Sharp Plane Bounds Give Linear Energy for Finite Commuting Sources

The positive Hodge-gated flux guide can be tested beyond an ideal infinite tube. For fixed, opposite commuting sources with a source-free interval of length $R$, the same action gives a lower bound $T_{\min}R$. Finite cap trials give an upper bound $T_{\min}R+C$ with an $R$-independent positive constant. The bounds therefore share a linear asymptotic coefficient without requiring an exact finite-end solution.

This is a prescribed-source variational result for a commuting normal-connection channel. It is not a proof about quantum color representations, dynamical source creation or full non-Abelian screening. The same exact Hodge gate and ordinary spatial sector response are retained; no independently adjusted particle coefficients enter.

The Tube Is a Sharp Transverse Minimum

For physical depth $d=3,4$, let $n=d-1$ and let $v$ be the positive ground profile of $-\Delta v+v=v^3$ on $\mathbb R^n$. Put $P=\int v^4$. The established sharp interpolation characterization of this elliptic profile gives

$$\int u^4\le\frac{\Omega^{n-4}}P S[u]^2, \qquad S[u]=\int\left(|\nabla u|^2+\Omega^2u^2\right).$$

For example, the sharp Gagliardo–Nirenberg form is $\int u^4\le C_nK^{n/2}M^{(4-n)/2}$. Maximizing its right-hand side at fixed $K+\Omega^2M$ sets $K/(\Omega^2M)=n/(4-n)$. The ground profile saturates that ratio and gives the displayed constant. Holmer and Roudenko, section 3, gives the three-dimensional sharp inequality and profile identities; Weinstein's sharp-interpolation work is the general variational reference. These are mathematical inputs, not IDWT particle-frequency laws.

At fixed longitudinal flux $Q$, Cauchy–Schwarz then gives

$$T[u]\ge\frac{Q^2P\Omega^{4-n}}{2\kappa S[u]^2}+\frac12S[u].$$

Minimizing over the positive scalar $S$ yields

$$S_*^3=\frac{2Q^2P\Omega^{4-n}}\kappa, \qquad \boxed{T_{\min}=\frac34S_*.}$$

The rescaled ground profile with its flux-balanced amplitude attains equality. This establishes a global transverse minimum within the stated neutral-mode commuting-flux class, not merely a stationary radial solution.

The unchanged all-sector comparison remains explicit: depth two has a zero geometric dielectric; five/six/ten also have that exact zero, and their transverse dimensions independently fail the regular quartic profile identity. Independent nonzero color sources are constrained away there in this action. The finite-source calculation therefore concerns active three/four while retaining controls at all occupied depths.

Slice the General Positive Hamiltonian

Allow arbitrary longitudinal dependence, transverse displacement and magnetic fields. In a source-free interval, ordinary Abelian Gauss's law gives

$$\frac{d}{dz}\int_{\mathbb R^n}D_z(y,z)\,d^ny=0,$$

provided the lateral boundary flux vanishes. Thus each plane carries the same $Q$. The full positive Hamiltonian includes transverse scalar gradients and restoring energy, $D_z^2/(2\kappa u^4)$, longitudinal scalar gradients, transverse electric displacement and magnetic energy. Dropping the latter positive terms and applying the sharp bound on each plane gives

$$\boxed{E\ge T_{\min}R.}$$

The bound permits nonradial shapes, bends and variation along the interval. Its conserved flux and boundary domain are essential. In non-Abelian Gauss's law, connection commutators also carry color. A fixed component of color flux is not automatically a conserved gauge-invariant quantity, so this proof cannot be transferred to that case without a further reduction.

Explicit Finite Caps Have the Same Linear Coefficient

Take the ground-profile amplitude $A$ and constant electric field $E_z$ of the tube. Let $w(z)=1$ throughout an interval of length $R$ and join it to zero in two caps of fixed width $L$. Use

$$u(y,z)=A v(\Omega y)w(z),\qquad D_z=\kappa A^4v(\Omega y)^4w(z)^4 E_z, \qquad D_\perp=0,$$

with zero magnetic field. The electric field is constant, so its spatial curl vanishes. The required fixed source is

$$\rho(y,z)=Q f(y)\partial_z(w^4),\qquad f(y)=\frac{\Omega^n v(\Omega y)^4}P,\qquad\int f=1.$$

The two caps carry total source $+Q$ and $-Q$. Their transverse profiles decay rapidly, and both source shapes remain fixed as $R$ changes. The deformation is an admissible trial in the caps, rather than a claimed exact stationary solution there.

For a unit cap coordinate $s\in[0,1]$, choose

$$w(s)=10s^3-15s^4+6s^5.$$

Its value and first two derivatives match the constant exterior pieces. Define the exact positive integrals $\alpha_j=\int_0^1w^jds$ and $\alpha_{\nabla}=\int_0^1(w')^2ds$. Direct polynomial integration gives $\alpha_2=181/462$, $\alpha_4=2549/8398$ and $\alpha_{\nabla}=10/7$. If $M=\int v^2$, one cap has energy

$$C_{\rm cap}=T_{\min}L\left(\frac{\alpha_4}3+\frac{2\alpha_2}3\right) +\frac{A^2\Omega^{-n}M}{2L}\alpha_{\nabla}.$$

The first two terms retain the electric and transverse scalar energy; the last is the longitudinal deformation gradient. Exact polynomial integration and an independent numerical integral check the cap factors. Therefore, for these fixed source cores,

$$\boxed{T_{\min}R\le E_{\min}(R) \le T_{\min}R+2C_{\rm cap}.}$$

No numerical endpoint fit determines the slope. The existence and regularity of an actual minimizing endpoint solution, and any source's own kinetic energy, remain separate questions. The bounds concern the field energy with prescribed sources.

The Complete Abelian Quadratic Energy Retains Stability

The scalar minimum is also a lower bound for the whole commuting electric second variation. In profile units, take $u_0=v$, $D_{z,0}=\kappa v^4E_z$ and $2\kappa E_z^2=1$. Write $h=\delta u$, $p=\delta D_z$ and $g=4\kappa E_zv^3$. Directly differentiating $D_z^2/(2\kappa u^4)$ and completing its square gives

$$H_2=\frac12\int\left(\dot h^2+hL_+h+ \frac{(p-gh)^2}{\kappa v^4}\right) +\text{positive remaining field terms},$$

where $L_+=-\Delta_y+1-3v^2$. The remaining terms include the longitudinal scalar gradient, transverse electric energy and magnetic energy. At fixed plane flux, $\int p=0$. Minimizing the square under that constraint yields

$$H_2\ge\frac12\langle h,L_+h\rangle +\frac4P\left(\int v^3h\right)^2.$$

This is precisely half the nonnegative fixed-flux Hessian already established for the scalar profile. Variations of the electric displacement therefore do not restore its negative local-cubic direction. Positive longitudinal and other Abelian field terms cannot lower this bound. The argument assumes regular finite-energy perturbations with the stated Gauss and lateral boundary conditions.

It establishes a nonnegative quadratic Hamiltonian in the commuting connection sector. It does not settle nonlinear evolution, charged non-Abelian fluctuations, endpoint response or the degenerate vacuum Cauchy problem.

A Separate Net-Source Obstruction

In full physical dimension three or four, finite scalar $H^1$ energy implies $u\in L^4$. With $\epsilon=\kappa u^4$, finite electric energy then gives

$$\int|D|\le \left(\int\frac{|D|^2}{\epsilon}\right)^{1/2} \left(\int\epsilon\right)^{1/2}<\infty.$$

If $\nabla\cdot D=\rho$ with a compact smooth source, integrate against a cutoff equal to one inside radius $R$, with gradient bounded by $C/R$. Integration by parts bounds the total cutoff source by $(C/R)\int|D|$, which tends to zero. Consequently every such finite-energy source has $\int\rho=0$. A nonzero isolated net commuting source cannot have finite energy in this action.

This is an ordinary divergence and Sobolev argument. It does not identify a classical source component with an observed quark charge, prove full non-Abelian confinement or forbid neutral sources with nontrivial spatial response.

Verification

The checker and report preserve 231 passing checks across 15 cases, with exact code dependencies, occupied-sector controls, polynomial cap identities, independent complete cap integrals, Gaussian comparisons with the sharp bound, fixed-source separation slopes and directly completed electric second variations. The original tube checker supplies its independently verified ground profiles. No particle simulation, particle-specific retuning or calculation at unoccupied depths is used.

Full Classical Connection Has a Compact Alternative

The explicit non-Abelian Gauss construction supplies finite-energy compact trials for fixed one-signed mean-color source cores under this same neutral dielectric. Two opposite disjoint cores have a separation-independent field-energy upper bound. Their commutator source removes the conserved commuting plane flux used above. These results therefore establish the commuting theorem’s actual domain and exclude transferring its linear lower bound to the unrestricted classical prescribed-source problem. They do not screen quantum fundamental triality or derive a localized matter source. The inactive Hodge source constraints survive unchanged.