Two localized cores and regular binding

Summary: Two independently solved cores have an exact reciprocal initial interaction and a binding-current cancellation; a smooth inverse binding removes that specific singularity while retaining localized simplex-frequency branches.

Start with two solved spatial fields

The localized current action supports a common spatial core at different simplex frequencies. Its finite probe identity applies to arbitrary probe fields. A sharper test prepares both objects as independently solved localized cores, retaining the complete field equation after they overlap.

Use natural units $\hbar=c=1$ and one ordinary spatial depth $d$. Prepare distinct grades $N>M$ with unit canonical norm:

$$\begin{aligned} \Psi_N(x)&=\begin{pmatrix}\phi(x-X)\\0\end{pmatrix} \otimes\overline z^{\,N},\\ \Psi_M(x)&=\begin{pmatrix}\phi(x-Y)\\0\end{pmatrix} \otimes\overline w^{\,M},\qquad \|z\|=\|w\|=1. \end{aligned} \tag{1}$$

Each isolated field oscillates at its assigned $\omega_R=\omega_{0,d}\binom{R+d}{d}$. These are initial data for a pair, not a claim that their overlapping sum is a stationary solution. The grades label distinct components of the same field. At the same grade, one must instead retain coherent interference in one summed field.

Write $\phi^\dagger\phi=\rho I_{r_d}$ for the core at $X$, use $\sigma$ for the corresponding density at $Y$, and let

$$s=|z^\dagger w|^2,\qquad 0\le s\le1. \tag{2}$$

The scalar $s$ summarizes the two coherent coefficient rays in this particular preparation. Their actual radial spinor textures remain in $\phi(x-X)$ and $\phi(x-Y)$ and enter the complete binding source. It is not a replacement for spatial polarization in a general encounter.

The exact initial cross energy

At every shared current order $k\le M$, the complete mass current is

$$R_k=I_{r_d}\otimes(\rho P_{z,k}+\sigma P_{w,k}),\qquad \operatorname{Tr}(P_{z,k}P_{w,k})=s^k. \tag{3}$$

Subtract the two isolated potentials from the combined potential. Unshared orders cancel. Both upper-only fields have $J=0$, so the binding potential has zero value on this initial set. The full initial cross energy density is therefore

$$\boxed{\mathcal V(x)=\omega_{0,d}r_d \sum_{k=1}^{M}\binom{k+d-1}{d-1} \left[\sqrt{\rho^2+\sigma^2+2\rho\sigma s^k} -\rho-\sigma\right].} \tag{4}$$

The order-zero term vanishes because $s^0=1$. The factor $r_d$ includes the common root normalization and every partner in the canonical density. Equation (4) does not depend on the larger target grade $N$. Differentiating it with respect to either center gives the reciprocal initial translation forces, in agreement with the complete field gradients.

For coincident cores, $\rho=\sigma$ and $r_d\int\rho=1$. Their integrated cross energy has the profile-independent value

$$\boxed{V(X=Y)=\omega_{0,d} \sum_{k=1}^{M}\binom{k+d-1}{d-1} \left[\sqrt{2+2s^k}-2\right].} \tag{5}$$

This is an interaction energy of the prepared pair, not a new isolated mass formula. A common phase of either grade does not change it. General relative phases within the two polarized rays affect $s$ and the full shape response.

The root contribution gives an initial attractive force

Every bracket in (4) is nonpositive. For $t=s^k$, define its positive magnitude

$$F_t(\rho,\sigma)=\rho+\sigma -\sqrt{\rho^2+\sigma^2+2t\rho\sigma}. \tag{6}$$

Direct differentiation gives

$$\frac{\partial^2F_t}{\partial\rho\partial\sigma} =\frac{\rho\sigma(1-t^2)} {(\rho^2+\sigma^2+2t\rho\sigma)^{3/2}}\ge0. \tag{7}$$

Also $F_t(0,\sigma)=F_t(\rho,0)=0$. Thus $F_t$ is the double integral of the nonnegative weight (7), over density thresholds below $\rho$ and $\sigma$. The spatial integral is a positive weighted sum of volumes where the two density superlevel sets overlap.

For the nodeless radial cores, (12) of the core calculation gives $\rho'=-4fh-2(d-1)h^2/r<0$ away from the origin. The superlevel sets are balls. Their intersection volume decreases as their centers separate. Consequently the magnitude of the negative cross energy decreases with separation: the root contribution is initially attractive for nonaligned rays. It vanishes identically for $s=1$.

Translation invariance of the integrated energy gives equal and opposite center forces. Their accelerations differ because the isolated inertial coefficients are $\omega_N$ and $\omega_M$ at $Q=1$. This is a conditional initial force law of the inverse model, not a derivation of observed electric charges or a solution of the ensuing encounter.

Zero initial energy does not mean zero coupling

Take coincident cores with the same coherent ray. The root cross energy and both initial translation forces vanish. However, their binding currents add before the interaction is varied:

$$J_{\mathrm{total}}=0,\qquad K_{\mathrm{total}}=2K_\phi, \qquad K_\phi=\phi^\dagger\beta\phi. \tag{8}$$

For the original fractional binding function $f(K)=\operatorname{sgn}(K)|K|^a$, the complete source on each field differs from its isolated source by

$$\boxed{\Delta\mathcal N_R =\begin{pmatrix}0\\-g(2^a-1)\rho^a\phi\end{pmatrix} \otimes\overline z^{\,R},\qquad R=N,M.} \tag{9}$$

The shared gap term is linear and cancels in this difference. The root source stays equal to its isolated value. Equation (9) is nonzero: both objects immediately develop lower-component motion. Restricting the variation to the upper-only initial family would miss it.

This supplies a response from the same binding law that sustains each core. Its normalized spatial coefficient is common across the grades, although subsequent dynamics can depend on their different frequencies. It does not establish automatic alignment, repulsion or identical trajectories. The full evolving spatial shapes must be retained to decide those questions.

Two regular isolated currents can cancel

For the radial matrix core $\phi=fI+ih\alpha_{\hat x}$, Clifford multiplication gives

$$K_\phi=(f^2-h^2)\beta+2ifh\beta\alpha_{\hat x}. \tag{10}$$

Its eigenvalues are $\pm\rho$, so it is invertible at every finite point of a nonzero isolated core. Now put two identical cores on opposite sides of a midpoint. Their radial directions at the midpoint are opposite. At equal distance $r$ from their centers,

$$K_{\mathrm{total}}=2(f(r)^2-h(r)^2)\beta. \tag{11}$$

The solved profiles have a radius $r_*$ at which $f=h$. Therefore their combined current vanishes at the midpoint when the center separation is $2r_*$. The fractional core values in $m_c=1$ units are:

This is a specific allowed pair preparation. It is not a claim that every trajectory reaches that separation or polarization. It does show that invertibility of each isolated current does not ensure invertibility of their sum in a coupling calculation.

An explicit failure of the fractional variation

At the midpoint write the two upper matrices as $\phi_\pm=A(I\pm i\alpha_1)$ and put $\rho=2A^2$. Introduce small, allowed upper and lower variations of these two grade fields:

$$\Psi_\pm(\epsilon,t)= \begin{pmatrix} \phi_\pm+\epsilon\beta\phi_\pm\\ t\beta\phi_\pm \end{pmatrix}. \tag{12}$$

The symbol $t$ in (12) is a dimensionless variation parameter, not time. The coefficient rays have unit norm and drop out of the traced binding currents. Exact matrix multiplication gives

$$\begin{aligned} K_{\mathrm{total}}&=4\epsilon\rho I,\\ J_{\mathrm{total}}&=4t\rho I,\\ C_{\mathrm{total}}&=2\rho(1+\epsilon^2-t^2)I. \end{aligned} \tag{13}$$

For small $|t|$, the mass current stays strictly positive. The problem here is specifically the fractional binding law. Its potential along this path is

$$U_{\mathrm{bind}}=-g r_d t(4\rho)^{1+a} \operatorname{sgn}(\epsilon)|\epsilon|^a. \tag{14}$$

The gap trace vanishes because $\operatorname{Tr}\beta_p=0$. At every nonzero $t$ and $0<a<1$, the derivative with respect to $\epsilon$ diverges as $|\epsilon|^{a-1}$. Such configurations occur arbitrarily close to the upper-only cancelling pair.

At the exact point $t=\epsilon=0$, the binding potential has zero first derivative, since its leading joint variation has degree $1+a>1$. That special value does not provide a differentiable neighborhood: the nearby configurations with $t\ne0$, $\epsilon=0$ have no finite derivative. The ordinary quadratic kinetic and gap terms cannot cancel (14).

This is a concrete regularity obstruction in the written fractional action. It is stronger than checking isolated tails or assigning a value to $f(0)$ alone. No particle lifetime or actual collision outcome is inferred from it.

A smooth inverse binding retains the frequency branch

A positive current scale $\kappa$ supplies an explicit smooth comparison:

$$\boxed{f_\kappa(K)=K(K^2+\kappa^2I)^{(a-1)/2},\qquad \kappa>0.} \tag{15}$$

Replace $f$ by $f_\kappa$ in the same binding potential and vary it completely. The source is still

$$\mathcal N_{\mathrm{bind}}= m_cB_1\Psi\beta_p-g\left[B_1\Psi f_\kappa(K) +B_2\Psi Df_{\kappa,K}(J)\right]. \tag{16}$$

For a scalar eigenvalue $x$,

$$f_\kappa'(x)=(\kappa^2+a x^2) (x^2+\kappa^2)^{(a-3)/2},\qquad 0<f_\kappa'(x)\le\kappa^{a-1}. \tag{17}$$

The off-diagonal matrix divided differences obey the same bound by the scalar mean-value theorem. Hence the full derivative has Hilbert–Schmidt operator norm at most $\kappa^{a-1}$, including repeated, opposite-sign and zero eigenspaces. In particular,

$$Df_{\kappa,0}(J)=\kappa^{a-1}J. \tag{18}$$

The cancelling-current pair now has a finite complete source and variation. This removes the specific defect (14). A small chosen $\kappa$ can still give a large response coefficient; finite does not mean weak or stable. The separate zero mass-current root and the polarized kinetic limitation of the proposed rest-frame carrier have not been repaired by (15).

The matrix function is odd and respects unitary conjugation. Thus the radial isotropic core reduction and the original boost argument still apply. Its radial binding coefficient is now

$$F(\rho)=g\rho(\rho^2+\kappa^2)^{(a-1)/2}, \qquad f'=-(1+F)h,\quad h'=(F-1)f-\frac{d-1}{r}h \tag{19}$$

in $m_c=1$ units. A spatial solution of this equation gives the same exact current-hierarchy frequency $\omega_N=\omega_{0,d}D_N$ and the same canonical inertia $\hbar\omega_NQ/c^2$. The binding profile can change without changing the isolated mass ratios.

Two verified smooth depth-six cores

Use central-normalized profiles with $f(0)=1$ and set $a=1/6$. Let $\delta$ be the current scale in those temporary shape units, and solve (19) for its unknown coefficient $p$. If $q$ is that shape's full canonical norm, the common unit-norm physical profile has

$$g=pq^{1/6},\qquad \kappa=\delta/q. \tag{20}$$

The two independently refined numerical examples are:

Each row specifies one action used for both the electron and muon grades. Its field magnitude solves the fixed point at $Q=1$; it is never identified as either particle's mass. The two rows are alternative inverse bindings, not independent tuning of the electron and muon.

The new current scale has units of field density. A proposed common-gap scale law can write $\kappa=\widehat\kappa(m_c/c)^d$ with dimensionless $\widehat\kappa$, but selecting that coefficient is additional physical input. The original fractional size factors remain results of their original law. Smoothing changes them, as this table demonstrates.

The smooth spatial functional uses the vacuum-subtracted primitive $H(\rho)=g[(\rho^2+\kappa^2)^{(1+a)/2}-\kappa^{1+a}]/(1+a)$. Writing $R=\int r_d\rho F(\rho)$ and $V=\int r_d H(\rho)$, its independent virials are $T+S=R$ and $T=d(R-V)$ at unit gap. These are fixed-point identities. This auxiliary functional is not added to the full particle energy.

Full component response of the smooth core

The complete occupied-ray Hessian uses the same matrix reduction as the original localized action, with its actual smooth potential and stiffness:

$$F(\rho)=g\rho(\rho^2+\kappa^2)^{(a-1)/2},\qquad 2\rho F'(\rho)=2g\rho(\kappa^2+a\rho^2) (\rho^2+\kappa^2)^{(a-3)/2}. \tag{21}$$

Independent arbitrary complex variations of the full source verify this ordinary/conjugate Hessian, retaining both spinor blocks and every partner. The $\delta=10^{-4}$ core was then tested in all constant-Clifford angular multiplets $k=0,\ldots,6$, with 26 and 40 radial basis functions, at $\omega=18564$ and $3838380$. The complete unsymmetrized matrices and their canonical structure are retained.

The largest displayed electron real growth part is approximately $2.32\times10^{-11}$ in core units. Direct muon diagonalization produces an apparent growth $1.48873194\times10^{-4}$ in the 40-function $k=6$ block. It fails the exact slow-block matching equation: the smallest matching singular value there is $2.68574013\times10^{-8}$, rather than zero. The true slow squared growth of that block is $-4.69417325\times10^{-9}$.

At 26, 40 and 55 radial functions, the independently matched muon slow oscillation frequencies are $1.87198029\times10^{-3}$, $6.85140369\times10^{-5}$ and $9.38787993\times10^{-6}$, approaching the phase-symmetry zero mode. Thus the apparent growing result is a numerical cancellation artifact for this finite block. Other tested real parts remain at numerical precision. These checks find no resolved instability in their stated scope; they do not cover higher angular representations, all radial wavelengths, finite encounters or nonlinear stability.

Verification and remaining scope

localized_core_pair_checks.py verifies 122 conditions for the initial pair: full tensor-current energies, both reciprocal source variations, the nonzero aligned shape response, ordinary spatial energy integrals, the three current cancellations, and the exact fractional derivative exponents. Its arrays retain the actual currents, gradients and spatial interaction values. The scalar integral implementation is localized_core_pair.py.

smooth_current_core_checks.py checks 110 conditions for independent smooth matrix and field variations, including the cancelling pair, both refined localized profiles and their virials, the unchanged assigned frequencies, full boosted equations and every stress component, full ordinary/conjugate Hessians, bounded spatial spectra and the exact muon slow-block comparison. Its data retain the misleading direct eigenvalue as well as the independently matched result. The action and solver are in smooth_current_core.py.

The pair is still an initial preparation. A complete nonlinear encounter, observed interaction channels, cross-depth maps and neutrino suppression remain to be derived. The smooth binding establishes a local repair and new spatial witnesses; its master origin, parameter selection and complete spatial stability remain open. The physical oscillator-mode carrier is also still required before this inverse construction can be identified with particles.

The later trace-current comparison keeps this smooth core and its binding law, while adding a positive trace contraction to the mass norm. The resulting initial pair energy remains independent of the larger target grade, with an explicitly changed angular coefficient. Its improved constant far-field response and additional spatial tests apply to that declared comparison.