A local homogeneous current that closes the spatial mass equation

A local interaction can preserve the selected spatial frequency without requiring the field density to be spatially constant. The construction below uses the direction of a same-field current, while retaining its complete variation. It supplies exact standing waves of a positive spatial action at every maintained depth.

This is an inverse construction. The proposed ordinary spatial geometry and its retained component chain come from the whole-depth carrier. They have not been derived from the original scalar-current master interaction. The construction does not yet select a supporting radius, the physical coupling scales, the required shared six/ten frequency unit, or a complete particle response to motion.

One local current and one generating functional

On the stated spatial geometry, let $H_0$ be the first-order carrier with its adjacent component coupling removed. If $J$ is the matrix with ones immediately below its diagonal, define the Hermitian channel

$$K=\begin{pmatrix}0&J^\dagger\\J&0\end{pmatrix}. \tag{1}$$

It acts on the retained components of the same field and is the identity on the spatial spinor or form indices. Those components do not add physical spatial dimensions or describe additional particles.

For the local density and real current, write

$$\rho=\Psi^\dagger\Psi,\qquad j=\Psi^\dagger K\Psi,\qquad k=j/\rho. \tag{2}$$

The quotient is used where $\rho>0$. Choose one positive strength $g$, with frequency units, and a dimensionless $\eta>0$, common to the modes being compared. Specify

$$\boxed{ E_1[\Psi]=\langle\Psi,H_0\Psi\rangle -g\int\sqrt{j^2+\eta^2\rho^2}\,dV. } \tag{3}$$

Here $E_1$ is the real generating functional for the first-order map. The complete positive wave energy introduced below is a different quantity. A negative term in (3) is not negative particle inertia.

Set $s=\sqrt{k^2+\eta^2}$. The complete complex gradient, in the convention $\delta E_1=2\operatorname{Re}\langle F,\delta\Psi\rangle$, is

$$\boxed{ F[\Psi]=H_0\Psi+G(k)K\Psi+C(k)\Psi, \quad G=-\frac{gk}{s},\quad C=-\frac{g\eta^2}{s}. } \tag{4}$$

The scalar term $C$ is required by the variation of the density in the denominator of $k$. Merely inserting $G(k)$ into the old carrier omits it and gives a different equation.

The interaction in (3) is homogeneous of degree two in the field. Its gradient is degree one and obeys $F[e^{i\theta}\Psi]=e^{i\theta}F[\Psi]$. The local source extends continuously by zero at a field zero. Its generating density is continuously differentiable there, but is not generally twice differentiable at the origin. The source itself need not be differentiable there. Statements about higher variations at nodes need the corresponding weak-variation or regularity hypothesis; $\eta>0$ does not erase this qualification.

The actual spatial source closes

Let $u_n(x)$ be any eigensection of the spatial parent in the selected eigenspace. Keep the full eigensection, including its spatial polarization. For a normalized component vector $\phi$, take

$$\Psi(x)=\phi\otimes u_n(x),\qquad \phi^\dagger\phi=1. \tag{5}$$

Then $k=\phi^\dagger K\phi$ is constant wherever the field is nonzero. Its density need not be constant. The coefficients in (4) are therefore spatially constant, and the full nonlinear source stays in the same spatial eigenspace. The statement holds for an arbitrary linear combination within that eigenspace, not just a single highest-weight example.

Let $\epsilon_n(G)>0$ be the exact smallest singular frequency of the complete component chain. Hellmann--Feynman differentiation gives $k=\epsilon_n'(G)$. The remaining finite fixed point is

$$\boxed{ G=-\frac{g\epsilon_n'(G)} {\sqrt{\epsilon_n'(G)^2+\eta^2}},\qquad \lambda_n=\epsilon_n(G) -\frac{g\eta^2}{\sqrt{\epsilon_n'(G)^2+\eta^2}}. } \tag{6}$$

The reviewed near-saturation branches have $\lambda_n>0$. The selected particle frequency is $\omega_n=\lambda_n$ on those branches, or $|\lambda_n|$ for the positive action generally. No eigenfunction amplitude is being used in place of this frequency.

The finite carrier still has the leading simplex spectrum

$$\epsilon_n(G)=\frac{c}{R} \frac{d!}{(GR/c)^{d-1}}\binom{n+d-1}{d} \left[1+O_{d,n}\big((GR/c)^{-2}\big)\right]. \tag{7}$$

Equation (6) supplies a further, calculable correction under a common current law. It is not an exact finite-gap derivation of the bare mass formula, and the allowed standing branches do not by themselves select the occupied particle indices.

The simplex scaling constrains the current law

The nearly saturated current is not interchangeable with an arbitrary power of the same normalized current. This follows directly from the leading carrier. Write

$$r=d-1,\qquad \mathcal P_{n,d}=\left(\frac cR\right)^d d!S(n,d),\qquad \epsilon=\frac{\mathcal P_{n,d}}{G^r},\qquad k=-\frac{r\mathcal P_{n,d}}{G^d}. \tag{7a}$$

Here $\mathcal P_{n,d}$ is a spectral coefficient, not a field amplitude. Consider the alternative homogeneous generating density $\rho f_p(k)$, with $f_p(k)=-a|k|^p$, $p\ge1$, and one fixed frequency coefficient $a>0$ for the depth. On its negative current branch, the complete fixed point and scalar offset obey

$$G=a p|k|^{p-1},\qquad C=a(p-1)|k|^p,\qquad D=1+d(p-1),\qquad G^D=a p(r\mathcal P_{n,d})^{p-1}. \tag{7b}$$

Including $C$ in the frequency gives

$$\boxed{ \begin{aligned} \omega&=\frac Dp\,[a p r^{p-1}]^{-r/D} \mathcal P_{n,d}^{\,p/D},\\ \omega_{n,d}&\propto S(n,d)^{\,p/[1+d(p-1)]}. \end{aligned} } \tag{7c}$$

For every maintained depth $d>1$, the original linear simplex dependence requires $p=1$ in this family. A quadratic current, for example, changes the count exponent to $2/(d+1)$. A common depth coefficient cannot undo that changed dependence on the index. These are leading spectral statements; the finite chain has its separately calculable corrections.

There is also a functional version. Under the additional requirement of exact proportionality $\lambda=\Lambda_d\mathcal P$ over a continuous leading-order family, the complete stationary derivative is

$$\frac{d\lambda}{d\mathcal P}=G^{-r}. \tag{7d}$$

All induced derivatives of $G$ and $k$ cancel. Proportionality therefore requires a constant $G$ on that family, and hence a linear $f(k)=Gk$ over the sampled negative-current interval, with no additive frequency offset. The continuous-family requirement is stronger than matching a finite set of assigned indices, so this is not a uniqueness theorem for every possible interaction.

The even square-root law (3) approaches that linear behavior when $|k|\gg\eta$, while retaining a differentiable generating density through $k=0$ at nonzero field. Its finite regularizer accounts for the shifts in (6). The competing-branch calculation then tests the dynamics of the small-current alternative under the same specified law.

The complete positive action has exact standing waves

Define the action on the full spatial field by

$$\boxed{ L_+=\|\dot\Psi\|^2-\|F[\Psi]\|^2, \qquad \ddot\Psi+(DF[\Psi])^*F[\Psi]=0. } \tag{8}$$

The star here denotes the adjoint for the real Hilbert inner product. Since $F$ is a real gradient, $DF$ is self-adjoint wherever the variation is defined. Homogeneity gives $DF[\phi]\phi=F[\phi]$. Consequently,

$$F[\phi]=\lambda\phi \quad\Longrightarrow\quad \Psi(t)=e^{-i|\lambda|t}\phi \text{ solves (8).} \tag{9}$$

This varies the entire local interaction. It neither freezes its coefficients before squaring nor discards a spatial complement.

For a positive-frequency standing wave with norm $N=\|\phi\|^2$, the complete conserved energy and canonical phase charge are

$$E_+=2\omega^2N,\qquad \mathcal Q=-2\operatorname{Im}\langle\Psi,\dot\Psi\rangle =2\omega N,\qquad \boxed{E_+=\mathcal Q\omega.} \tag{10}$$

The phase charge is the action's normalization, not an assignment of electric charge. At fixed $\mathcal Q$, different modes have energies proportional to their selected mass-frequencies. Equation (10) includes the interaction in (3). It does not yet derive a complete Lorentz energy--momentum relation for a moving, self-supported particle.

Full variation and spatial energy flow

Write $f(k)=-g\sqrt{k^2+\eta^2}$, $v=(K-kI)\Psi$, and

$$f''(k)=-\frac{g\eta^2}{(k^2+\eta^2)^{3/2}}. \tag{11}$$

The real derivative of the map is

$$\boxed{ DF[\Psi]h=(H_0+GK+C)h +\frac{2f''}{\rho}v\operatorname{Re}(v^\dagger h). } \tag{12}$$

This retains variations of the current numerator, the density denominator, and the complex conjugate field. The local last term is real self-adjoint. The spatial stability calculation uses it before projecting the positive square.

For the spatial principal part $-icA^a\nabla_a$, put $\Pi=\dot\Psi$ and $W=F[\Psi]$. The equations give $\dot\Pi=-DF\,W$ and $\dot W=DF\,\Pi$. The local self-adjoint terms cancel in the energy balance:

$$\partial_t e+\nabla_a\mathcal J^a=0,\qquad e=|\Pi|^2+|W|^2,\qquad \mathcal J^a=2c\operatorname{Im}(\Pi^\dagger A^aW). \tag{13}$$

For a unit normal $\nu$, the Dirac symbol obeys $(A^\nu)^2=I$, so

$$\boxed{|\mathcal J^\nu|\le c e.} \tag{14}$$

This is a causal energy-flow bound for sufficiently regular solutions, or weak solutions that retain the stated energy balance. It is not a global existence or uniqueness theorem at the non-smooth field origin. The same principal-symbol calculation applies to the actual depth-two differential-form parent.

Why a cubic density source behaves differently

A conventional local cubic source $\rho\Psi$ does not have the closure property of (4). For the explicit highest-weight spinor $u_n=z^n\zeta$, $z=x_1+ix_2$, choose $(\Gamma_1+i\Gamma_2)\zeta=0$. On the unit $d$-sphere, $q=|z|^2$ has the probability distribution $\operatorname{Beta}(1,b)$, $b=(d-1)/2$. Set

$$M(k)=\frac{\Gamma(k+1)\Gamma(b+1)}{\Gamma(k+b+1)},\qquad I_p=\frac{M(pn)}{M(n)^p}. \tag{15}$$

For the normalized density, the cubic source's squared norm fraction outside its original ray is exactly

$$\boxed{\mathcal L_{n,d}=1-\frac{I_2^2}{I_3},\qquad \lim_{n\to\infty}\mathcal L_{n,d}=1-(3/4)^{(d-1)/2}.} \tag{16}$$

Its actual spatial expansion contains degrees $n,n+2,\ldots,3n$. The complete Jacobi decomposition and direct Dirac derivatives agree with (16). Representative leakage fractions are about 0.133 at the depth-two indices, 0.111 for the depth-three down control, 0.247 for the depth-four up control, 0.387 for the first depth-five control, 0.463 for the electron, and 0.679 for the tau. These are fractions of the source's squared norm for this stated polarized family, not probabilities that an arbitrary nonlinear particle state fails.

Numerical scope

At the illustrative common strength $gR/c=1000$, all maintained indices at depths two, three, four, five, six and ten pass the complete fixed-point and finite component variation checks. A single common $\eta$ across depths is checked separately from a control that keeps $\eta$ common only within each depth. Neither normalization selects a physical sector scale.

For the latter control, setting $\eta$ to one hundredth of the smallest bare current magnitude within the depth gives the following relative shifts from the fixed-$g$ frequency at its lowest tested index:

These are model corrections under a declared common law, not fitted particle masses or measured interaction strengths.

Verification

homogeneous_current_checks.py passes 483 checks of the complete variation, fixed points, finite component stability, homogeneity, and positive standing-wave equation. spatial_current_closure_checks.py passes 181 checks of the actual cubic spatial source and its full angular decomposition. positive_current_flux_checks.py passes 266 checks of the spatial energy current and its sharp bound. The corresponding reports preserve exact executed source hashes. current_power_mass_scaling_checks.py checks the complete power-current fixed points and their leading count exponents against the finite chain, using common coefficients within each depth. It compares powers one, 1.2 and two at two reference gaps.