Compact Non-Abelian Gauss Data Delimit the Flux-Confinement Result

The linear separation bound is valid for its commuting connection sector. Allowing the full connection of the proposed action changes the prescribed-source variational problem: commutators can carry the required color density inside each source core. An explicit finite-energy construction gives a separation-independent upper bound for two opposite compact sources. A universal positive linear lower bound is therefore unavailable in this larger classical domain.

This is a constructive limitation of that action, not a disproof of the sector geometry or of a quantum color theory. The Hodge gate continues to remove an independent source at the inactive occupied depths. A connection induced by a more restrictive geometric reduction, a fully dynamical matter source, or a quantum fundamental-source sector defines a different problem and needs its own derivation.

The Same Positive Hamiltonian and Full Gauss Law

Use the neutral scalar and Hermitian SU3 connection variables already proposed, with $c=1$ and positive $\kappa,\Omega$:

$$\epsilon=\kappa h_du^4,\qquad \mathcal E=\frac12(|\nabla u|^2+\Omega^2u^2) +\frac1{2\epsilon}\sum_i\|D_i\|^2 +\frac\epsilon2\sum_{i<j}\|F_{ij}\|^2.$$

The generator metric is $\|X\|^2=2\operatorname{tr}(X^\dagger X)$, so it agrees with components in the convention $\operatorname{tr}(T_AT_B)=\delta_{AB}/2$. The full constraint is

$$\sum_i\left(\partial_iD_i-ig[A_i,D_i]\right)=\rho, \qquad F_{ij}=\partial_iA_j-\partial_jA_i-ig[A_i,A_j].$$

At active physical depths three/four, $h_d=1$. Unlike ordinary divergence, this covariant divergence does not make a fixed color component of plane flux conserved.

One Commutator Produces a Fundamental Mean-Color Direction

For a unit fundamental color ray $e_1$, its classical density matrix in the Lie algebra is

$$H=\sum_A(e_1^\dagger T_Ae_1)T_A =\frac12\left(e_1e_1^\dagger-\frac13I\right) =\operatorname{diag}(1/3,-1/6,-1/6).$$

There are traceless Hermitian matrices $X,Y$ with

$$-i[X,Y]=H,\qquad \|X\|^2=4,\qquad \|Y\|^2=\frac14.$$

An explicit construction uses the unitary three-point Fourier matrix $U$. The matrix $H'=UHU^\dagger$ has zero diagonal and off-diagonal entries $1/6$. Let $X'=\operatorname{diag}(-1,0,1)$, with entries $a_i$, and set

$$Y'_{ij}=\frac{iH'_{ij}}{a_i-a_j}\quad(i\ne j),\qquad Y'_{ii}=0.$$

Then $X=U^\dagger X'U$, $Y=U^\dagger Y'U$ have the stated commutator and metric. This avoids restricting the source to one SU2 root, whose diagonal eigenvalues would not equal those of a pure SU3 fundamental mean source.

Compact Divergence-Free Displacement

Choose a nonnegative compact radial function $w$ on the unit ball and a constant antisymmetric matrix $J$ rotating its first two ordinary spatial directions. Put

$$f=w^6,\qquad v=J\nabla f.$$

Because $J$ is antisymmetric and the Hessian of $f$ is symmetric,

$$\nabla\cdot v=\operatorname{tr}(J\nabla^2f)=0.$$

Take

$$u=a w,\qquad A_i=\alpha v_iX,\qquad D_i=\beta v_iY, \qquad a,\alpha,\beta>0.$$

All connection components are proportional to $X$, so their mutual commutators vanish. Their derivative curvature is generally nonzero. The full Gauss law becomes exactly

$$\rho=g\alpha\beta |v|^2H.$$

Let $I_0=\int|v|^2$ and fix the source independently of $a$ by

$$\alpha\beta=\frac{Q}{gI_0},\qquad \rho(x)=\frac{Q}{I_0}|v(x)|^2H.$$

In the chosen source gauge, $\int\rho=QH\ne0$. The source has one fixed color orientation and a nonnegative scalar density. Its covariant Gauss charge is carried locally by the connection commutator, while every field strength vanishes outside the ball. A gauge transformation changes its color orientation, not the gauge-invariant vanishing of the exterior field strengths.

Complete Energy Is Finite

Define the positive shape integrals

$$I_e=\int\frac{|v|^2}{w^4},\qquad I_m=\int w^4\sum_{i<j}(\partial_iv_j-\partial_jv_i)^2, \qquad K=\int|\nabla w|^2,\quad M=\int w^2.$$

The complete energy is

$$E=\frac{\|Y\|^2 I_e}{2\kappa}\frac{\beta^2}{a^4} +\frac{\kappa\|X\|^2 I_m}{2}\alpha^2a^4 +\frac{a^2}{2}(K+\Omega^2M).$$

At the fixed product $\alpha\beta=Q/(gI_0)$, balance the first two terms. Their minimum in this trial family is

$$\boxed{E_A=\frac{Q}{g}\frac{\sqrt{I_eI_m}}{I_0}},$$

using $\|X\|^2\|Y\|^2=1$. The complete energy adds the positive finite neutral deformation term. Every finite $a>0$ gives finite connection amplitudes and energy; no singular limit is needed for the separation counterexample. As $a\to0$, $A$ grows as $a^{-2}$ while $D$ decreases as $a^2$. That limiting family has no asserted regular minimizer or master-field interpretation.

The executable polynomial example is

$$w=(1-r^2)^4_+,\qquad f=(1-r^2)^{24}_+.$$

The neutral mode is $C^3$ across the boundary; the connection, displacement and curvature vanish to much higher order. The electric field $D/\epsilon$ also approaches zero there. All displayed integrals are finite. Exact beta integrals and independent radial quadrature verify their values. Smooth bump functions provide the same analytic construction without the polynomial example's finite differentiability.

Opposite Fixed Cores Defeat a Universal Linear Lower Bound

For each core, keep the same shape, radius, source strength and constitutive parameters. Reverse $\alpha$ in the second core to reverse its source. Translate the two cores to separation $R>2$, so their supports are disjoint. Every local term in the complete Hamiltonian is then additive, and combined Gauss has no cross-commutator support. Hence

$$\boxed{E_{\min}^{\mathrm{full\ classical}}(R)\le2E_{\mathrm{core}} \quad\text{for every }R>2.}$$

There is no displacement flux through the plane separating the balls. This upper bound excludes a positive $T R$ lower bound in the unrestricted classical prescribed-source domain for these fixed sources. The original commuting theorem remains valid because these fields do not belong to its commuting domain.

The trials are admissible constrained Hamiltonian data, not exact static solutions. A variational upper bound requires admissible data; it does not require the trial to be a stationary solution. Global evolution in the degenerate vacuum is still unproved. The calculation also does not include the source's own kinetic and supporting energy. If localized master-field matter of fixed shape can be supplied with finite local energy, that energy must be included and checked; it has not been assumed away as a particle result here.

Center Charge and Classical Mean Color Are Different

The connection variables are adjoint and are unchanged by the SU3 center. A fundamental vector is multiplied by $e^{2\pi i/3}$, which no tensor product of center-neutral adjoints cancels. Therefore this classical mean-density construction does not screen a quantum fundamental's triality or construct a quantum gauge-invariant isolated quark. The source matrix $H$ is adjoint-covariant data and contains less information than a fundamental quantum source state.

Ordinary integrated color components are gauge dependent; vanishing exterior field strength and finite Hamiltonian energy are the relevant verified statements here. Historical classical-source work already warns that connection contributions can change naive color screening conclusions. Sikivie and Weiss's primary paper gives examples; a subsequent primary analysis emphasizes the gauge interpretation. Their ordinary-dielectric examples are comparisons, not the derivation of this quartic-dielectric trial. The first preprint's text was accessed in full; the subsequent paper was available only as its publisher abstract, so its detailed arguments have not been used.

Sector Scope and Verification

The same Hodge operator is recomputed at all occupied depths. It is active at three/four and zero at two/five/six/ten. At the inactive depths with a surviving normal algebra, finite energy still requires $D=0$, and the declared action still constrains its source to zero. At ten this selected carrier has no independent normal algebra. The compact nonzero-displacement construction is consequently permitted only in the active sectors of this action.

The checker and report preserve 846 Gauss and energy checks across eight cases, plus 18 Hodge dependency checks. They verify actual SU3 matrices, source direction and normalization, divergence, complete curvature, positivity, independent shape integrals, fixed-source amplitude variations, disjoint-core Gauss and every occupied gate. No particle simulation or particle-specific fit is used.

The surviving sector result is a conditional selection of an independent classical color-source channel at three/four and positive commuting flux guides there. Full physical confinement now specifically needs a source-sector or geometric restriction excluding this admissible classical alternative, rather than a further refinement of the commuting tube profile.