Geometric Flux Guides Survive a Quadratic Dielectric Correction

The finite-width commuting flux result does not require the dielectric to be exactly quartic. Adding a positive quadratic deformation term preserves localized guides above a calculable flux threshold. One action gives different onset behavior at the two active occupied depths: the depth-three guide widens continuously at threshold, whereas depth four has a finite-width binding transition and a metastable branch.

The result concerns the neutral deformation and independently varied connection action. It does not restore full non-Abelian confinement or resolve its zero-dielectric vacuum ambiguity. Its geometric value is a robustness test of the conditional three/four guide mechanism, including all scalar shapes rather than only a chosen radial family.

The Same Sector Gate, With an Additional Allowed Invariant

Let $u$ be a real color-neutral deformation mode and retain positive canonical kinetic, gradient and restoring energy. Replace the previous dielectric with

$$\epsilon_d(u)=h_d(\kappa_2u^2+\kappa_4u^4), \qquad \kappa_2,\kappa_4,\Omega>0.$$

The common $h_d$ is the same real middle-Hodge coefficient on the neighboring even carrier. It is one at occupied three/four and zero at two/five/six/ten. This retains the declared odd-sector carrier premise. The quadratic term still vanishes at $u=0$; it is not an ordinary positive vacuum floor.

For an independently varied connection, the inactive source constraints therefore remain at two/five/six. At ten the selected parent has no independent normal connection. The complete retained action is

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

The coefficients are common constitutive parameters, not fits for separate particles. A common-parent canonical reduction is still required to compare their physical units across depths. $\Omega$ denotes the neutral restoring clock; assigning it the sector fundamental remains an additional physical hypothesis.

Complete Fixed-Flux Variation

In an active sector let $n=d-1$ be the transverse spatial dimension of a long commuting flux guide. Write

$$K=\int|\nabla u|^2,\qquad M=\int u^2,\qquad P=\int u^4, \qquad I=\kappa_2M+\kappa_4P.$$

At fixed classical displacement flux $Q>0$, electric minimization again gives constant transverse electric field $E=Q/I$. The complete energy per length is

$$\boxed{T[u]=\frac{Q^2}{2I}+\frac12(K+\Omega^2M).}$$

Variation gives

$$-\Delta u+(\Omega^2-\kappa_2E^2)u=2\kappa_4E^2u^3.$$

For a nonzero regular subcritical ground profile define

$$t=\frac{\kappa_2E^2}{\Omega^2}\in(0,1),\qquad \mu=\Omega\sqrt{1-t},\qquad A^2=\frac{\kappa_2(1-t)}{2\kappa_4t}.$$

Then $u(y)=A v(\mu y)$, where the same positive cubic ground profile satisfies $-\Delta v+v=v^3$. Denote its moments by $M_n,K_n,P_n$. The identities $K_n=nM_n/(4-n)$ and $P_n=4M_n/(4-n)$ determine all needed normalization factors. In particular $M_2\simeq11.70089652$ and $M_3\simeq18.89725130$ are numerical profile integrals, not exact rational constants.

The regular profile still requires $n<4$ when $\mu>0$. Actual higher occupied sectors also have zero $h_d$. No calculation at unoccupied depths is needed for this comparison.

Why the Shape Minimization Is Exact

For $n=2,3$, the sharp interpolation inequality is

$$P\le C_nK^{n/2}M^{(4-n)/2},\qquad C_n=\frac{4(4-n)^{n/2-1}}{n^{n/2}M_n}.$$

Scaled copies of $v$ attain equality. This is the standard sharp Gagliardo–Nirenberg optimizer result, recalled with its equality statement in section 2, equation (2.3), of Soave's primary paper. The nonlinear Schrödinger evolution in that paper is not the present flux dynamics.

Here the energy decreases as $P$ increases at fixed $K,M$. Thus every admissible shape obeys

$$T[u]\ge g(K,M) =\frac{Q^2}{2[\kappa_2M+\kappa_4C_nK^{n/2}M^{(4-n)/2}]} +\frac12(K+\Omega^2M).$$

This bound is attainable for every positive pair $K,M$: an amplitude and a spatial dilation of $v$ independently set those two moments. Therefore minimizing $g$ over the positive quadrant gives the exact arbitrary-shape energy infimum. Radial symmetry is an optimizer property, not a restriction imposed on the search.

If a bounded-energy minimizing sequence has $K\to0$, its quartic moment also tends to zero. Optimizing its remaining mass moment yields the diffuse threshold

$$T_{\rm diffuse}=\frac{Q\Omega}{\sqrt{\kappa_2}}.$$

The other escapes cannot lower a bounded energy: $K$ or $M$ tending to infinity raises the positive deformation term, and $M\to0$ with bounded $K$ makes the electric term diverge. Any energy strictly below $T_{\rm diffuse}$ consequently attains an interior moment minimum and hence a finite profile. At the depth-four binding equality an interior minimum also exists, although diffuse minimizing sequences coexist.

A strict local minimum of $g$ is also a full scalar energetic local minimum. The moments vary continuously in $H^1$ at these dimensions, and the interpolation deficit only raises the energy. This is an energetic statement at fixed commuting flux, with translations retained; it is not a nonlinear orbital stability theorem for the singular connection action.

Universal Branch Curves

Define the dimensionful flux scale

$$Q_0=\frac{\kappa_2^{3/2}\Omega^{1-n}M_n} {2\kappa_4(4-n)}.$$

Direct substitution of the profile moments gives

$$\boxed{\frac Q{Q_0} =(1-t)^{1-n/2}t^{-3/2}[2+(2-n)t],}$$

$$\boxed{\frac T{T_{\rm diffuse}} =\sqrt t\,\frac{3+(1-n)t}{2+(2-n)t}.}$$

Its width is $\mu^{-1}=1/[\Omega\sqrt{1-t}]$. Thus the neutral restoring clock affects both the width and the flux threshold. These equations do not identify $\Omega$ with a known particle frequency or determine a measured strong coupling.

Depth Three: Continuous Widening at Threshold

For $n=2$, define

$$Q_c=\frac{\kappa_2^{3/2}M_2}{2\kappa_4\Omega}.$$

The exact branch relation is $t=(Q_c/Q)^{2/3}$. A finite guide exists and is the global scalar minimum precisely for $Q>Q_c$. Its tension ratio is $\sqrt t(3-t)/2<1$, and its moment Hessian is positive. For $Q\le Q_c$, the infimum is $T_{\rm diffuse}$ without a finite profile attaining it. As $Q\downarrow Q_c$, the width diverges and the amplitude tends to zero.

For unit coefficients, $Q_c\simeq5.850448262$. This is a dimensionless diagnostic normalization, not a predicted physical color charge.

Depth Four: A Finite-Width Binding Transition

For $n=3$, write $q=Q/Q_0$. Then

$$q(t)=\frac{2-t}{\sqrt{1-t}\,t^{3/2}}.$$

The stationary branches first appear at

$$t_f=\frac{9-\sqrt{33}}4\simeq0.8138593384, \qquad q_f\simeq3.744480221.$$

The lower-$t$ branch is a local scalar energy minimum; the upper branch has exactly one negative direction in the full scalar Hessian. Global binding begins at

$$t_b=\frac{9-\sqrt{17}}8\simeq0.6096117968, \qquad q_b\simeq4.675280667.$$

This follows directly from the factorization

$$t(3-2t)^2-(2-t)^2=(t-1)(4t^2-9t+4).$$

At $q=q_b$, a finite profile is already a global minimum with $T=T_{\rm diffuse}$. For $q>q_b$ it binds strictly below that threshold. Between $q_f$ and $q_b$, the lower-$t$ guide is metastable: it is a local scalar minimum, but diffuse configurations have lower limiting energy. The upper branch has an energetic instability.

At the fold the Hessian is nonnegative, but the profile curve $u_t$ at fixed evaluated flux $Q_f$ has

$$\left.\frac{d^3T_{Q_f}[u_t]}{dt^3}\right|_{t_f} =-E'(t_f)Q''(t_f)<0.$$

The first and second derivatives vanish. The cubic changes sign across the fold, proving that the fold is not a local energy minimum.

For completeness, the full scalar Hessian index follows from the sharp inequality rather than from a restricted shape ansatz. Write $\delta=P_{\max}(K,M)-P\ge0$. At the ground profile, $\delta=\delta'=0$ and $\delta''\ge0$. The Hessian is the pullback of the two-moment Hessian plus this nonnegative deficit contribution. It has no more negative directions than the moment Hessian. Amplitude and dilation preserve equality and vary both moments independently, so every negative moment direction is realized. The lower branch has index zero and the upper branch index one. This statement does not classify the nullspace or prove dynamical stability.

At the binding point the width is finite:

$$\mu_b^{-1}\simeq\frac{1.6004851804}{\Omega}.$$

With unit coefficients, the binding flux is approximately $44.17497684$. Its units differ from the depth-three flux normalization, so those raw numbers are not a cross-sector charge ratio.

These are consequences of one action. No observed distinction between quark families is identified from the onset behavior without a physical source-flux assignment and normalization.

Complete Stress, Inertia and a Neutral Frame

For the stationary profile, variation in transverse width gives

$$\frac Kn-\frac12(K+\Omega^2M)+\frac{QE}{2}=0.$$

Therefore the complete transverse boost identity remains $2T_E+K/n=T$, with $T_E=QE/2$. The tube retains positive transverse inertia per length $T/c^2$ in this scalar/connection action. Envelope differentiation also gives $dT/dQ=E$. Both statements retain all electric, gradient and restoring energy.

The same extension fits the ordinary neutral-frame construction. For two real spatial deformation directions $a,b$, put $u^2=|a|^2+|b|^2$ and

$$\epsilon=\kappa_2u^2+\kappa_4|a\wedge b|^2.$$

The exact area bound $|a\wedge b|^2\le u^4/4$ and the frame gradient bound imply the scalar mixed-dielectric energy bound with effective quartic coefficient $\kappa_4/4$. An equal orthogonal constant frame attains it. The quadratic correction therefore preserves the tested frame-polarization completion; it does not require deleting orientation or imbalance modes.

The finite-guide shape calculation derives two degenerate positive-gap imbalance channels from this actual neutral frame. Their exact endpoint response adds short-range outward force for mismatched shapes to the existing tension. The same energy produces alignment torque, and a norm bound puts small-distortion radial balance inside the guide width. These are conditional reciprocal response results, not a stable composite or full non-Abelian confinement.

The relaxed frame-twist continuation supplies exact uniformly rotating bulk guides with the shifted restoring coefficient. Complete profile relaxation keeps their static twist force inward, and its separation slope equals the reduced longitudinal phase stiffness. A full-frame tilt channel and a rotating Hamiltonian square bound connect internal motion to energetic protection and positive bulk momentum response. The finite endpoint solution, complete particle inertia and master-field reduction remain open.

What This Adds to the Color Construction

The common Hodge zeros remain, and finite-width commuting guides survive an additional lower-order invariant. The threshold and stability distinctions strengthen the model calculation by exposing its domain rather than assuming every nonzero flux localizes. The same cubic profile mathematics supplies the arbitrary-shape bound, thresholds, stress and inertia.

The independently varied connection still has the exact-zero vacuum problem. The compact non-Abelian Gauss alternatives also remain admissible with finite energy: adding the positive quadratic term does not make those compact fields singular. Thus this is a stronger commuting flux-guide result, not a proof of full classical or quantum color confinement. Restricting to an induced geometric connection would require a new variation and cannot automatically retain these source constraints.

The checker and report retain 374 checks across 50 cases, plus 26 profile and Hodge dependency checks. They include all occupied-sector zeros, exact threshold radicals, direct complete-energy and Hessian differences, independent two-moment minimization, transverse stress and boost identities, three common coefficient choices, and the frame-area bound. No particle simulation or measured force fit is used.

An independent review adds 30 symbolic and high-precision checks of the branches, threshold endpoints, full scalar Hessian index argument and nonzero fold cubic. Its frozen inputs and source hashes are preserved with the private review evidence.

Large-Deformation Growth Is Also Substantive

The growth analysis proves that unrestricted positive powers above six permit zero-tension concentration at active depth four. The critical quadratic/sextic law has an exact positive but unattained boundary infimum. A bounded nonlinear completion can retain a local positive sextic correction while staying between two quartic laws. Thus the present quadratic correction is a controlled robustness result, not permission to extrapolate arbitrary higher polynomials globally.