A Positive Hodge-Gated Dielectric Supports Finite-Width Color Flux at Three and Four

One positive action can combine the existing geometric color gate with an ordinary spatial flux guide. Under its explicit neutral-mode constitutive premise, a long tube of commuting color flux has a finite transverse width and positive tension at three and four. The same scaling identity excludes a regular finite-width profile at the other higher occupied depths, while the common Hodge factor removes depth two. The profile is a minimum of its transverse fixed-flux scalar energy, rather than a negative-energy supporting medium.

This is a conditional construction, not a derivation of the master field's dielectric, complete non-Abelian confinement or observed hadron spectra. Its value is that source constraints, the independent geometric color carrier, spatial flux support and complete retained energy can now be examined in one stated action.

The Neutral-Mode Premise

Retain the actual normal gauge algebra commuting with every active spatial Clifford direction. In the selected ten-dimensional parent it is $\mathfrak{su}(3)$ at two/three/four, residual $\mathfrak{su}(2)$ at five/six and zero at ten. Its generators act on spinor components associated with ordinary spatial normal rotations, not additional hidden spatial dimensions.

Let $u$ be a real, color-singlet deformation mode of the same underlying field, with canonical positive kinetic energy. Define

$$\epsilon_d(u)=\kappa h_du^4,\qquad h_d=\frac{\operatorname{Tr}(H^\dagger H)}{\dim\Lambda^m}, \qquad H=\frac12(*+*^\dagger),\qquad\kappa>0.$$

The neighboring even carrier is $2m=d$ at even depth and $2m=d+1$ at odd depth. Thus the existing even-shadow premise is retained. Ordinary Euclidean middle-form Hodge algebra gives $h_d=1$ at occupied three/four and zero at two/five/six/ten.

In units with common wave speed $c=1$, take

$$\mathcal L=\frac12\left(\dot u^2-|\nabla u|^2-\Omega^2u^2\right) +\frac12\epsilon_d(u)\sum_AE_A^2 -\frac14\epsilon_d(u)\sum_{A,i,j}F_{A,ij}^2, \qquad \Omega>0.$$

The connection and deformation are retained geometric modes of the one field. They introduce no exchanged particles or external cavity wall. $\Omega$ is the neutral deformation's restoring frequency. Identifying it with a sector fundamental or selected particle mass-frequency would require an additional derivation.

This neutral mode must be distinguished from a charged spinor amplitude. The earlier candidate $\epsilon(\Psi)=\alpha\sum_A\|HJ_A(\Psi)\|^2$ motivates quartic homogeneity, but a charged amplitude has covariant time and spatial energy. Simply naming that amplitude $u$ would omit those terms. The neutral-mode action here is a separate coherent proposal; its reduction from the master field remains open.

The retained Hamiltonian is nonnegative. Where $u\ne0$ and $h_d>0$, its principal wave cones are ordinary. At $u=0$ the gauge kinetic coefficient degenerates, so a global Cauchy problem remains to establish. In sectors with identically zero $h_d$, the Gauss construction instead imposes zero connection sources. Its explicitly checked singlet branches preserve every active spatial Clifford direction.

Minimize the Complete Energy at Fixed Flux

Take an infinitely long tube along one ordinary coordinate $z$, in one commuting color direction. The transverse coordinates $y$ have dimension

$$n=d-1.$$

For a transverse deformation $u(y)$, fix the longitudinal displacement flux $\int D_z(y)\,d^ny=Q$. Cauchy–Schwarz gives

$$\frac12\int\frac{D_z^2}{\kappa u^4}\,d^ny \ge\frac{Q^2}{2\kappa\int u^4\,d^ny}.$$

Equality holds at $D_z=\kappa u^4E_z$ with constant transverse $E_z$. The electric field therefore obeys the electrostatic curl condition. The exact positive remainder for any other displacement is $\frac12\int(D_z-D_{z,*})^2/(\kappa u^4)$.

The complete energy per unit length becomes

$$T[u]=\frac{Q^2}{2\kappa I_4[u]} +\frac12\int\left(|\nabla u|^2+\Omega^2u^2\right)d^ny, \qquad I_4[u]=\int u^4d^ny.$$

Both the electric and deformation contributions are retained. Varying the free amplitude and shape gives

$$-\Delta u+\Omega^2u=a u^3, \qquad a=\frac{2Q^2}{\kappa I_4[u]^2}.$$

Writing $u(y)=A v(\Omega y)$ reduces this to

$$-\Delta v+v=v^3, \qquad A^6=\frac{2Q^2\Omega^{2n-2}}{\kappa I_4[v]^2}.$$

The amplitude is fixed by flux balance in the proposed action, not fitted to a particle mass. The resulting tension is

$$\boxed{T=\frac34\left(\frac{2Q^2I_4[v]}\kappa\right)^{1/3} \Omega^{(4-n)/3}.}$$

It gives a linear bulk separation cost $TR$ for a long tube segment of length $R$. $Q$ is classical commuting color flux. This nonlinear tension does not establish quantum Casimir scaling, and the familiar quadratic meson/baryon coefficients cannot be inserted into it without a new derivation.

An Exact Dimensional Exclusion

Let $K=\int|\nabla v|^2$, $M=\int v^2$ and $P=\int v^4$. Multiplication of the profile equation by $v$ and integration gives $K+M=P$. Independent spatial scaling gives

$$\frac{n-2}{2}K+\frac n2M=\frac n4P.$$

Together,

$$\boxed{(4-n)K=nM.}$$

For $\Omega>0$, a nonzero regular finite-energy profile requires $n<4$, hence $d<5$. This exclusion does not depend on assuming a radial shape. Positive ground profiles exist for $n=1,2,3$. The common geometric gate then gives:

At the critical transverse dimension four, the identity forces $M=0$. Above it, the two sides have incompatible signs. These conclusions concern this regular finite-width flux-guide mechanism. They do not prove absence of every possible higher-sector interaction or singular flux state.

The cubic elliptic profile and its Pohozaev identities are standard mathematical tools; see Holmer and Roudenko, section 3. Their nonlinear Schrödinger dynamics is different from the fixed-flux action here and is not imported.

Profiles and Their Restricted Stability

Independent radial boundary-value solves at cutoffs $15,20,25$ give:

The first profile is exactly $\sqrt2\operatorname{sech}r$ and serves as a nongated mathematical control. The others are numerical roots, not exact fractions. Cutoff changes agree at about $10^{-10}$, and the integrated equation and scaling residuals are below $10^{-9}$.

The scalar Hessian of the complete energy at fixed flux, after the profile rescaling, is

$$\mathcal H=L_++\frac8P|v^3\rangle\langle v^3|, \qquad L_+=-\Delta+1-3v^2.$$

The positive nonlocal term follows from varying the dielectric integral. Dropping it would change the energy being tested. Standard ground-state variational positivity gives $\langle f,L_+f\rangle\ge0$ when $\langle v^3,f\rangle=0$; for a direct three-dimensional statement see Lemma 3.1 of Li and Yang. Decompose any real shape variation as $h=av+f$ with that orthogonality. Since $L_+v=-2v^3$,

$$\langle h,\mathcal Hh\rangle =6a^2P+\langle f,L_+f\rangle\ge0.$$

Translations remain zero modes because $\int v^3\partial_i v=0$. Thus the scalar fixed-flux shape has nonnegative second variation, with translation freedom retained.

Finite-volume radial discretizations at 400, 800 and 1,600 cells independently test the radial, translation and next angular channels. At the finest mesh the lowest radial values change from approximately $-5.413$ to $0.9782$ for $n=2$, and from $-15.30$ to $0.9989$ for $n=3$, after the required flux term. The translation eigenvalues approach zero, and the angular-two channels are positive. Direct finite differences of the complete energy also reproduce the Hessian.

This is transverse scalar stability at fixed commuting flux. It does not establish stability under general non-Abelian perturbations, source creation, longitudinal dynamics or the degenerate vacuum evolution.

Complete Energy and Transverse Inertia

The same identities fix the energy fractions:

$$\frac{T_E}{T}=\frac13,\qquad \frac{T_{\nabla u}}T=\frac n6,\qquad \frac{T_{\Omega u}}T=\frac{4-n}{6}.$$

At physical depth three the three retained contributions are equal. At four the gradient fraction is $1/2$, restoring fraction $1/6$ and electric fraction $1/3$. These are exact model identities and require no calibrated field amplitude.

For every transverse axis the integrated stress is

$$\int T_{ii}d^ny=\frac Kn-\frac{K+M}{2}+\frac P4=0$$

in profile units. The longitudinal stress is $-T$, as appropriate for positive tension. A transverse boost of the complete scalar and electric fields gives $E'=\gamma T$ and $P_i'=\gamma v_iT/c^2$ per unit uncontracted tube length. Hence the tube has positive transverse inertia $T/c^2$. The checker retains the transformed magnetic field and the contracted scalar gradient; using the scalar contribution alone would miss this result. This is a flux-guide property, not an identification of the tube with an elementary particle.

Which Premises Matter

A positive ordinary floor $\epsilon=\eta+\kappa h_du^4$, $\eta>0$, removes the exact inactive source constraint. It also permits fixed flux to spread across an arbitrarily large transverse area with $u=0$ and energy tending to zero. Absolute minimization no longer forces a tube. The exact vacuum zero therefore needs physical protection; a small coefficient is not an adequate substitute.

The exponent is also substantive. Replacing $u^4$ by $u^p$, $p>2$, gives the necessary support identity

$$[2p-n(p-2)]K=n(p-2)M.$$

The nongated three/four support and occupied higher-depth exclusion persist for $4\le p<6$. Powers below four can admit a depth-five profile; at six a depth-four profile is critical and excluded. The quartic choice is motivated by the prior current's homogeneity, while its actual neutral-mode realization remains unproved. The Hodge gate itself is unchanged in these controls.

Finally, the transverse operator is assumed to act on the stated ordinary sector geometry. A tube localized in some directions but free to spread through a larger ambient response space need not retain the same dimensional argument. Finite endpoints, reciprocal matter sources, gauge-invariant non-Abelian flux, the vacuum Cauchy problem and reduction of $u$ from the master field remain the next physical questions. No external wall, compactification, independent particle-specific coefficient or exchanged particle enters this calculation.

Verification

The single checker and report preserve 256 passing checks across 55 cases. They include every occupied depth, actual Hodge matrices, exact scaling and width identities, nine profile solves, cutoff convergence, direct flux minimization, complete energy and boost response, independently varied fixed-flux Hessians, 18 spectral grids, exponent controls and the ordinary-floor limit. The report records exact executed code hashes and numerical dependency versions.

Finite Sources Share the Same Linear Energy Bound

The sharp separation comparison now applies the transverse minimum plane by plane. Opposite fixed commuting sources have energy bounded below by tension times their source-free separation; explicit finite caps give an upper bound with the same slope. Completing the full Abelian electric quadratic energy also retains the fixed-flux stability bound. Full non-Abelian screening and dynamical endpoints remain outside this result.

A Neutral Geometric Frame Supplies the Quartic Power

The neutral-frame construction realizes the active dielectric as the squared area of two ordinary deformation directions. Exact area and gradient bounds recover this scalar minimum with coefficient kappa/4, while retaining and testing relative amplitude, angle and orientation modes. Its common-parent reduction and full spacetime tensor completion remain premises; the scalar boost result is not automatically a complete frame boost law. Absolute and cross-sector strengths also await the canonical physical normalization.

Robustness Under a Quadratic Correction

The mixed dielectric extension adds a positive quadratic invariant while preserving the exact-zero vacuum and occupied-sector Hodge gate. A sharp interpolation bound reduces full scalar shape minimization exactly to two moments. At depth three a finite guide requires flux above a continuous widening threshold; at four a metastable branch precedes finite-width global binding. The fold is not a local minimum, and the upper branch has exactly one negative scalar Hessian direction. These results retain the commuting restriction and do not override the compact full non-Abelian alternatives.