A Fourth-Order Static Response Has a Three-and-Four Neutral-Pair Window

A common static pole can distinguish the quark-assigned depths without inserting a separate dimensional coefficient. The result concerns a specific positive source-energy functional. It does not establish that the normal color connection has that pole, that its sources have the required mobility, or that higher sectors have no forces.

The coupled geometric wave action provides a local positive finite-speed completion of a fourth-order displacement stiffness. Its rotation and shear channels have different static poles. This distinction is central to a proposed color application.

1. One Kernel, Two Infrared Requirements

For smooth finite-core sources with adjoint-color components $\rho^A$, consider

$$E[\rho]=\frac{g^2}{2}\sum_A\int\frac{d^dk}{(2\pi)^d} \frac{|\widehat\rho^A(k)|^2}{|k|^4}.$$

This assumes a positive constrained-source energy, with an appropriate source contraction for quantum singlet carriers where used below. It is not automatically the effective energy obtained by minimizing an elastic displacement under an imposed force: that minimization has a different sign. The physical source constraint and reciprocal action must be specified.

An isolated source has nonzero $\widehat\rho(0)$, so its infrared radial integral is $\int_0 k^{d-5}dk$ and diverges for $d\le4$. A generic neutral pair has $\widehat\rho(k)=O(k)$, so its energy is $\int_0 k^{d-3}dk$ and is finite for $d>2$. Both conditions hold exactly when

$$\boxed{2<d\le4.}$$

At three the separated pair cost grows linearly; at four it grows logarithmically. Higher-depth costs approach a finite ceiling for finite source cores. Finite isolated cost permits other interactions; it does not prove their absence or weakness. This conditional distinction concerns an electrostatic confinement criterion, not the entire observed strong interaction.

2. Exact Kernels and Positive Core-Regularized Energies

With $(-\Delta)^2G_d=\delta$, the checked kernels are

$$G_2(r)=\frac{r^2\log r}{8\pi},\quad G_3(r)=-\frac r{8\pi},\quad G_4(r)=-\frac{\log r}{8\pi^2},$$

$$G_5(r)=\frac1{16\pi^2r},\quad G_6(r)=\frac1{16\pi^3r^2},\quad G_{10}(r)=\frac1{8\pi^5r^6}.$$

Their source flux is exactly one. Additive homogeneous terms and subtraction conventions must be fixed when individual kernels are infrared divergent. Negative kernel values at three/four do not make the complete neutral field energy negative.

For a Gaussian Fourier core $e^{-\sigma^2k^2}$, define the unit neutral-pair energy

$$I_d(R,\sigma)=\int\frac{d^dk}{(2\pi)^d} \frac{e^{-\sigma^2k^2}[1-\cos(k\cdot R)]}{k^4}.$$

Using $1/k^4=\int_0^\infty t e^{-tk^2}dt$ gives the positive representation

$$I_d=\frac1{(4\pi)^{d/2}}\int_0^\infty \frac{t[1-e^{-R^2/(4(t+\sigma^2))}]}{(t+\sigma^2)^{d/2}}dt.$$

It agrees with an independent angular Fourier/Bessel integral for every occupied $d>2$. At three,

$$I_3=\frac1{8\pi}\left[ \left(R+\frac{2\sigma^2}{R}\right)\operatorname{erf}\frac R{2\sigma} +\frac{2\sigma}{\sqrt\pi}e^{-R^2/(4\sigma^2)} -\frac{4\sigma}{\sqrt\pi}\right].$$

At four, with $x=R^2/(4\sigma^2)$,

$$I_4=\frac{\gamma+\log x+E_1(x)-1+(1-e^{-x})/x}{16\pi^2}.$$

At $d>4$, the positive pair energy approaches

$$I_d(\infty,\sigma)= \frac{\sigma^{4-d}}{(4\pi)^{d/2}(d/2-1)(d/2-2)}.$$

The depth-two neutral dipole remains logarithmically infrared divergent. With low wave-number cutoff $\epsilon$, its divergent coefficient is $R^2\log(1/\epsilon)/(8\pi)$. A subtracted logarithmic four-dimensional Green kernel also has the linear three-dimensional marginal: integrating its radial derivative over one ordinary noncompact direction gives $-1/(8\pi)$.

3. Actual Normal-Color Singlets Have the Required Dipole Order

Use the independently retained normal triplet, normalized by $\operatorname{tr}(T_AT_B)=\delta_{AB}/2$ and $C_F=4/3$. The conjugate-pair singlet has $T_1\cdot T_2=-4/3$ and the three-triplet determinant has $T_i\cdot T_j=-2/3$. Their positive source energies are

$$E_{q\bar q}=\frac43g^2 I_d(R,\sigma),\qquad E_{qqq}=\frac23g^2\sum_{i<j}I_d(R_{ij},\sigma).$$

Total singlet charge vanishes, but spatial separation leaves a nonzero color dipole norm. For the determinant,

$$\left\langle\left|\sum_i r_iT_i\right|^2\right\rangle =\frac23\sum_{i<j}|r_i-r_j|^2.$$

It vanishes only when all three positions coincide. Thus ordinary separated meson and baryon singlet sources have the generic dipole infrared order used in the window. This calculation does not identify a three-body configuration-space dimension with a particle's sector depth, or establish composite mass from the static energy.

4. Ambient Propagation Can Remove the Window

The inverse operator must act on the assigned sector geometry. Localizing a source normally inside a ten-dimensional field does not automatically replace the ambient propagator by a lower-dimensional one.

Let the tangent core width be $\sigma$ and the finite Gaussian normal width $L$, with $c=10-d$ ordinary normal directions. The ambient response gives

$$I_{10,d}(R)=\frac1{(4\pi)^5}\int_0^\infty \frac{t[1-e^{-R^2/(4(t+\sigma^2))}]} {(t+\sigma^2)^{d/2}(t+L^2)^{c/2}}dt.$$

Every finite $L>0$ has finite isolated cost and a finite pair-energy ceiling because its large-$t$ isolated integrand is $t^{-4}$. When $L=\sigma$, this is exactly the ordinary ten-dimensional isotropic response, independently of how the source's tangent labels were grouped.

Multiplying by the effective normal volume $(4\pi)^{c/2}L^c$ and then taking $L\to\infty$ recovers $I_d$ for $d>2$. The scaled integrand increases monotonically toward the sector-only expression and is dominated by its finite neutral integral. This is a cylindrical continuation per normal volume, not a finite-width localized source or a cancellation of a normalized infinite-volume probability. At depth two the corresponding neutral sector limit diverges.

Consequently a finite normal profile in a freely propagating ambient ten-dimensional fourth-order operator does not yield the three/four long-distance confinement window. Bound-within/free-without continuation, a sector-restricted response, or another actual spatial mechanism must supply the lower-dimensional operator.

5. Derivative Vertices, Higher Multipoles and Lower Stiffness

A stiffness $k^4$ with $v$ spatial derivatives at each source vertex produces an effective source pole $k^{-(4-2v)}$. The generic isolated-cost/neutral-dipole window becomes $2-2v<d\le4-2v$. With no derivative it is three/four; with one derivative it contains only occupied depth two; with two it contains no occupied depth. The bending action checks this distinction directly: its displacement responds as $1/(bk^4)+1/(\kappa k^2)$, rotation as $1/(bk^2)$ and shear as $1/\kappa$.

The depth-two dipole obstruction also does not exclude every neutral composite. A three-site source with weights $(1,-2,1)$ has zero charge and zero dipole, so its Fourier amplitude starts at $k^2$. Its positive Gaussian energy is finite at every occupied depth. At two it is computed independently by Fourier/Bessel and Schwinger integrals. The window specifically concerns generic neutral pairs and the separated color singlets above.

A permitted lower stiffness can cut off confinement. For inverse response $1/[k^2(k^2+m^2)]$ in three dimensions, the point-core neutral energy is

$$I_{3,m}(R)=\frac1{4\pi m}-\frac{1-e^{-mR}}{4\pi m^2R}.$$

It starts as $R/(8\pi)$ at short separation but saturates at $1/(4\pi m)$. Absence of that lower term requires a symmetry or another physical explanation. The flat constant-tilt symmetry of the bending action supplies one candidate protection within that action; the actual source and curved geometry still need to respect it.

6. The Source Constraint and Mobility Remain Physical Questions

Pretko's rank-two scalar-charge model, sections III A and IV, gives an established example in which a double-derivative Gauss constraint produces a fourth-order static source energy. It also conserves dipole moment and prevents isolated charges from moving freely through local processes. Those consequences are part of that model, not optional details to discard when borrowing its static pole. Its U(1) tensor theory is not an SU(3) sector derivation.

IDWT needs a same-field source constraint, a compatible normal-color current, complete reciprocal dynamics and the intended mobile field patterns. Positive elastic stiffness, a positive constrained source energy and the force between dynamical color patterns must be connected explicitly. The conditional dimensional window is useful evidence for investigating that construction, not an established derivation of confinement or the absence of all higher-sector forces.

Verification

The checker and report preserve 361 checks across 129 cases. Exact unit-flux kernels and singlet contractions are paired with independent core-regularized integral representations, all occupied depths, normal-width controls, derivative vertices, neutral quadrupoles and the lower-stiffness limit. Core widths and source amplitudes remain diagnostic parameters, not fitted particle sizes or charges. No particle simulation is used.

Ordinary Gauss Response and Its Signal Cone

The gradient-electric Gauss comparison now realizes this positive static pole with ordinary charge continuity, without imposing dipole conservation. Its homogeneous waves have finite speed, but a conserved finite-energy quadrupole produces a nonzero gauge-invariant response outside their wave cone at every occupied depth. This candidate therefore does not yet provide a consistent physical color completion. A complementary finite positive ordinary-channel bound shows why merely adding ordinary gauge modes cannot generate a double infrared pole. The conditional dimensional window above remains distinct from these dynamical requirements.