A localized core determined by polarization ratios

A reconstructed local current action admits a localized spatial core whose shape is determined by polarization ratios. Its defining equation is independent of the overall field magnitude. The same coefficients and profile support every coherent grade of the existing simplex current hierarchy. In that hierarchy the assigned electron and muon frequencies remain different, while their common spatial size is unchanged.

This is a specified mathematical model with testable consequences. The binomial current weights, their physical spatial carrier and the common unit of frequency remain premises. This action has not been identified with the separate gapped normal operator. Neither its nonlinear stability nor the physical selection of occupied particle grades is assumed below. Mass remains the selected frequency in equivalent units.

One local potential and its complete source

Use the doubled Dirac field and partner currents of the localized-current construction on ordinary physical R^d. The matrices satisfy

$$A_a=\sigma_x\otimes\alpha_a,\qquad B_0=\sigma_z\otimes I_r,\qquad B_1=\sigma_x\otimes\beta,\qquad B_2=I_2\otimes\beta,$$

where the Hermitian alpha_a and beta anticommute and square to one. Here r is a representation rank, not a count of physical dimensions. The partner matrix beta_p is the same numerical beta on the right index. For the 2r by r field matrix Psi, define Hermitian currents

$$C=\Psi^\dagger B_0\Psi,\qquad J=\Psi^\dagger B_1\Psi,\qquad K=\Psi^\dagger B_2\Psi.$$

Retain the canonical first-order kinetic term and the spatial operator -i A_a partial_a in units hbar=c=1. For one coherent grade, write

$$\begin{aligned} F_q(C)&=\sqrt{qr\,\operatorname{Tr}(C^2) +(1-q)[\operatorname{Tr}C]^2},\qquad 0<q\le1,\\ U&=\omega_N F_q(C)+m_c\operatorname{Tr}(J\beta_p) -\kappa\frac{\operatorname{Tr}(\beta_p K) \operatorname{Tr}(JK)}{\operatorname{Tr}(K^2)}. \end{aligned} \tag{1}$$

The coefficient omega_N in this isolated-grade expression is supplied by the common graded current action described below, rather than an independent parameter for each particle. The core coefficients obey kappa > m_c > 0. The ratio q is the existing trace/exchange current comparison; q=1 and q=1/(r+1) leave the displayed branch unchanged.

The last term depends on matrix polarization ratios. It is a local interaction of the retained components of the same field. No independently interacting scalar field, Yukawa coupling or exchanged particle is present. The denominator requires K to be nonzero. Its behavior at a vanishing current remains a domain question for general interacting configurations.

Put s=Tr(K^2), b=Tr(beta_p K), w=Tr(JK), and define

$$\begin{aligned} A_C&=\frac{qr C+(1-q)\operatorname{Tr}(C)I_r}{F_q(C)},\\ F(K)&=\frac{bK}{s},\\ Z(J,K)&=\frac ws\beta_p+\frac bs J-\frac{2bw}{s^2}K. \end{aligned} \tag{2}$$

The full local source in the field equation is

$$\mathcal S(\Psi)=\omega_N B_0\Psi A_C +m_c B_1\Psi\beta_p -\kappa[B_1\Psi F(K)+B_2\Psi Z(J,K)]. \tag{3}$$

The second term in the square brackets comes from varying K. It cannot be omitted merely because the particular stationary branch has J=0. All complex and conjugate field variations are retained in (3).

Every term of U is homogeneous of degree two in Psi. Consequently S(a Psi)=a S(Psi) for real a>0. Multiplying a solved profile by a common constant changes its canonical norm without changing its frequency, shape, or coefficients. This does not derive a quantized norm.

Reduce the spatial equation to a polarization ratio

Take the upper-block branch

$$\Psi_N(t,x)=e^{-i\omega_Nt} \begin{pmatrix}\phi(x)\\0\end{pmatrix},\qquad \phi=f(r_x)I_r+i h(r_x)\alpha_{\hat x},\qquad r_x=|x|. \tag{4}$$

The field is regular at the origin when h(0)=0. Write rho=f^2+h^2. Then phi^dagger phi=phi phi^dagger=rho I_r, J=0, K=rho S with S^2=I_r, and

$$t=\frac{\operatorname{Tr}(\beta_p K)}{r\rho} =\frac{f^2-h^2}{f^2+h^2},\qquad \beta\phi F(K)=t\phi. \tag{5}$$

The upper field equation gives the selected frequency. The complete lower field equation becomes

$$-i\alpha\cdot\nabla\phi+m_c\beta\phi\beta_p-\kappa t\phi=0,$$

or equivalently

$$f'=-(m_c+\kappa t)h,\qquad h'=(\kappa t-m_c)f-\frac{d-1}{r_x}h. \tag{6}$$

Unlike a density-dependent core law, this potential depends on f and h only through their ratio. Let y=h/f and a=kappa-m_c. Direct subtraction of the two equations gives the Riccati equation

$$\boxed{y'=a(1-y^2)-\frac{d-1}{r_x}y.} \tag{7}$$

Its regular solution is explicit:

$$\boxed{y(r_x)=\frac{I_{d/2}(a r_x)}{I_{d/2-1}(a r_x)}.} \tag{8}$$

The modified-Bessel recurrences in DLMF 10.29 verify (7) by direct differentiation. They are used as mathematical identities, not as an imported physical mechanism. The remaining profile is the positive quadrature

$$f(r_x)=A\exp\left[-\int_0^{r_x} \left(m_c+\kappa\frac{1-y(s)^2}{1+y(s)^2}\right)y(s)\,ds\right], \qquad h=yf. \tag{9}$$

Thus there is no shooting adjustment of the central magnitude to select a localized solution. A controls its canonical norm and is absent from (7). The field's mass is still read from omega_N.

Regularity and localization

The small-argument Bessel limit gives

$$y(r_x)=\frac{a r_x}{d}+O(r_x^3),\qquad f(r_x)=A\left[1-\frac{(m_c+\kappa)a}{2d}r_x^2+O(r_x^4)\right]. \tag{10}$$

For a>0 the positive Bessel ratio lies between zero and one. Its large-argument expansion gives

$$y=1-\frac{d-1}{2a r_x}+O(r_x^{-2}),\qquad t=\frac{d-1}{2a r_x}+O(r_x^{-2}). \tag{11}$$

Substituting (11) into (6) cancels the coefficient a in the leading power-law prefactor:

$$f(r_x)=C r_x^{-(d-1)/2}e^{-m_c r_x}[1+O(r_x^{-1})], \qquad h/f\longrightarrow1. \tag{12}$$

The norm and all positive spatial moments are finite. Equation (12) is for fixed kappa>m_c; the limit kappa approaching m_c is not uniform. At kappa=m_c the regular ratio is y=0 and the core is constant, so it is not a localized member of this family.

A common shape and frequency hierarchy

For the complete graded action replace omega_N F_q(C) in (1) by

$$U_{\rm mass}=\omega_{0,d}\sum_{k\ge0} \binom{k+d-1}{d-1}F_q(R_k), \tag{13}$$

where R_k is the existing partial-current hierarchy and r in F_q is the partner rank. The binding currents J and K sum over all grades before (1) is varied. On a single coherent grade N, R_k=rho I_r tensor P_(z,k) for k<=N, with a normalized coherent ray z. Their contractions give

$$\omega_N=\omega_{0,d}\sum_{k=0}^N\binom{k+d-1}{d-1} =\omega_{0,d}\binom{N+d}{d}. \tag{14}$$

The spatial equation (6) contains neither N nor omega_N. Thus the electron N=12 and muon N=34 at depth six have the same profile at fixed m_c and kappa, while their simplex factors remain 18564 and 3838380. The depth-ten comparison N=22 has factor 64512240. These assignments have not changed. The carrier and the binomial weights in (13) remain model premises.

For kappa/m_c=2 the refined unit-norm profile calculations give L_rms*m_c approximately 2.72054 at depth five, 3.21819 at depth six, and 5.21361 at depth ten. The odd-depth comparison uses a sufficient Clifford representation without introducing another spatial coordinate. The independent report below verifies their integration and domain refinement. Restoring units gives L_rms=c times the dimensionless factor divided by m_c. Identifying m_c with the shared frequency unit is an additional common coefficient choice, not a consequence of the profile equation.

Energy, phase and complete inertial response

Because the full Hamiltonian is homogeneous of degree two, a regular stationary solution with canonical norm Q satisfies E=omega Q. This follows by varying its overall magnitude in the stationary equation; it does not identify that magnitude with mass.

For (4), the binding and spatial kinetic energy densities vanish on the solution while their variations sustain (6). The complete stress is T^00=omega_N Tr(phi^dagger phi), with T^0a=T^ab=0. Every B_j anticommutes with A_a, so the displayed scalar currents are invariant under the same spinor boost as in the maintained localized-current action. The boosted solution therefore has

$$E(v)=\gamma\hbar\omega_NQ,\qquad \mathbf P(v)=\gamma\frac{\hbar\omega_NQ}{c^2}\mathbf v. \tag{15}$$

This is the momentum of the full declared action. For Q=1 its positive inertial coefficient is the selected mass-frequency value. The field continues oscillating in its zero-total-momentum frame. The independent report below checks the ordinary spatial derivatives, full source and complete boosted stress of this new binding law.

The statement about boosts uses the existing coefficient-frame assumptions. A retained dynamical frame can preserve this branch when its added terms and first variations vanish there, but that does not identify coefficient grades with the actual spatial oscillator spectrum. A complete physical mode carrier, interaction-domain control, stability and occupation selection remain separate tests of the proposed model.

A full tail response limits a stability claim

The first independent report, polarization-core-20260914T120746_793183Z.json, passes 405 checks of the complete source, its real Hessian, ordinary Cartesian derivatives, profile refinement, relative-phase energy, physical scaling and complete boosted stress. The radius factors above are stable under the documented outer-domain and integration refinements.

The relative phase is explicit. For an upper field phi=f I_r+exp(i vartheta) h alpha_(xhat), at fixed f and h,

$$U_{\rm mass}=r\omega_N \sqrt{(f^2+h^2)^2+4q f^2h^2\cos^2\vartheta}. \tag{16}$$

The quadrature values vartheta=+/-pi/2 minimize this restricted energy when q>0. The full radial stationary equation chooses the displayed normalizable branch. Equation (16) does not assign an antiparticle interpretation to the other sign or prove full dynamical stability.

A complete frozen tail calculation retains all spinor and partner variations, including directions outside the radial ansatz. At y=1, the binding changes the real Hessian by rank two. Its smallest invariant space under that change, the remaining local terms, and Fourier motion parallel and perpendicular to the radial direction is eight-dimensional. An exact rational-matrix construction verifies that closure.

The leading slow disturbance frequencies in that block, at large omega_N and fixed spatial wave vector p, satisfy

$$\nu_\pm^2=q\left[p_\parallel^2+ \left(\sqrt{p_\perp^2+(\kappa-m_c)^2}\pm\kappa\right)^2\right] +O(\omega_N^{-2}). \tag{17}$$

These are frequencies of disturbances to the oscillating pattern; omega_N remains its assigned mass-frequency. In particular, the exact finite-omega generator has a zero eigenvalue when

$$p_\parallel=0,\qquad p_\perp^2=m_c(2\kappa-m_c). \tag{18}$$

At m_c=1, kappa=2 and q=1/9, exact ranks give a one-dimensional kernel and a two-dimensional kernel of the squared generator, including at the electron frequency. This is a nontrivial zero-frequency Jordan block: a frozen linear disturbance can have a secular contribution proportional to time. It is not negative physical particle inertia.

The exact characteristic polynomial and full invariant subspace are checked in polarization-tail-20260914T125458_161142Z.json. Its 61 checks include exact Jordan ranks at both assigned depth-six frequencies. Since localization requires kappa>m_c, the critical wave number in (18) is real throughout this family. Moving those two coefficients does not remove that critical direction.

The formal full-space linearization is not uniformly bounded

The tail calculation can be carried beyond a frozen-point diagnostic. For m_c=1, kappa=2 and p=sqrt(3) e_perpendicular, let A_infinity(p) be the complete frozen generator. Its exact affected subspace contains vectors u and v with

$$A_\infty(p)u=v,\qquad A_\infty(p)v=0,\qquad \langle u,v\rangle=0. \tag{19}$$

The inner product is the canonical real-field norm inherited from the complete spinor and partner space. Exact algebra gives

$$\boxed{\frac{\|v\|}{\|u\|} =c_J=\frac{2(q\omega_N^2+1)}{\sqrt{\omega_N^2+1}},\qquad \frac{\|u+t v\|}{\|u\|}=\sqrt{1+c_J^2t^2}.} \tag{20}$$

The full depth-five and depth-six Hessians give the same expression through their common two-direction Clifford subspace. At q=1/9, omega_N=18564 gives c_J approximately 4125.33 in the stated core time units; omega_N=3838380 gives approximately 852973.33. These are amplification coefficients, not negative inertias or new mass values. Equation (20) also has a nonzero continuous limit at q=0.

To connect this result to the localized core, normalize a smooth compact packet chi and put

$$\eta_{R,L}(x,t)=L^{-d/2} \chi\!\left(\frac{x-R e_1}{L}\right) e^{ip\cdot x}(u+t v). \tag{21}$$

Take R much larger than L. On the packet support the radial direction approaches e_1 and h/f approaches one. The Hessian of the degree-two potential is independent of the common magnitude of the background; on this branch its coefficients are smooth bounded functions of those polarization ratios. Their difference from the frozen generator is O((L+1)/R), while derivatives of the envelope contribute O(1/L). For each fixed time interval 0<=t<=T the full linearized residual obeys

$$\|\mathrm{residual}(t)\| \le C\left(\frac1L+\frac{L+1}{R}\right)(1+t). \tag{22}$$

The formal linearized operator has the constant symmetric Dirac principal part plus bounded local coefficients. Its linear evolution therefore exists on the canonical L2 space with an exponential finite-time norm bound. Apply its variation-of-constants formula to (22). Taking L and R sufficiently large for the fixed T makes the difference between the exact linear solution and (21) arbitrarily small. It follows that

$$\boxed{\|e^{T\mathcal A}\|\ge\sqrt{1+c_J^2T^2}.} \tag{23}$$

Real perturbations follow by combining the opposite Fourier vectors. The finite nonzero wave number distinguishes these packets from a global phase change or a common translation. Their overlaps with the finite localized symmetry directions tend to zero as R grows.

Equation (23) rules out uniformly bounded formal linearized evolution for the full unbounded spatial core. It is stronger than merely finding a Jordan block in a frozen matrix. The report polarization-packet-20260914T131301_562773Z.json passes 32 exact embedding, norm and amplification checks. The packet argument itself is analytic; the report is not a nonlinear time simulation.

A nonlinear stability theorem would additionally have to handle the interaction's current-zero domain. Compact packets can be scaled small relative to the background on their support, but that does not provide uniform nonlinear control of arbitrarily distant tails in the L2 norm. The existence and positive total inertia results remain valid for the stated branch, while a stable physical particle has not been obtained. The physical mode carrier and coefficient frame also remain model inputs.

A structural test of more general polarization binding

Changing the power of the binding ratio does not automatically repair the response. A broader homogeneous local potential is

$$U_{\rm bind}=-\frac{\sqrt r\,g(t)}{\sqrt{\operatorname{Tr}K^2}} \operatorname{Tr}(JK),\qquad t=\frac{\operatorname{Tr}(\beta_pK)}{\sqrt{r\operatorname{Tr}K^2}}. \tag{24}$$

For g(t)=kappa t, this is (1). On the same conformal-unitary upper field, the radial equation becomes

$$y'=(1+y^2)[g(t)-m_ct]-\frac{d-1}{r_x}y,\qquad t=\frac{1-y^2}{1+y^2}. \tag{25}$$

A limiting polarization t_infinity in [0,1) satisfies g(t_infinity)=m_c t_infinity. Write s_infinity=sqrt(1-t_infinity^2). The fixed point attracts the radial ratio when g'(t_infinity)>m_c.

There is an explicit neutral direction in the full frozen equation at

$$\boxed{p_*^2=m_c s_\infty^2[2g'(t_\infty)-m_c].} \tag{26}$$

To verify it, let alpha_1 point radially and alpha_2 transversely, put Xi=i alpha_1 alpha_2, and normalize the limiting upper field phi. Define the two real-space perturbation coefficients

$$\begin{aligned} F_c&=\phi(2m_ct_\infty I-i m_cs_\infty\alpha_1),\\ F_s&=i\phi\,p_*(t_\infty\alpha_2-s_\infty\Xi). \end{aligned} \tag{27}$$

The upper disturbance F_c cos(p_*x_2)+F_s sin(p_*x_2), with zero lower block, has only a scalar variation of C. The rotating-frame root-current variation therefore vanishes. Substitution into the complete lower equation leaves zero precisely under (26). This retains the sine, cosine, complex-conjugate and local norm variations together. It does not change the assigned background mass-frequency.

The canonical symplectic product of these two coefficients is zero. When the full frozen kernel is one-dimensional, that condition supplies a generalized zero vector and hence a secular mode. A degenerate kernel requires its own analysis; (26) alone is not a universal instability theorem for every possible function g.

For g(t)=kappa t^p with p=3 or 5 and kappa>m_c, t_infinity=(m_c/kappa)^(1/(p-1)). At m_c=1 and kappa=2 the critical wave numbers are approximately 1.58114 and 1.62359. Both retain the one-dimensional kernel and two-dimensional squared kernel in full depth-six checks. The resolved singular values agree with the explicit vector (27).

polarization-power-20260914T134233_510923Z.json passes 63 checks of the complete variational source, real Hessian, homogeneity, explicit neutral vector, symplectic product and numerical Jordan ranks for the linear, cubic and fifth-power laws. These are specified mathematical controls, not additional fitted particle laws. The attracting-tail condition makes (26) real, so this neutral direction is a structural constraint on this polarization-only binding family. A joint model with spatial geometry must address it explicitly.

The five exact symbolic identities in polarization_tail_identity.py verify (26)-(27), both real-space quadratures, the scalar current variation and the symplectic product for arbitrary m_c, p and limiting ratio y. The representation includes both Clifford irreducible signs; the calculation does not infer physical dimension from its matrix rank.