_2(m)$ counts the ones in the binary expansion. The identity follows from $v_2(n!)=n-s_2(n)$. Hence

$$k_m\text{ odd}\quad\Longleftrightarrow\quad m\text{ is a power of two}.$$

The allowed depth pairs are $(1,2),(3,4),(7,8),(15,16),\ldots$. This is an all-dimensional parity statement; binary notation computes divisibility and is not a physical numeral-base hypothesis.

Among the maintained depths, the non-Abelian full-channel version of this one-family construction passes the parity test only at three/four. Higher controls prevent claiming uniqueness over all dimensions. This constrains a proposed full-channel quark-like weak family; it does not prove that an arbitrary extra vector-like force cannot exist. A geometric field model must calculate its complete observed chiral spectrum before applying the anomaly count.

6. What Has Been Connected

One candidate rank now controls the complete color carrier, its eight generators, a volume-to-pair interaction, conditional electric fractions and a global weak-consistency test. The same-field vertex and occupation law remain explicit physical inputs to derive. The calculation gives several linked restrictions on that completion rather than independent tunable assignments.

The finite-sawtooth extension adds an explicit temporal source under a separable preparation. Its positive pair overlap multiplies $I-\mathsf S$, preserving the symmetric kernel and antisymmetric channel while making the scalar response depend on relative phase. The higher-rank three-form contraction still carries $k-2$; temporal weighting alone supplies no additional exclusion of higher-sector forces.

The common checker contracts volume tensors explicitly through rank six, checks the three-form extension through rank eight, verifies $SU(3)$ generator and pair identities, and tests charge polynomials and geometric parity across depths 2 through 16.