A Hodge-Gated Gauss Constraint Preserves Complete Spatial Singlet Branches

In the four-plus-six spatial reference split, the base has four ordinary spatial dimensions and its perpendicular complement has six. Time is a separate coordinate in the infinite-dimensional physical spacetime. This four-plus-six split concerns ten spatial directions in the matter domain.

A vanishing quartic contact does not remove a color source already present in a covariant kinetic term. A different common action can address that gap: put the geometric Hodge factor in the connection's actual electric-energy coefficient. Its exact zero then imposes a Gauss constraint on the source. There are nontrivial color-singlet branches satisfying that constraint without losing any active spatial direction.

This is a conditional action construction in the selected ten-dimensional parent. It is not a derivation of that coefficient from the master field, a complete strong-force law or an exclusion of all larger containing carriers.

The Gauge Algebra Must Preserve the Active Spatial Symbol

Use the four-dimensional spatial base/six-normal construction,

$$\Sigma_{10}=\Sigma_4\otimes\Lambda^*\mathbb C^3.$$

The six normal coordinates are ordinary spatial coordinates. The exterior factor describes spinor components, not additional spatial dimensions. The normal rotations have Hermitian generators $T_A$ with fundamental normalization $\operatorname{tr}(T_AT_B)=\delta_{AB}/2$.

At physical depth $d$, retain the independent normal algebra

$$\mathfrak g_d=\{T\in\mathfrak{su}(3):[T,\gamma_i]=0\text{ for all }i\le d\}.$$

It is the full eight-dimensional algebra at two, three and four. At five and six it is the three-dimensional $\mathfrak{su}(2)$ acting on the last two unused normal complex directions. At ten it is zero. The full normal rotation action at higher depth can still have a geometric spin/orbital role; it is not an independent color action commuting with that entire tangent symbol.

This step avoids gauging a fixed internal color matrix that would erase or rotate an active spatial derivative without transforming the geometry. It does not exclude other equivalent-mode multiplicities or other containing spatial parents.

Put the Common Factor in the Actual Connection Action

For the neighboring even source carrier $2m$, define

$$H=\frac12(*+*^\dagger).$$

On ordinary Euclidean middle forms, $*^2=(-1)^m$ and $*^\dagger=(-1)^m*$. Consequently $H=*$ for even $m$ and $H=0$ for odd $m$. Odd physical depths use the existing even-shadow premise; an intrinsic odd-dimensional middle-degree star is not being substituted.

Two prospective dielectrics share the exact geometric zeros:

$$\epsilon(\Psi)=\alpha\sum_A\|HJ_A(\Psi)\|^2, \qquad J_{A,I}=\frac12\Psi^\dagger\{K_I,T_A\}\Psi,$$

or

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

The first uses the actual Hermitian current. Its scalar norm is gauge invariant at three/four and vanishes identically in the inactive cases. The second uses the operator norm itself and a real color-singlet deformation mode $u$ of the same field. It is a separate neutral-mode constitutive proposal, motivated by the quartic homogeneity of the current norm. Reducing a charged spinor amplitude to that neutral scalar has not been derived.

For either declared dielectric, take a collective normal connection $A_\mu\in\mathfrak g_d$ with

$$L_A=\frac12\epsilon\sum_A E_A^2 -\frac{c^2}{4}\epsilon\sum_{A,i,j}F_{A,ij}^2.$$

Matter couples through the same gauge-covariant kinetic term, and its variation defines $\rho,j_i$. These are retained modes of the one field, not additional exchanged particles. The metric construction supplies a possible geometric connection; the reduction determining its dynamical coefficient and source vertex remains a physical question.

Where $\epsilon>0$, the electric displacement is $D=\epsilon E$, and the gauge Hamiltonian is positive:

$$\mathcal H_A=\frac1{2\epsilon}\sum_A D_A^2 +\frac{c^2\epsilon}{4}\sum_{A,i,j}F_{A,ij}^2.$$

Ordinary gauge invariance gives covariant current continuity. Variation in $A_0$ gives

$$\mathcal D_iD_i=\rho.$$

When $H=0$ identically, the dielectric is zero for every field preparation. A finite extended Hamiltonian then requires $D=0$, so Gauss requires $\rho=0$. Since the entire connection action vanishes identically there, variation in its spatial components also requires $j_i=0$. Variations of the prefactor with respect to the underlying amplitude vanish identically in those sectors as well.

This is a source constraint in a singular zero-dielectric limit. It is stronger than disappearance of one local contact term. It must not be described as an ordinary weak-coupling limit or inferred solely from a small numerical coefficient.

Explicit Singlets Retain Every Active Spatial Direction

Let

$$C_d=\sum_{a\in\mathfrak g_d}T_a^2,\qquad P_d=\operatorname{Proj}\ker C_d.$$

Each $C_d$ is positive semidefinite, so its kernel is the joint kernel of the actual gauge generators. These projectors can be calculated on the full rank-32 spatial carrier.

At two, the normal $SU(3)$ singlets are exterior degrees zero and three. They retain the full base spinor factor, giving rank eight. At five/six, the residual $SU(2)$ singlets have occupation zero or two in the last two unused normal modes; the first active normal mode remains unrestricted. Their rank is sixteen. At ten there is no remaining independent normal algebra in this carrier, so $P_{10}=I_{32}$.

The complete matrices satisfy

$$[P_d,\gamma_i]=0\quad(i\le d),$$

$$(P_d\gamma_iP_d)(P_d\gamma_jP_d) +(P_d\gamma_jP_d)(P_d\gamma_iP_d) =2\delta_{ij}P_d.$$

No active kinetic direction is erased. In particular this residual-singlet projection at five/six differs from projecting the whole higher-depth field onto the fixed normal triplet, which killed its added normal Clifford directions.

For any field in a singlet branch, $T_aP_d=0$. Its density and spatial kinetic currents therefore vanish:

$$\Psi^\dagger T_a\Psi=0,\qquad \Psi^\dagger\gamma_iT_a\Psi=0.$$

Since the projector commutes with every active constant Clifford matrix, this branch is preserved by the corresponding free spatial kinetic operator. Observer spin is preserved as well. These are sufficient nonzero solutions of the algebraic current constraints. A general classical state with zero moment map need not itself lie in a linear fibre-singlet representation; quantum Gauss constraints and occupation require separate analysis.

A Positive Floor Tests the Exact-Constraint Premise

Replace $\epsilon$ by $\eta+\epsilon$, with $\eta>0$. In an otherwise inactive sector, a nonzero smooth neutral source now has finite ordinary Gauss energy proportional to $1/\eta$. To test this without an infrared-divergent preparation, use

$$\widehat\rho(k)=k_1^2e^{-\sigma^2k^2/2}.$$

It has zero total charge and dipole. Direct angular and radial integration gives

$$E_\eta=\frac1{2\eta}\int\frac{d^dk}{(2\pi)^d} \frac{k_1^4e^{-\sigma^2k^2}}{k^2} =\frac3{4\eta(d+2)(4\pi)^{d/2}\sigma^{d+2}}.$$

This is finite and strictly positive at every occupied depth for positive $\eta$, and diverges as $\eta\to0$. The exact source constraint requires the exact geometric zero. An allowed ordinary connection term would remove it, so its absence needs a physical justification.

Physical Scope

The common coefficient and the actual normal Clifford commutant now supply a coherent mechanism for excluding an independent color source in the other maintained sectors while preserving their spatial kinematics. Higher-dimensional form counts alone do not generate higher color forces in this action. The zero source follows from a declared dynamical coefficient and a constraint, rather than from the component count alone.

Several parts remain conditional. The symmetric Hodge choice is a premise; complementary contacts and an ordinary positive floor are not forbidden by the existing symmetry argument. Zero dielectric degenerates the connection's vacuum kinetic term, so a complete Cauchy formulation remains to establish. Positive gauge energy does not by itself prove positivity of an unspecified complete matter action. Active three/four has an ordinary second-order static Gauss response when the positive dielectric is frozen; its nonlinear dynamics, attraction, flux support, confinement and empirical strength must be calculated. The earlier fourth-order confinement window is a different response law and has not been transferred to this action.

Verification

The checker and report preserve 1,121 passing checks across 12 cases. They retain every occupied depth, the full spatial Clifford matrices, actual normal commutants, exact exterior singlet branches, all compressed kinetic anticommutators, zero singlet sources, observer-spin and gauge covariance, and independent finite-floor source integrals. No particle simulation or particle-specific fitted coefficient is used.

A Positive Neutral-Mode Flux Guide

The quartic neutral-mode flux construction now supplies finite-width commuting-color tubes in the active sectors. Its exact transverse scaling identity requires physical depth below five; the same Hodge factor removes depth two. Complete positive tension, transverse stress, boost response and restricted scalar stability are verified. The neutral dielectric is a separate constitutive premise rather than a derived charged-spinor reduction. Vacuum well-posedness, finite endpoints and full non-Abelian stability remain open.

Zero Source Is Weaker Than a Fibre Singlet

The complete classical-source comparison constructs nonsinglet zero-current positive modes using these same matrices. A complex-linear conjugate-representation line preserves arbitrary free spatial dynamics at two through four. Residual SU2 at five/six permits canceled positive modes, while generic coherent interference restores a source; imposed antiunitary reality instead degenerates the first-order symplectic form. The singlet branches above remain sufficient, but are not exhaustive classical solutions of zero mean source. Electric characters, localized dynamics and quantum Gauss occupation require their own tests.

The Exact Vacuum Needs a Connection Rule

The vacuum-history test constructs distinct smooth local driven solutions with identical initial raw fields and an identical later neutral probe. Connection histories cost zero energy while the neutral carrier vanishes, but their settled curvature changes its subsequent activation. This identifies a local constitutive Cauchy ambiguity in the independently varied action. It is not a finite-total-energy particle counterexample; a derived geometric connection or vacuum matching condition is needed. Adding a positive floor excludes the history but loses the exact source constraint.