Dielectric Growth, Critical Concentration and the Three/Four Guide Window

The mixed quadratic/quartic law supports finite-width commuting guides above derived thresholds. Its robustness has a precise boundary. In the active four-dimensional sector, a positive dielectric term growing faster than the sixth power admits smooth fixed-flux configurations whose complete tension tends to zero. A small coefficient does not protect an unrestricted polynomial law at arbitrarily large deformation.

This observation constrains a candidate constitutive action. It does not disprove a small-deformation expansion, and it does not establish an empirical force law. The neutral two-frame area-squared construction supplies a geometrically motivated quartic invariant; its large-deformation completion still needs to be specified.

One Fixed-Flux Functional

Retain the neutral scalar deformation and independently varied connection action. In an active occupied sector let $n=d-1$ be the transverse dimension and fix $Q>0$:

$$T_Q[u]=\frac{Q^2}{2I[u]}+\frac12(K+\Omega^2M), \quad I[u]=\int\epsilon(u), \quad K=\int|\nabla u|^2,\quad M=\int u^2.$$

Here $\Omega>0$ is the neutral restoring clock. It is not identified with a particle mass-frequency. All dielectric coefficients below are positive. Smooth decaying trial functions are admissible; for a supercritical polynomial the domain explicitly includes the corresponding finite higher moment as well as $H^1$ regularity.

The same declared middle-Hodge factor multiplies the entire dielectric. It is active at occupied depths three/four and zero at two/five/six/ten. The higher-depth comparisons below identify properties of an ungated mathematical functional, not additional physical forces. At ten the selected parent also has no normal gauge connection.

A Smooth Concentration Test

Suppose $\epsilon(u)$ includes $\kappa_p|u|^p$ with $n>2$. Use a fixed smooth Gaussian $\phi$ and write

$$u_L(y)=L^{-a}\phi(y/L),\qquad L\downarrow0.$$

Its moments scale as

$$K_L=L^{n-2-2a}K_\phi, \quad M_L=L^{n-2a}M_\phi, \quad P_{p,L}=L^{n-pa}P_{p,\phi}.$$

Both deformation terms tend to zero while the dielectric integral diverges if

$$\frac np<a<\frac{n-2}{2}.$$

Such an interval exists exactly when

$$\boxed{p>p_*(n):=\frac{2n}{n-2}.}$$

Consequently $T_Q[u_L]\to0$. Since every nonzero admissible finite shape has positive energy, the infimum is zero and is unattained. Lower positive quadratic and quartic terms do not remove this sequence. The result holds for every fixed positive $\kappa_p$, however small.

At physical depth four, $n=3$ and $p_*=6$. For example, with an eighth-power correction choose $a=7/16$. Then $K_L\propto L^{1/8}$, $M_L\propto L^{17/8}$ and $P_{8,L}\propto L^{-1/2}$. The complete energy tends to zero with no sign-indefinite contribution.

In two transverse dimensions every finite polynomial moment is controlled by an $H^1$ interpolation inequality. If both $K$ and $M$ tend to zero, every such moment does too. A finite positive polynomial dielectric therefore cannot realize the zero-energy concentration sequence just exhibited. This is a distinction between the two active transverse geometries under the same functional, not a calibration of quark charge.

The Sixth Power Is a Genuine Critical Boundary

For $n=3$, the sharp Sobolev inequality is

$$P_6\le S K^3,\qquad S=\frac{16}{27\pi^4}.$$

The sharp inequality and rescaled equality family are established in Talenti's primary paper, opening theorem. In the present normalization an optimizer in homogeneous $H^1$ is

$$b(r)=\frac1{\sqrt{1+r^2}},\qquad K_b=\frac{3\pi^2}{4},\quad P_{6,b}=\frac{\pi^2}{4}.$$

Their ratio gives $S$ directly. This bubble is not square integrable on all of $\mathbb R^3$, so it is not itself an admissible finite restoring-energy guide.

Smooth cutoffs followed by concentration nevertheless approach its gradient and sixth moment with vanishing mass and quartic moment. For example set $b_R=b\exp[-r^2/(2R^2)]$, take $R=L^{-1/2}$ and $u_L=A L^{-1/2}b_R(y/L)$. Then $K_L\to A^2K_b$, $P_{6,L}\to A^6P_{6,b}$, while $M_L=O(L^{3/2})$ and $P_{4,L}=O(L)$. Every member is smooth and has finite complete energy.

The limiting energy at sixth-order coefficient $\kappa_6$ is optimized over the remaining gradient moment:

$$\min_{K>0}\left\{\frac{Q^2}{2\kappa_6S K^3}+\frac K2\right\}, \qquad K_*^4=\frac{3Q^2}{\kappa_6S}.$$

It gives the exact critical concentration level

$$\boxed{T_{\rm critical}(Q)=\frac23K_* =\frac{\pi\sqrt Q}{\kappa_6^{1/4}}.}$$

For a law containing positive quadratic, quartic and sextic terms, this is an escape level, not a formula for the full minimum. A finite profile might lie below both this level and the diffuse level. The quartic and critical sharp inequalities generally have different optimizers, so their separate upper bounds do not supply the earlier exact two-moment reduction simultaneously.

An Exact Quadratic/Sextic Infimum at Depth Four

For the specific critical law $\epsilon=\kappa_2u^2+\kappa_6u^6$, with no quartic term, the arbitrary-shape problem can be solved exactly. Define

$$Z=K+\Omega^2M>0.$$

For every nonzero $H^1$ shape, $0<K<Z$. The sharp inequality gives

$$I\le\frac{\kappa_2}{\Omega^2}(Z-K)+\kappa_6S K^3.$$

At fixed $Z$ this right side is strictly convex in $K$. Its interior value is strictly below the larger endpoint value:

$$I<\max\left\{\frac{\kappa_2Z}{\Omega^2},\kappa_6S Z^3\right\}.$$

Thus the energy is strictly larger than the smaller of the two corresponding boundary expressions at that $Z$. Minimizing those expressions independently yields

$$T_Q[u]>\min\left\{\frac{Q\Omega}{\sqrt{\kappa_2}}, \frac{\pi\sqrt Q}{\kappa_6^{1/4}}\right\}.$$

Diffuse fixed-mass dilations approach the first boundary, and the cutoff-bubble sequence approaches the second. Therefore

$$\boxed{\inf T_Q=\min\{T_{\rm diffuse},T_{\rm critical}\}, \qquad\text{no finite shape attains the infimum}.}$$

The boundary branches cross at

$$Q_\times=\frac{\pi^2\kappa_2}{\Omega^2\sqrt{\kappa_6}}.$$

Below that flux diffuse configurations set the infimum; above it concentration does. At equality both types of minimizing sequence exist. This conclusion concerns the precise two-term critical law; it is not transferred to the three-term mixed law.

The Same Sextic Term Behaves Differently at Depth Three

At $n=2$, sixth-power growth remains below the Sobolev critical exponent. Let $v$ be the positive quintic ground optimizer, $-\Delta v+v=v^5$, and write $M_v=\int v^2$. Its identities are $K_v=2M_v$ and $P_{6,v}=3M_v$. Hence the sharp interpolation constant is $C=3/(4M_v^2)$ in $P_6\le C K^2M$.

For $\epsilon=\kappa_2u^2+\kappa_6u^6$, amplitude and dilation independently realize every positive pair $K,M$ at equality. Minimizing the sharp energy first over $M$ therefore leaves the exact one-variable problem

$$\inf_{K>0}\left\{\frac{Q\Omega}{\sqrt{\kappa_2+\kappa_6C K^2}}+\frac K2\right\}.$$

Define $t=\kappa_2/(\kappa_2+\kappa_6C K^2)$ and $Q_*=\kappa_2/(2\Omega\sqrt{\kappa_6C})$. The stationary curve and its tension ratio are

$$\frac Q{Q_*}=\frac1{t\sqrt{1-t}},\qquad \frac T{T_{\rm diffuse}}=\sqrt t(2-t).$$

The fold is $t_f=2/3$, and binding occurs at $t_b=(3-\sqrt5)/2$. The exact factorization is $t(2-t)^2-1=(t-1)(t^2-3t+1)$. Thus this higher invariant changes the depth-three onset from continuous widening to finite-width binding, whereas its pure quadratic/sextic depth-four counterpart has only unattained boundary minima. The geometric dimension and constitutive growth jointly determine the result.

A Positive Local Sextic Correction Need Not Imply Critical Growth

A low-deformation Taylor coefficient does not determine the complete action. Consider the bounded nonlinear completion

$$\epsilon(u)=\kappa_2u^2+ \kappa_4u^4\left(1+\frac{\delta u^2}{u_*^2+u^2}\right), \qquad\delta,u_*>0.$$

It has a positive local sixth-order correction $\kappa_4\delta u^6/u_*^2$, yet its global growth is quartic. Pointwise,

$$\kappa_2u^2+\kappa_4u^4\le\epsilon(u) \le\kappa_2u^2+\kappa_4(1+\delta)u^4.$$

The associated fixed-flux energies obey the reverse inequalities, and so do their infima. At the active depths three/four, the two mixed quartic bounds have strictly positive infima for $Q>0$, excluding the zero-energy concentration above. A trial guide from the lower quartic law also gives an energy below the diffuse level whenever that law binds. These bounds alone are not a compactness or existence theorem for the nonlinear completion.

This is a coherent repair to test if higher local powers arise from a reduction of the same field. Its new deformation scale and bounded constitutive function remain explicit assumptions. A polynomial extrapolated outside its valid amplitude range and a global polynomial action are different proposals.

Scope and Verification

The growth analysis sharpens which geometric flux-guide claims are robust. The quartic neutral-frame invariant and its positive quadratic extension occupy a controlled range; an arbitrary positive higher polynomial is not harmless. The same sharp-inequality and scaling mathematics supplies the thresholds, critical escape and completed-growth bounds.

These results retain the independently varied, commuting-flux restriction. They do not remove the vacuum connection ambiguity or the compact full non-Abelian Gauss alternatives. Neither an energetic escape sequence nor an attained scalar profile identifies an observed quark, a physical charge unit or the master-field action.

The checker and report preserve 165 checks across 48 cases: all occupied depths, exact critical integrals, explicit eighth-power concentration at three coefficient sizes, cutoff-bubble convergence, critical capacity convexity, exact crossover, sextic depth-three roots and bounded nonlinear growth. No particle evolution, fitted amplitude or measured force calibration is used.

An independent review adds 118 checks, including a different rational-cutoff recovery sequence and the exact strict chord deficit $\kappa_6S K(Z-K)(Z+K)$. Its full proofs and frozen inputs are preserved separately.