A scalar current law for self mass and aligned pair response

This is a second inverse candidate for a shared self and pair interaction. It keeps the existing Lorentz-scalar matrix current $C=\Psi^\dagger B_0\Psi$, which can be indefinite, and avoids the instantaneously selected rest frame used in the earlier positive-current trial. The candidate still uses the prior action's separate same-field spatial binding term, shell weights and frequency unit. It is not yet the physical mass operator.

A positive function of every nonzero Hermitian current

For the complete retained current at one order, set $T=\operatorname{Tr}C$, $p_2=\operatorname{Tr}C^2$, $p_3=\operatorname{Tr}C^3$, and $N=\sqrt{r p_2}$, where the partner rank is $r\ge2$. Define $\mathcal G_r(0)=0$ and, for $C\ne0$,

$$\boxed{\mathcal G_r(C) =\frac{T^2}{N} -\left(1-\frac1{2r}\right)T +\left(1-\frac1r\right)N +\frac r2\frac{p_3}{N^2}.} \tag{1}$$

All terms have degree one in the current, hence degree two in the field. The coefficients are fixed functions of the representation rank, not retuned by particle index. Positivity holds even if $C$ has negative eigenvalues. Put $z=T/N$ and $w=p_3/N^3$. The Schatten inequality gives $|w|\le r^{-3/2}$, and completing a square gives the dimension-independent matrix-size bound

$$\frac{\mathcal G_r(C)}N =(z-1)^2+\left(1+\frac1{2r}\right)z -\frac1r+\frac r2w \ge\frac34-\frac3{4r}-\frac1{16r^2} -\frac1{2\sqrt r}>0\quad(r\ge2). \tag{2}$$

The weakest stated lower bound is $0.0058216094$ at $r=2$. Thus (1) is real, positive and smooth on every nonzero Hermitian current, including currents with zero trace or mixed signs. At the full matrix cancellation $C=0$, its value is continuous but its gradient is not generally defined. That boundary needs a physical completion.

On a coherent core $C_0=\rho I_r\otimes P$ with $P$ unit rank, $T=N=r\rho$ and $p_3=r\rho^3$. Direct variation gives

$$\mathcal G_r(C_0)=r\rho, \qquad \left.\nabla_C\mathcal G_r\right|_{C_0}P=P. \tag{3}$$

Consequently the one shell action $U_{\rm mass}=\omega_{0,d}\sum_k h_{k,d}\mathcal G_{r_d}(C_k)$, with the same $h_{k,d}=\binom{k+d-1}{d-1}$ at every grade, has the same conditional coherent frequencies $\omega_N=\omega_{0,d}\binom{N+d}{d}$. The maintained particle index is $n=N+1$. The action's full first variation is $B_0\Psi\nabla_C\mathcal G_r$; both upper and lower components are retained. It agrees with the previous localized solution's mass source on the coherent upper-only core. Because $C$ is a Lorentz scalar matrix, the previous boosted solution and its canonical inertial-response identity continue conditionally with the unchanged binding law and canonical norm. This is an action-level replacement, not a selection of the shell weights or physical $Q=1$.

At fixed trace, a small partner split with eigenvalues $\rho(1\pm\varepsilon)$ has $\mathcal G_r(C)-r\rho =\rho\varepsilon^2/r+O(\varepsilon^3)>0$. This checks the local self-current sign; it does not prove stability of all spatial spinor directions.

Exact prepared-pair sign

For two same-depth upper-only cores of distinct grades $N>M$, write one shared-order current as $C_k=I_r\otimes(\rho P_{z,k}+\sigma P_{w,k})$ and $s=|z^\dagger w|^2$. Put $L=\rho+\sigma$ and

$$x=\sqrt{1-\frac{2\rho\sigma}{L^2}(1-s^k)}, \qquad \frac1{\sqrt2}\le x\le1.$$

Substitution into (1) gives one exact pair term, without a separate interaction coefficient:

$$\boxed{\mathcal G_r(C_k)-rL =L(1-x)\left[ \frac{r(1-x)}x +\frac{4x^2-x-1}{4x^2}\right]\ge0.} \tag{4}$$

The final inequality follows because $4x^2-x-1>0$ on the stated interval. Alignment $s=1$ is an angular energy minimum. Expanding for $s=\cos^2\theta$ gives $\Delta\mathcal G_r=\rho\sigma k\theta^2/[2L] +O(\theta^4)$. For equal coincident unit-norm cores, the full coefficient of $\theta^2$ divided by the smaller core frequency is $dM/[4r(d+1)]>0$ when $M>0$. The depth-three index-one state has $M=0$ and no positive-order mass-current pair term.

At larger misalignment, the first term in (4) grows with $r$. The positive aligned curvature does not establish a small physical encounter rate. Spatial overlap, complete deformation, moving states, and the cross-depth vertex for neutrinos remain unknown.

Full frozen source controls and the failed rest lift

The executed batch and report evaluate the complete first variation of (1), including every complex partner direction. Independent central energy differences agree with its source to better than $8.0\times10^{-9}$ on six depths. Random indefinite currents satisfy (2); random prepared pairs satisfy (4) to $6.8\times10^{-14}$. The scalar matrix $C$ is boost invariant to machine precision in the explicit six-depth spinor controls.

The finite frozen field calculation retains all real and imaginary spinor/partner directions, positive and negative frequency sectors, the optional constant core gap, and one or two oscillator columns. At depths two, four and six, gap-to-frequency ratios zero and one, and momenta $0.1,1,10$ in frequency units, its largest imaginary frequency is below $1.6\times10^{-14}$, numerical resolution. This is a constant-coefficient control, not stability of a spatially varying core or every oscillator grade.

The earlier positive-current law (1) of the linked note used an algebraic timelike rest frame. Its frozen response has a growing block for any positive partner-current stiffness. Write that stiffness as $A>0$, rank $r$, and momentum $p$. In a pair of Clifford directions transverse to the momentum, the high-momentum real-linear equations give the exact characteristic factor

$$(\Omega^2-c^2p^2)^2+ \frac{8A}{r}\omega^2c^2p^2=0, \qquad \lim_{|p|\to\infty}|\operatorname{Im}\Omega| =\omega\sqrt{\frac{2A}{r}}. \tag{5}$$

For that trial $A=1/2$; the limiting growth is $\omega/\sqrt r$. Full numerical linearizations at ranks two, four and eight approach this value. The growing block is independent of the pair-alignment coefficient, and a finite algebraic core gap does not change its high-momentum limit because the gap anticommutes with the principal spatial matrix. This is a frozen-field defect of the instantaneous rest-current lift, not a no-go for (1).

Localized depth-six spatial response

The complete first variation of (1) can also be differentiated on the existing nonconstant smooth localized core. For a field perturbation $\eta=(U,L)$, writing the upper core as $\phi=\sqrt\rho V$ with unitary $V$, its mass part in the rotating frame is exactly the previous trace-free partner block with curvature $q=1/r$:

$$\delta\mathcal N_{\rm mass}-\omega\eta =\omega\begin{pmatrix} \frac1r V[\Delta-\operatorname{Tr}(\Delta)I_r/r]\\ -2L \end{pmatrix}, \qquad \Delta=V^\dagger U+U^\dagger V. \tag{6}$$

This is a formula for the full upper/lower field source at the coherent core, not a fixed-profile or scalar-only perturbation. An independent central variation of the matrix action at a nonconstant core point agrees with (6) to relative error $1.3\times10^{-9}$. The binding source is unchanged, so the existing finite Clifford angular machinery can test the new law by setting its derived mass curvature to $q=1/8$ at depth six.

The spatial script and selected report retain all seven constant-Clifford form degrees $k=0,\ldots,6$, both assigned depth-six frequencies $18564$ and $3838380$ in common core units, and two radial basis sizes and quadratures $(18,140)$ and $(26,200)$. All electron cases have no resolved growth above $6.4\times10^{-9}$, and every muon case except one cancellation-sensitive pseudoscalar diagonalization is below $1.2\times10^{-8}$. The exceptional raw value at $k=6$, basis size 26, is $3.42549\times10^{-4}$.

The independent slow-block Schur equation rejects that apparent growth: its squared growth is negative, $-4.38414\times10^{-8}$ at basis size 26 and $-2.43742\times10^{-9}$ at size 34. The corresponding real slow frequencies are $2.09383\times10^{-4}$ and $4.93702\times10^{-5}$, approaching the phase zero as the basis grows. This resolves the finite precision eigenvalue without declaring an instability or a stable isolated particle.

These are finite radial and Clifford angular controls in a stipulated flat six-dimensional spatial carrier. Arbitrary angular degree, coefficient-ray deformation, the physical nested geometry and full nonlinear evolution remain untested. In particular, this calculation does not remove the current-cancellation singularity of (1).

Full noncommuting current and oscillator variation

The local current derivative used above is not limited to eigenvalue changes. For $X=X^\dagger$, put $dT=\operatorname{Tr}X$, $dN=r\operatorname{Tr}(CX)/N$, and $dp_3=3\operatorname{Tr}(C^2X)$. The Hermitian gradient of (1) is

$$G(C)=a_0I+a_1C+a_2C^2, \quad a_0=2T/N-1+1/(2r), \quad a_1=r[(1-1/r)/N-T^2/N^3-rp_3/N^4], \quad a_2=3r/(2N^2). \tag{7}$$

Its complete derivative includes the noncommuting term $a_2(CX+XC)$, the direct $a_1X$, and derivatives of all three scalar coefficients. The complete-variation script and report check the value derivative and complete gradient derivative on indefinite Hermitian matrices at ranks 2, 4, 8 and 32. The largest reported relative gradient-derivative error is $2.5\times10^{-9}$.

The same source is inserted into the full finite oscillator tower at depth six, with arbitrary complex perturbations of every spinor, partner and oscillator coefficient. Grades two and three have isolated coherent energy counts 28 and 84 in that finite test. Their complete source-Hessian central differences agree to better than $5.2\times10^{-11}$ relative. Both larger target grades give the same initial pair energy $6.13116648$ against the same grade-one probe; the exact shared-order formula (4) agrees within $1.8\times10^{-15}$. These values are dimensionless checks at one prepared spatial point, not physical coupling constants or solved encounters. They verify that the pair term survives the actual retained-mode maps and full source variation.

Cancellation, scale and smooth-polynomial boundary

The full-zero-current boundary is physically accessible, including when all three scalar-insertion matrices vanish. Let the partner involution $\beta$ have one positive unit vector $u$ and one negative unit vector $v$, with $u^\dagger v=0$. In one oscillator column take the nonzero doubled field $\Psi=(u,v)^\mathsf T$. The currents defined in the localized action obey

$$C=u^\dagger u-v^\dagger v=0,\qquad J=u^\dagger\beta v+v^\dagger\beta u=0,\qquad K=u^\dagger\beta u+v^\dagger\beta v=0. \tag{8}$$

The boundary script and report verify (8) at the representation ranks used for depths 2, 3, 4, 5, 6 and 10, alongside their maintained particle indices. Equation (8) is an algebraic witness, not a stationary particle. Adding a function of $J$ and $K$ does not automatically exclude simultaneous cancellation.

There is a general regularity obstruction for the current-only class. If a function $F$ of Hermitian $C$ is positively homogeneous of degree one and differentiable at zero, then for every Hermitian $X$, $F(X)=DF(0)[X]$. It must be linear. A linear function cannot be strictly positive on both $X$ and $-X$. Thus no globally positive, degree-one function of the indefinite current has a differentiable source at the full cancellation point. The obstruction applies to equation (1), not to every possible same-field interaction. Indeed, varying the positive amplitude of $u$ in (8) makes $C=tP$ for either sign of small $t$; the two field derivatives of (1) disagree.

A second restriction concerns spatial support. Suppose the full interaction is pointwise algebraic and homogeneous of degree two in $\Psi$, the spatial carrier is first-order Dirac, and the canonical norm is fixed. Under the physical dilation $\Psi_R(x)=R^{-d/2}\Psi(x/R)$, its potential integral is independent of $R$, while the spatial kinetic energy scales as $R^{-1}$. Consequently

$$E(R)=K_1/R+V_2,\qquad E'(R)=-K_1/R^2. \tag{9}$$

A finite stationary radius requires $K_1=0$, in which case every radius along this dilation has the same energy. This class cannot give a strict finite-radius minimum. The original localized action escapes (9) through its nonlinear off-diagonal binding current, which scales with a different field degree. Also, a potential depending only on $C$ has zero lower-spinor source on the upper-only core, leaving the free lower equation $-i\alpha\cdot\nabla\phi=0$; a finite-norm flat-space core then vanishes. These statements identify terms the shared law still must generate. They do not rule out derivative interactions or a physical spatial geometry with its own scale.

The simplest smooth quartic field substitute has a separate sign obstruction. Every quadratic invariant of one Hermitian current is $U(C)=a(\operatorname{Tr}C)^2+b\operatorname{Tr}C^2$. On an accessible traceless current, positivity forces $b\ge0$. For two prepared coherent rays, its mixed term is $2r\rho\sigma(ar+b s^k)$, where $s=\cos^2\theta$. The coefficient of $\theta^2$ at alignment is $-2br\rho\sigma k\le0$. No positive member of this restricted quartic class makes alignment a strict angular minimum.

A higher polynomial shows the restriction's boundary. In units with a fixed current scale, let $H(C)=(\operatorname{Tr}C)^2+ [\operatorname{Tr}C^2-(\operatorname{Tr}C)^2/r]^2$. It is smooth and nonnegative. Its prepared-pair angular excess is $4r^2\rho^2\sigma^2(1-s^k)^2$, so alignment is a minimum, but its angular Hessian vanishes there. The aligned mixed energy is already $2r^2\rho\sigma$, and its coherent source coefficient is $2r\rho$, varying with spatial density. It therefore does not retain the constant coherent mass source or the established common profile. The fixed scale needed between its two field degrees is an additional physical input.

Together these results focus the next construction on a regular same-field action with a scale-setting nonlinear off-diagonal source, while preserving the useful scalar-current self and pair identities.

Mixed scale and lower-spinor Hessian of the analytic core

The gapless analytic profile of the localized action has an additional constrained energetic direction. Let $d>2$, $a=1/(d-1)$, and let $\phi$ be its nonconstant normalized upper-spinor solution with $m_c=0$. Write its lower equation as $D\phi-W(\phi)=0$, where $D=-i\alpha\cdot\nabla$ and $W(\phi)=g\beta\phi f(\phi^\dagger\beta\phi)$. The physical dilation $\phi_s(x)=s^{-d/2}\phi(x/s)$ keeps its canonical norm fixed. Homogeneity gives the exact lower residual

$$\mathcal B(\phi_s)(x) =s^{-d/2}\bigl(s^{-1}-s^{-da}\bigr) D\phi(x/s),\qquad \left.\frac{d\mathcal B(\phi_s)}{ds}\right|_{s=1} =\frac{D\phi}{d-1}\ne0. \tag{10}$$

For every such upper-only coherent field, including the scalar-current replacement (1), $J=0$, the binding energy vanishes, and the mass energy is exactly $\omega Q$. Hence the constrained energy along $(\phi_s,0)$ is flat. Yet the full energy's derivative in a lower spinor direction $\zeta$ is the nonzero residual (10) when $s\ne1$. Its mixed Hessian with the scale direction is $2\operatorname{Re}\int\zeta^\dagger D\phi/(d-1)$. The scale-scale Hessian is zero, and choosing a small lower direction with opposite overlap makes a negative quadratic direction. Thus this particular gapless analytic core is a constrained energetic saddle, for both its original mass norm and (1). This is stronger than merely showing an energy unbounded below, but it does not by itself show exponential time growth of a Hamiltonian perturbation.

The direct source script and report check (10) with the complete nonlinear field source at depths 2, 4, 6 and 10. The central derivative agrees within $2.7\times10^{-9}$ relative. At depths 4, 6 and 10, the chosen profiles have canonical norm one. The depth-two calculation is only a local equation check: this analytic profile has divergent full-plane norm there, so the finite-norm saddle conclusion does not extend to it. Odd-depth spatial representations, the nonzero-gap smooth core, physical geometry and dynamical stability still need independent tests.

Extension to a nonzero core gap

The same mixed-Hessian argument extends to any nonzero finite-norm coherent upper-only solution of the specified fractional binding law with a constant core gap $m_c$, provided its dilation lies in the energy's form domain. Write $M\phi=\beta\phi\beta_p$. The lower stationary equation is $D\phi+m_cM\phi-gW(\phi)=0$. The exact fixed-norm dilation residual is

$$\mathcal B(\phi_s)=s^{-d/2}\left[ (s^{-1}-s^{-da})D\phi+(1-s^{-da})m_cM\phi \right]\!(x/s),\quad \mathcal B'(1)=\frac{D\phi+d m_c M\phi}{d-1}. \tag{11}$$

For a nonzero square-integrable $\phi$, this derivative cannot vanish: $D+d m_cM$ has square $-\Delta+d^2m_c^2$ because $D$ and $M$ anticommute. It has no nonzero square-integrable zero mode on flat $\mathbb R^d$. The constrained energy is still flat on the coherent upper-only dilation, so the mixed block makes its Hessian indefinite. This rules out a constrained energy minimum for that class of upper-only cores with this action. It does not establish dynamical growth or rule out a different core with occupied lower components, spatial geometry, or an additional scale-setting same-field term.

The general scale script and report verify (11) in local source controls at depths 2, 4, 6 and 10 for $m_c=0$ and $0.3$. The nonzero-gap controls assign a local derivative satisfying the equation; they are not global profiles. For the actual normalized gapless analytic profiles at depths 4, 6 and 10, direct radial integration independently gives the exact nonzero mixed-source norm

$$\|\mathcal B'(1)\|_{L^2}^2 =\frac{d^2(d-2)}{4(d-1)^3} =\begin{cases} 0.2962962963,&d=4,\\ 0.2880000000,&d=6,\\ 0.2743484225,&d=10. \end{cases} \tag{12}$$

For a lower variation chosen as $\zeta=t\mathcal B'(1)$, the scale-lower entry of the constrained energy Hessian is twice (12). The zero scale-scale entry and nonzero off-diagonal entry force a negative eigenvalue regardless of the finite lower-lower entry. This exact result sets a concrete design requirement for a complete one-law action: it must add a coherent upper-field restoring energy, change the occupied core structure, or alter the spatial carrier.

General exponent and the smooth depth-six core

The preceding calculation used $a=1/(d-1)$. The same dilation argument for any positive homogeneous exponent $a$ gives

$$\mathcal B'(1)=(da-1)D\phi+da m_cM\phi. \tag{13}$$

The operator on the right has square $(da-1)^2(-\Delta)+(da m_c)^2$. Thus its action on a nonzero square-integrable flat-space core cannot vanish unless $da=1$ and $m_c=0$. For the actual smooth depth-six core in the localized action, $a=1/d$ and $m_c>0$, so the mixed scale source is simply $m_cM\phi\ne0$. Its constrained energy Hessian is therefore also indefinite under the stated coherent upper-only ansatz. The finite real-frequency scans do not supply an energy minimum, and an indefinite energy Hessian alone does not imply exponentially growing time evolution. A local source check at depths 4, 6 and 10 with $a=1/d$ and $m_c=0.3$ verifies the residual scaling; these are algebraic controls, not newly solved profiles.

Remaining physical work

The scalar-current law (1) improves the inverse candidate on positivity, prepared-pair orientation and the tested frozen principal response. It still inherits the separate nonlinear off-diagonal binding term that supports the localized core. No full spatial polarization or geometry Hessian, nonlinear collision, common cross-depth normalization, unique master-field reduction or observed neutrino vertex has been established. The cancellation $C=0$ can occur for nonzero complete fields, and the action is not smooth there. A complete physical mass operator must resolve that boundary and derive the entire interaction from one same-field law.