Exact-Zero Dielectric Requires a Rule for Vacuum Connection Histories

The exact zero in $\epsilon=\kappa h_du^4$ does more than constrain a source: it also removes the vacuum connection equation. Smooth connection histories can then differ despite identical raw initial data. The histories are distinguishable by the response to an identical later neutral excitation. The action therefore needs a geometric restriction or matching rule; ordinary characteristic speeds where $\epsilon>0$ do not settle this problem.

This is a local homogeneous driven-field calculation, with complete energy density and source work. It is not a finite-total-energy localized solution, a particle simulation or a universal obstruction to IDWT. It tests the specific independently varied neutral-mode/connection action already proposed.

Explicit Smooth Histories Share the Same Initial Data

Choose a commuting generator $T$ with $\|T\|^2=1$ and two ordinary spatial coordinates. In temporal gauge set

$$A_2(t,x)=b(t)x_1T,\qquad A_i=0\ (i\ne2).$$

Then

$$F_{12}=b(t)T,\qquad E_2=-\dot b(t)x_1T.$$

Both fields come from one smooth connection and satisfy the Bianchi identity. Let $b(t)=0$ initially, change it smoothly only in $1/4<t<3/2$, and hold it at a constant $B$ thereafter. A standard flat-ended smooth step makes all derivatives match. Different $B$ have the identical initial $A,\dot A,F,E=0$.

Take $u=\dot u=0$ until $t=2$. During this entire vacuum interval,

$$\epsilon=\dot\epsilon=0,\qquad \frac{\partial\epsilon}{\partial u}=4\kappa h_du^3=0.$$

The connection equations and its contribution to the scalar equation consequently vanish for every smooth history $b$. Connection energy is exactly zero. Gauss has zero source: $D_2$ depends only on $x_1$, so its ordinary divergence is zero; all matrices commute. No energy source is needed to select a raw connection history in this constitutive action.

The Same Later Neutral Probe Distinguishes the Histories

Add the same specified color-neutral forcing $f(t)u$ to the Lagrangian. It is zero up to $t=2$ and is a smooth pulse in $2<t<3$. This is a prescribed probe of the action, not a proposed external driver of a particle or a derivation of the other modes supplying it.

After $t=2$, $b=B$ is constant and the electric field vanishes. A spatially homogeneous $u(t)$ obeys

$$\boxed{\ddot u+\Omega^2u+2\kappa h_dB^2u^3=f(t).}$$

The full connection equation still holds: $\epsilon$ depends only on time, the magnetic field is constant, and all color commutators vanish. Thus these are exact smooth local solutions of the full driven scalar/connection equations, rather than an oscillator substituted without its gauge equation.

Their complete energy density and work law are

$$\mathcal E=\frac12\dot u^2+\frac12\Omega^2u^2 +\frac12\kappa h_dB^2u^4, \qquad \dot{\mathcal E}=f(t)\dot u.$$

For active three/four, the solutions depend on $B$ after activation. The physical neutral amplitude and weighted magnetic energy differ. Identifying every raw connection in the empty vacuum cannot, by itself, make their later probe responses equivalent.

The local Cauchy problem with that smooth specified forcing has the same initial raw fields and forcing but distinct smooth solutions in this unrestricted class. Spatially homogeneous excitation and a linear vector potential are used, so total energy over all ordinary space is not finite. The result establishes the local constitutive ambiguity; a separate finite-energy global construction would be needed for that stronger domain claim. It also does not claim that an unspecified geometric parent permits arbitrary independent connection histories.

Complete Energy and Clock Checks

In dimensionless diagnostics take $\kappa=6/5$, $\Omega=9/10$ and

$$f(t)=\begin{cases} \exp\!\left[4-\dfrac1{(t-2)(3-t)}\right],&2<t<3,\\ 0,&\text{otherwise}. \end{cases}$$

Independent adaptive integration, a tighter tolerance pass, finite-difference acceleration, and accumulated source work check the full equations. Representative active-sector responses are

The work differs slightly because the same force acts on different velocities. The work-energy error is below $5\times10^{-14}$ in these cases. No energy is attributed to the preceding vacuum connection change.

A separate unforced control starts at $u=0$, $\dot u=v$. Put $c=\kappa h_dB^2$. Complete energy gives

$$\Omega^2a^2+ca^4=v^2,\qquad \nu^2=\Omega^2+2ca^2,\qquad k^2=\frac{ca^2}{\nu^2}.$$

Its Jacobi elliptic period is $4K(k^2)/\nu$, using the parameter convention of the checker. Independent zero crossings reproduce it. With $v=4/5$, the periods at $B=0,1,3$ are approximately $6.981317008$, $4.917382231$ and $3.111923495$. This is the neutral deformation clock under a specified curvature preparation, not the particle mass-frequency or an identified sector fundamental.

Occupied Sectors and a Positive Floor

The actual Hodge operator is evaluated at depths two, three, four, five, six and ten. Active three/four have the demonstrated vacuum/probe ambiguity. At two and five/six, $h_d=0$ identically, so the independent raw connection remains absent from neutral-mode dynamics even after the probe. Their source constraints remain unchanged. At ten there is no independent normal connection in the selected ten-parent; its neutral scalar is the harmonic control.

For a positive dielectric floor $\eta$, the vacuum connection equation for this trial would require $\eta\ddot b\,x_1=0$. A nontrivial flat-ended $b(t)$ violates that equation without an additional connection source. The floor therefore excludes the freely varying history. It also removes the exact inactive source constraints and absolute commuting flux minimum already derived, so those earlier conclusions cannot be retained under that replacement.

An exact rule requiring flat curvature in an undeformed open vacuum would exclude these histories without adding a floor. That is an additional admissibility condition to derive from the same geometric field, not a consequence of the present action. Its evolution, boundary matching and compatibility with the existing tube and finite-cap constructions must be checked. A requirement of asymptotically vanishing raw curvature is stronger again: the current tube has constant raw electric field in exponentially small dielectric tails. The compact non-Abelian Gauss trials, in contrast, already have flat exterior vacuum, so vacuum flatness alone does not remove that separate confinement limitation.

Reproduction and the Next Physical Requirement

The checker and report retain 472 passing vacuum, full-equation, work and clock checks across 16 cases, plus 18 Hodge dependency checks. There is no particle-specific adjustment or calculation at unoccupied depth.

The exact-zero coefficient is useful for sector selection but does not yet define a complete physical connection theory. A common geometric reduction must determine how the connection exists and evolves where its deformation carrier vanishes, preserving whatever source and flux results survive that reduction. That requirement is sharper than either asserting causality from a frozen positive coefficient or abandoning the geometric zero without testing a completion.

Actual Induced Geometry Gives a More Specific Completion Test

The nested normal-curvature construction calculates a connection from one actual spatial embedding. Its flat undeformed graph has no independently selectable curvature. At four the graph generates full U3, retained by its declared three-sector pullback. Higher normal jets also generate U4 at two, so the embedding class still needs selection. Fixing an ambient metric and deriving the connection changes the available variations: the independent-connection Gauss constraints above do not automatically survive.