Positive current matrices and spatial localization

Summary: A scalar Dirac identity and spatial virial exclude localized flat-space stationary fields in the positive-current cone of the convex hierarchy, retaining all spinor and coefficient components.

A full-field condition on the self-mass action

The current hierarchy supplies an isolated self-source and reciprocal interactions from one specified action. Its spatial states still need to exist. The following exact condition tests that question beyond a radial profile or one coherent coefficient ray.

Use one grade $N$ on ordinary flat physical space $\mathbb R^d$, with the trivial connection on its coefficient bundle. Retain every complex spinor and coefficient component. Set

$$\mathscr D=-i\alpha\cdot\nabla,\qquad \{\mathscr D,\beta\}=0,\qquad \beta^2=I,\qquad \mathscr D^\dagger=\mathscr D. \tag{1}$$

The self-mass note defines the reduced scalar currents $R_k=\rho_{N,k}$, their positive adjoint maps $B_{Nk}$ and the weights $h_{k,d}$. For $r_k=\|R_k\|_{\rm HS}$, take the local interaction density

$$u(\Psi)=\sum_{k=0}^N h_{k,d}\Phi(r_k),\qquad \Phi(0)=0, \tag{2}$$

where $\Phi$ is nonnegative, nondecreasing and convex on the nonnegative axis. The quartic, homogeneous root and smooth saturating laws all have these properties for their stated positive coefficients. This is a condition on the written action, not a premise imported from conventional physics.

Where the source is defined, its complete variation is

$$\mathcal N(\Psi)=\beta A_\Psi\Psi,\qquad A_\Psi=\sum_{k=0}^N h_{k,d}\frac{\Phi'(r_k)}{r_k}B_{Nk}(R_k). \tag{3}$$

The zero-current values use the smooth limit when available. For a root law, they need a specified subgradient; any use of the theorem must check its positivity hypothesis there as well. The operator $A_\Psi$ acts on the coefficient indices and commutes with $\beta$. It is Hermitian, but it is not automatically positive for arbitrary signed scalar currents.

Let $\Psi(t,x)=e^{-i\omega t}\phi(x)$ be a stationary field with finite positive canonical norm, finite displayed integrals and enough regularity and decay for integration by parts and norm-preserving spatial dilation. These are explicit hypotheses. No ordinary charge interpretation is assigned to this norm.

Three exact identities

Define the integrated quantities

$$\begin{aligned} Q&=\int\phi^\dagger\phi,& S&=\int\phi^\dagger\beta\phi,\\ T&=\langle\phi,\mathscr D\phi\rangle,& U&=\int u(\phi),\\ P&=\int\phi^\dagger\beta A_\phi\phi =\int\sum_k h_{k,d}r_k\Phi'(r_k),& L&=\int\phi_-^\dagger A_\phi\phi_-, \end{aligned} \tag{4}$$

where $\phi_-=(I-\beta)\phi/2$. The equality for $P$ follows from the adjoint contraction relation, including every scalar spinor component. Multiplication of the stationary equation by $\phi^\dagger$ gives

$$\omega Q=T+P. \tag{5}$$

Under $\phi_\lambda(x)=\lambda^{d/2}\phi(\lambda x)$, the kinetic expectation becomes $\lambda T$ and the local currents become $\lambda^d R_k(\lambda x)$. Differentiation of the interaction therefore gives $d(P-U)$. The spatial virial is

$$T+d(P-U)=0,\qquad \omega Q=dU-(d-1)P. \tag{6}$$

There is also a scalar identity that does not use dilation. The real part of $\langle\phi,\beta\mathscr D\phi\rangle$ vanishes by (1). Multiplication of the same stationary equation by $\phi^\dagger\beta$ therefore gives

$$\boxed{\omega S=\int\phi^\dagger A_\phi\phi=P+2L.} \tag{7}$$

Equations (5) and (7) use the same full field and the same source. Subtracting (7) from (6) yields the decisive identity

$$\boxed{\omega(Q-S)=d(U-P)-2L.} \tag{8}$$

Convexity with $\Phi(0)=0$ gives $r\Phi'(r)\ge\Phi(r)$ pointwise, so $P\ge U$. If $A_\phi(x)$ is positive semidefinite almost everywhere, then $L\ge0$ as well.

What positivity excludes

Suppose first that $\omega>0$ and $A_\phi\ge0$. The left side of (8) is nonnegative, since $Q-S=2\|\phi_-\|^2$. The right side is nonpositive. Equality forces

$$P=U,\qquad L=0,\qquad \phi_-=0. \tag{9}$$

Projecting the stationary equation onto the negative-$\beta$ space now gives $\mathscr D\phi=0$. The trivial flat Dirac operator has no nonzero $L^2(\mathbb R^d)$ zero field: its squared Fourier symbol is $|p|^2I$, so an $L^2$ zero field would be supported only at $p=0$. Thus $\phi=0$, contrary to the assumed positive norm.

The quartic and smooth laws give an even more direct contradiction when the currents are nonzero. Their pointwise convexity deficits are

$$\begin{aligned} r\Phi'(r)-\Phi(r)&=\frac\kappa2r^2 &&\left(\Phi=\frac\kappa2r^2\right),\\ r\Phi_\sigma'(r)-\Phi_\sigma(r) &=\mu\sigma\left(1-\frac\sigma{\sqrt{r^2+\sigma^2}}\right)>0 &&\left(r>0,\ \sigma>0\right). \end{aligned} \tag{10}$$

For the root, $P=U$ identically. Equation (8) then supplies the missing restriction on the negative-$\beta$ components and gives the same flat zero-field conclusion. Homogeneity of the self-mass alone does not supply a spatially bound field.

A practical sufficient condition for $A_\phi\ge0$ is positivity of the full scalar mode current

$$C_N(x)=R_N(x)\ge0. \tag{11}$$

Every lower $R_k$ is its partial trace and is therefore positive. The maps $B_{Nk}(H)=T_{Nk}^\dagger(H\otimes I)T_{Nk}$ preserve positivity, and the coefficients in (3) are nonnegative. For nonzero current, (11) also gives $S>0$ and $\int\phi^\dagger A_\phi\phi>0$ for the three laws just specified. Equation (7) consequently forces $\omega>0$. The contradiction covers every real frequency in this nonzero-current cone. A smooth field with every current zero has zero source and would be a free flat Dirac eigenfield, which is also zero.

For a coherent coefficient field, $C_N=J P_{z,N}$. Positive scalar current $J$ is enough for (11). For a general field, positivity of its trace $J=\operatorname{Tr}C_N$ is weaker than (11); a positive trace can conceal negative current eigenvalues. Those fields cannot be excluded by substituting their trace into this theorem.

For example, take two orthogonal coherent rays $z,w$ with scalar mode current $C_N=2P_{z,N}-P_{w,N}$. Its scalar trace is positive. For the root law, however, the insertion on $w^N$ is

$$A_\phi w^N =\mu\left[1-\frac{D_N-1}{\sqrt5}\right]w^N. \tag{11a}$$

It is negative whenever $D_N>1+\sqrt5$. The order-zero current supplies the first term; every positive current order supplies the second through Hockey-Stick. This is an explicit signed matrix control, not a stationary spatial field.

A fixed sign decomposition gives the same obstruction

The argument is stronger than positivity in the original coefficient frame. Suppose there is a constant Hermitian coefficient involution $Z^2=I$ such that $[Z,A_\phi(x)]=0$ and $K(x)=ZA_\phi(x)\ge0$ at every point. Define $\Gamma=\beta Z$. Then

$$\Gamma^2=I,\qquad \{\mathscr D,\Gamma\}=0,\qquad \beta A_\phi=\Gamma K. \tag{12}$$

Replacing $\beta,A_\phi,S,L$ in (7)–(9) by $\Gamma,K,S_\Gamma,L_K$ proves the same nonexistence result at positive frequency. A fixed collection of positive and negative coefficient channels is therefore insufficient to evade it. This statement does not assign any particle interpretation to those signs.

A spatially changing sign projector contributes a new term

Where $A_\phi(x)$ has a smooth spectral sign, put $Z(x)=\operatorname{sgn}A_\phi(x)$, $K(x)=|A_\phi(x)|$ and $\Gamma(x)=\beta Z(x)$. For example, a smooth insertion with a gap about zero has this property. Crossings through zero require a separate analysis; the following derivative formula must not be applied across an unexamined discontinuity.

The source still equals $\Gamma K\phi$, but the kinetic anticommutator now contains the spatial change of its coefficient directions:

$$\{\mathscr D,\Gamma\} =-i\sum_j\alpha_j\beta\,\partial_jZ,\qquad G=\operatorname{Re}\langle\phi,\Gamma\mathscr D\phi\rangle =-\frac i2\int\phi^\dagger \left(\sum_j\alpha_j\beta\,\partial_jZ\right)\phi. \tag{13}$$

The integrand gives a real expectation: $\alpha_j\beta$ is anti-Hermitian and commutes with the Hermitian coefficient derivative. The scalar identity and its combination with the virial become

$$\begin{aligned} \omega S_\Gamma&=G+P+2L_K,\\ \boxed{G}&=\boxed{-\omega(Q-S_\Gamma)-d(P-U)-2L_K}, \qquad L_K=\int\phi_{\Gamma,-}^\dagger K\phi_{\Gamma,-}\ge0. \end{aligned} \tag{14}$$

Thus any positive-frequency stationary solution in this smooth-sign class must supply a nonpositive $G$. It must be strictly negative if any of the three nonnegative terms on the right is nonzero. A useful quantitative necessary bound is

$$\omega(Q-S_\Gamma)+d(P-U)+2L_K \le\frac12\int\rho(x) \left\|\sum_j\alpha_j\beta\,\partial_jZ(x)\right\|_{\rm op}dx, \qquad \rho=\phi^\dagger\phi. \tag{15}$$

The spatial sign projector is derived from the same field's mass insertion; it is not an added field or mediator particle. Equation (14) identifies where a changing polarization texture can enter a binding calculation. It does not prove that such a texture is dynamically selected or stable.

An independent trial illustrates the term without claiming a solution. At depth six and grade one, choose two active coefficient directions and opposite spinors $\chi_-=\alpha_1\chi_+$ with $\beta\chi_+=\chi_+$. Let $U(\theta)$ be a real two-by-two rotation through angle $\theta/2$, $\theta(x_1)=t\tanh x_1$, and set

$$\phi(x)=e^{-|x|^2/2} \left[\sqrt2\,\chi_+\otimes U(\theta)e_0 +i\chi_-\otimes U(\theta)e_1\right]. \tag{16}$$

The remaining coefficient directions are included with zero field. Its actual current insertion has positive and negative eigenvalues on the two active directions; its sign rotates with $U$. Direct six-dimensional integration gives

$$G=T=-\sqrt2\,\pi^{5/2} \int_{-\infty}^{\infty}e^{-x_1^2}\theta'(x_1)dx_1. \tag{17}$$

Reversing the twist reverses this contribution. For nonzero $t$, the kinetic expectation violates the root stationary virial $T=0$, so this field is explicitly not a stationary particle. It verifies the geometric term and shows why a texture must solve the complete equations, rather than merely have the desired orientation.

The resulting spatial search condition

This is a full-field conditional nonexistence result. It retains arbitrary spatial dependence, complex phases, coefficient entanglement and spinor polarizations. It does not assume a radial shape or a coherent ray. It does require the flat trivial connection, the stated local convex interaction and admissible virial variation.

At positive frequency, a localized solution of that same action must therefore escape every fixed sign decomposition, or fail another stated hypothesis. An indefinite scalar mode current is insufficient on its own. Spatial change of the sign subspaces, including possible eigenvalue-sign crossings outside the smooth class (13), is a concrete route to examine. The full source, stationary equation and stability still have to be calculated. Removing negative spinor components before doing that search would miss the very directions that could matter.

The radial tail calculation gives a different restriction. For smooth scalar contact on a constant coherent ray, it excludes all regular radial finite-norm solutions, including sign-changing ones, without relying on the above current-positivity condition. Its root extension retains the separate virial hypotheses.

Verification

positive_current_localization_checks.py checks the complete current source, its adjoint contraction, matrix positivity and the convexity deficits with arbitrary complex spinors and coefficient correlations. Independent physical Dirac integration and derivatives of the actual spectral sign check the extra texture term. Indefinite-current controls verify why positive scalar trace alone cannot replace the operator condition. The independent profile dilations in local_self_mass_localization_checks.py check the norm, root energy and scalar virial, including sign-changing profiles. The nonexistence conclusion is the argument (4)–(12), not an inference from numerical shooting.