A positive current law with mass and stable pair alignment

The quadratic-root current gives a nonzero self frequency while an aligned pair has zero initial cross energy. Its alignment is an angular maximum for every tested positive current-metric parameter at nonzero shared grade. Here a single different current function changes that sign without changing the isolated simplex frequency. This is an inverse action candidate, not a derivation from the IDWT master interaction.

One function of a positive same-field current

Let $R$ be a nonzero positive-semidefinite Hermitian retained current, $T=\operatorname{Tr}R$, $p_2=\operatorname{Tr}R^2$, and $p_3=\operatorname{Tr}R^3$. Let $r$ be the partner spinor rank, fixed by the sector's component representation. Define the degree-one function

$$\boxed{\mathcal F_r(R) =T+\frac{T}{2r}-\frac{p_2}{T} +\frac{r p_3}{2T^2}.} \tag{1}$$

This is one local interaction rule for both a state and a pair of states. The three moment coefficients in (1) are fixed by the algebraic construction, not independently fitted to any particle mass. On a coherent rank-$r$ current $R_0=\rho I_r\otimes P$, with a unit-rank coefficient projector $P$, one has $T=r\rho$, $p_2=r\rho^2$, $p_3=r\rho^3$, and

$$\mathcal F_r(R_0)=r\rho=T, \qquad (D\mathcal F_r)_{R_0}P=P. \tag{2}$$

The second identity means that its complete first variation acts on the occupied coherent field as the canonical density. In the same shell-current action as the localized construction, use one expression at every order,

$$U_{\rm mass}=\omega_{0,d}\sum_{k\ge0} \binom{k+d-1}{d-1}\mathcal F_{r_d}(R_k). \tag{3}$$

On an isolated coherent grade $N$, equation (2) and the Hockey-Stick sum again give $\omega_N=\omega_{0,d}\binom{N+d}{d}$. The maintained particle index is $n=N+1$, so its count is $S(n,d)=\binom{n+d-1}{d}$. The shell weights and common physical frequency unit are still inverse inputs; (3) does not derive them.

The sign of the new current energy is controlled exactly. For $a=p_2/T^2$ and $b=p_3/T^3$, Cauchy-Schwarz on the nonnegative eigenvalues gives $b\ge a^2$. Consequently

$$\frac{\mathcal F_r(R)}{T}-1 =\frac1{2r}-a+\frac r2 b \ge\frac r2\left(a-\frac1r\right)^2\ge0. \tag{4}$$

Equality holds for equal positive eigenvalues of rank $r$. Thus the isolated coherent current has minimum interaction energy at fixed trace against arbitrary positive current-matrix variations. This is a matrix statement, not a stability theorem for the complete spatial field and its indefinite kinetic energy.

For example, split two occupied partner eigenvalues as $\rho(1+\varepsilon)$ and $\rho(1-\varepsilon)$, leaving the other $r-2$ values equal to $\rho$. At fixed $T=r\rho$,

$$\mathcal F_r(R)-T =\frac{\rho}{r}\varepsilon^2+O(\varepsilon^3)>0. \tag{5}$$

Exact pair term and angular sign

Prepare two upper-field cores of distinct grades $N>M$ in one depth. At a shared order $k\le M$, let $R_k=I_r\otimes(\rho P_{z,k}+\sigma P_{w,k})$ and $s=|z^\dagger w|^2$. Directly taking the second and third traces in (1) gives the exact mixed density

$$\boxed{\mathcal F_r(R_k)-r\rho-r\sigma =\frac{\rho\sigma}{2(\rho+\sigma)}(1-s^k).} \tag{6}$$

The same function that generates the self frequency therefore gives a reciprocal nonnegative initial pair energy. Alignment $s=1$ is an angular minimum, not an assumed zero coupling rule. For coincident equal-norm cores with $r\int\rho=1$, the full prepared-pair value is

$$V(s)=\frac{\omega_{0,d}}{4r} \sum_{k=1}^M\binom{k+d-1}{d-1}(1-s^k). \tag{7}$$

At $s=0$, its ratio to the smaller isolated frequency is $[S(n,d)-1]/[4rS(n,d)]$, with $n=M+1$. For a small ray angle $s=\cos^2\theta$, the coefficient of $\theta^2$ relative to that frequency is $dM/[4r(d+1)]>0$ for $M>0$. These are potential-energy ratios for prepared coincident patterns, not scattering cross sections or encounter probabilities. The depth-three index-one state has $M=0$, so this mass-current pair term vanishes identically.

The six-depth moment batch tests all sixteen maintained indices, random positive current matrices, random pairs and finite boosts. Its report records a maximum pair-formula error of $4.33\times10^{-14}$. Selected upper bounds on the coincident pair-energy ratio are about $0.06247$ for depth-five index ten, $0.03125$ for depth-six index thirteen and $0.0078125$ for depth-ten index twenty-three. Those relative values follow from the partner ranks $4,8,32$ in this conditional construction. They do not imply that a depth-five neutrino reacts weakly with ordinary matter: that is a cross-depth interaction whose actual field embedding, overlap and dynamics are still missing.

A covariant positive-current lift

The original scalar mass current is Hermitian but can be indefinite, whereas (4) requires $R\ge0$. A conditional local lift can obtain a positive matrix from the same field without choosing an external rest frame. With the Hermitian Dirac principal matrices $A_a$, define

$$j^0=\operatorname{Tr}(\Psi^\dagger\Psi),\qquad j^a=\operatorname{Tr}(\Psi^\dagger A_a\Psi),\qquad u^\mu=\frac{j^\mu}{\sqrt{(j^0)^2-|j|^2}}, \tag{8}$$

where the local current is strictly future timelike. Set $H_u=u^0I-u^aA_a$ and form the retained partner/oscillator Gram matrices from $\Psi^\dagger H_u\Psi$, with the same spectator trace at each order. Since $(A\cdot\hat j)^2=I$, the eigenvalues of $H_u$ are $u^0\pm|u|>0$. Every retained current is therefore positive semidefinite. Under a common Lorentz boost, the spinor and its own $u$ transform together, making $\Psi^\dagger H_u\Psi$ an invariant matrix. No derivatives of $\Psi$ enter $H_u$, so the spatial principal symbol remains the Dirac one wherever this source is smooth.

On the isolated upper-only rest core, $j^a=0$, $H_u=I$, and the retained current is exactly $\rho I_r\otimes P$. Its first field variation also gives the same occupied mass source: the variation of $u$ multiplies $\Psi^\dagger A_a\Psi=0$ as a matrix there. The executed batch checks finite boosts and that first derivative at all six depths. Maximum matrix boost error is below $2.5\times10^{-15}$ in its normalized controls.

The lift is undefined when the total local current is null or zero. Its full arbitrary-field variation, complete Hessian, local well-posed domain and nonlinear spatial stability have not been established. The finite rest-current checks do not prove a global causal physical carrier or a same-field master origin.

Why this is still an inverse candidate

Equation (3) has the desired shared self/pair algebra and retains the conditional simplex mass frequency. Its positive matrix excess does not on its own sustain the displayed localized core. On the upper-only coherent branch, its first variation has no lower-spinor source, while the spatial Dirac derivative lies in that lower block. Removing the separate sustaining interaction from the prior localized action would leave $-i\alpha\cdot\nabla\phi=0$, whose finite-norm flat-space solution is trivial in this ansatz. The previous binding term remains an additional part of the candidate same-field action.

A complete one-law derivation must generate the off-diagonal sustaining source, the shell weights, the common normalization, stable spatial states, and cross-depth response from one physical interaction. The current construction isolates a useful algebraic condition: a positive same-field mass current can have a stable aligned-pair direction without sacrificing the assigned self-frequency, provided the current law uses more than quadratic purity and has a physically valid positive-current lift.

Frozen-field limitation of the instantaneous rest lift

The positive-current algebra above survives, but its algebraic same-field rest-frame lift has a growing high-momentum frozen block. For positive partner-current stiffness $A$, rank $r$, and particle frequency $\omega$, a transverse Clifford block obeys $(\Omega^2-c^2p^2)^2+8A\omega^2c^2p^2/r=0$. Its growth tends $\omega\sqrt{2A/r}$. In the displayed law $A=1/2$, so the limit is $\omega/\sqrt r$. The separate pair-alignment term does not alter this block. Finite complete frozen matrices at depths two, four and six confirm the limit. Thus the instantaneously selected rest-current lift is not a viable stable carrier as written. The scalar-current alternative keeps the self/pair sign while avoiding this specific frozen defect; its complete spatial stability and current-cancellation boundary remain open.