Local Current Ratios Close Spatial Shell Pairs but Compress Their Clocks

A Local Action with Its Complete Current Variation

The signed scalar-source calculation shows why a positive-band trial cannot freeze its opposite-sign source. Here a changed inverse action removes that source leakage for a different spatial family. It does not derive the master interaction or replace the maintained frequency law.

Let Φ have a full balanced set of spinor columns. Currents are summed over those columns before varying the action:

$$\rho=\operatorname{Tr}\Phi^\dagger\Phi,\quad j_a=\operatorname{Tr}\Phi^\dagger\alpha_a\Phi,\quad J=\operatorname{Tr}\Phi^\dagger\beta\Phi,\quad n=\sqrt{\rho^2-|j|^2}.$$

The carrier is an additional field-component premise, following the earlier balanced-multiplet comparison; the columns are neither extra spatial dimensions nor separate mediator particles. A reduction from the single field Ψ∞, and observed spin and charge, remain unestablished. In particular, balancing all columns cannot be assumed to preserve an observed particle's spin moments.

On the timelike-current domain n>0, use the ordinary Dirac kinetic term and the changed local potential

$$U=\frac{\kappa J^2}{2n},\qquad \kappa>0,\qquad E=\int\operatorname{Tr}\Phi^\dagger\mathscr D\Phi\,dv+\int U\,dv.$$

Units are ℏ=c=1. The field equation is i∂ₜΦ=DΦ+G, with

$$\boxed{ G=\left[\kappa\frac Jn\beta -\frac{\kappa J^2}{2n^3} \left(\rho I-\sum_a j_a\alpha_a\right)\right]\Phi .}$$

Both the scalar and the complete time/spatial current must be varied. The second term is essential. The summed Clifford bound J²+|j|²≤ρ² follows by bounding expectations of a unit Clifford combination and summing the column norms. Consequently

$$0\le U\le\frac{\kappa n}{2}\le\frac{\kappa\rho}{2}.$$

J and n are Lorentz scalars for the specified local Dirac carrier, so the potential is locally covariant. It has amplitude degree two; G has degree one. Neither fact proves global well-posedness, positive unrestricted Dirac energy, localization or a complete boost law for a particle. The round sphere used below remains a conditional actual ordinary spatial geometry of undetermined radius, without compactification. This classical internal field does not replace deterministic six-dimensional electron position q(t) or supply a position probability distribution.

An Exact Actual Spatial Shell Pair

Retain all occupied depths d=2,3,4,5,6,10. Even depths use intrinsic spinor grading; odd depths use the stated full graded double. Its ranks are r=2,4,4,8,8,32. Let K_l(y,x) be the actual positive sphere Dirac shell kernel from the signed-source calculation, normalized so

$$K_lK_l^\dagger=h_l(y)I,\qquad \int h_l\,d\nu=g_l,\qquad g_l=\binom{l+d-1}{d-1},\qquad \int d\nu=1.$$

Explicitly K_l=a_lI−i sinθ b_lγ(n̂), so h_l=a_l²+sin²θ b_l². It is strictly positive: in the interior, a simultaneous zero would make a Gegenbauer polynomial and its derivative vanish, forcing its solution of the regular second-order equation to vanish identically. At both endpoints a_l is nonzero.

Define E_l=K_l/√(r g_l). Then

$$\mathscr D E_l=hE_l,\qquad \mathscr D(\beta E_l)=-h\beta E_l,\qquad h=\frac{l+d/2}{R}.$$

These are normalized spatial shapes using dν. For physical volume V and canonical norm Q, multiply a unit pair shape by √(Q/V). All ratios and frequencies below are unchanged, and its physical integrated norm is Q.

β is the physical parallel grading. In the positive Killing frame its matrix varies with position; the negative partner uses that actual β, as in the preceding signed-source result.

For constant complex a,b, put Φ=aE_l+bβE_l. Full column traces give

$$\rho=(|a|^2+|b|^2)\frac{h_l}{g_l},\quad j_a=0,\quad n=\rho,\quad \frac Jn=\frac{2\operatorname{Re}(a^*b)}{|a|^2+|b|^2}.$$

The density varies spatially at excited l, but its scalar-current ratio is constant. The complete local source therefore stays within this actual opposite-sign shell pair. This is exact closure on this family, not a point-contact approximation or frozen-current variation.

Frequency, Complete Energy and Retained Persistence

At integrated norm Q=1 take a,b real and write x=2ab and c=a²−b², so x²+c²=1. The complete pair equation has matrix

$$H_{\rm pair}=h\sigma_3+\kappa x\sigma_1 -\frac{\kappa x^2}{2}I.$$

For κ>h, the nonzero branch is

$$c=\frac h\kappa,\qquad x=\pm\sqrt{1-h^2/\kappa^2},\qquad a=\sqrt{\frac{1+c}{2}},\quad b=\operatorname{sgn}(x)\sqrt{\frac{1-c}{2}}.$$

Substitution proves the full stationary field Φ(t,y)=exp(−iωt)Φ(y), with

$$\boxed{\omega=\frac\kappa2+\frac{h^2}{2\kappa},\qquad E=\omega Q.}$$

Amplitude rescaling leaves this frequency unchanged. The energy includes the bare spatial kinetic term and the interaction; it is not the scalar coefficient alone. Opposite Dirac signs and signs of x have no established antiparticle or electric-charge interpretation.

For complex pair amplitudes, the fixed-norm Bloch variables satisfy x²+y²+c²=1. Their energy is

$$E/Q=hc+\frac\kappa2(1-c^2-y^2).$$

The displayed solution is a strict local maximum on this Bloch sphere. Its Hamiltonian motion is elliptic, with shape frequency 2√(κ²−h²). The full four-real-component pair linearization has that imaginary eigenvalue pair and two zero phase/amplitude directions. Conserved pair energy controls distance from the strict maximum at fixed norm, proving orbital persistence within this invariant pair family. A negative constrained energy direction does not by itself establish growth. Perturbations outside the pair, full spatial stability, outgoing channels and lifetimes are untested. This result therefore does not yet meet the stronger persistence requirement for an identified electron.

A Ratio Bound for a Whole Positive Local Class

The compression is more general than the quadratic example. Replace U by n f(J/n), with one differentiable nonnegative f, and retain the same balanced spatial pair. At j=0 its complete source is

$$G=[f'(x)\beta+(f(x)-xf'(x))I]\Phi .$$

The pair energy is ω=E/Q=hc+f(x). On a smooth connected stationary branch parameterized by increasing h, fixed-norm stationarity removes the shape derivative, giving

$$\frac{d\omega}{dh}=c,\qquad \boxed{\frac{d(\omega/h)}{dh}=-\frac{f(x)}{h^2}\le0.}$$

For positive frequencies at h₂>h₁>0 it follows that ω₂/ω₁≤h₂/h₁. This assumes that the compared states lie on the same smooth connected branch of this family; it does not compare arbitrary disconnected solutions or different spatial profiles. A negative local potential, derivative interactions, additional responding components, or a different relation between cumulative cutoff and spatial shell would change the test. The theorem neither supplies nor selects those changes.

For the quadratic action, ω/h=(κ/h+h/κ)/2 is strictly decreasing when 0<h<κ. Therefore no choice of its common κ can repair the following maintained family comparisons under the test map l=N=n−1:

The single maintained tau assignment has no within-sector ratio test. All fourteen assigned comparisons use the same κR=256; no amplitude or per-particle parameter repairs the ratios. Cumulative mode count and individual shell degree remain different mathematical objects. Equating their cutoff labels here is an explicit test hypothesis. The depth-five solution has the same closure and ratio limitation as the other depths; there is no special suppression or neutrino identification.

The Null-Current Boundary Is a Cusp

The ratio expression has a denominator at n=0, but the bound U≤κn/2 proves a continuous extension U=0 there. The extension is not everywhere differentiable. Choose α₁u=u, ||u||=1, and the one-column path ψ_θ=cosθ u+sinθ βu. Direct anticommutation gives

$$\rho=1,\quad j_1=\cos2\theta,\quad j_{a>1}=0,\quad J=\sin2\theta,\quad n=|J|,\quad U=\frac\kappa2|\sin2\theta|.$$

For nonzero small θ, G=(κ/2)sgn(J)βψ_θ. Its one-sided limits differ by κβu. The global classical force therefore needs an additional rule or a changed action at this nonzero null-current field. The full matrix carrier admits this path by setting the other columns to zero. This limitation does not invalidate the explicitly timelike constructed backgrounds, but precludes a claim of a globally smooth action.

Reproduction and Remaining Scope

local-current-ratio-checks.py and its report check the complete complex variation, Lorentz-current transformation, homogeneity, explicit null-current limits, spatial equations and pair Jacobians across all six occupied depths and all fourteen maintained assignments. Spatial residuals are below 2.1×10⁻¹⁶ relative to their operator scale; the finite-difference variation error is below 1.4×10⁻¹⁰. Exact symbolic checks verify the general envelope and ratio identity. The final run passes 871 evaluations across thirty conditions. Two prior successful reports and their exact executed source versions are retained.

The earlier constant-current scalar-gap branch remains a different action with an established full spatial spectral-stability calculation. Neither result transfers automatically to this changed current-ratio action.

This construction establishes an exact source-closed spatial field and a class-wide clock restriction. It establishes no localized flat-space particle, observed spin or charge, master-field reduction, full spatial persistence, assigned-depth preference, or observed mass ladder.

The later inverse attractive-current benchmark evades the nonnegative-potential restriction by reconstructing a local constitutive function from the supplied count polynomial. It closes the actual spatial equation with E=ωQ and retained-pair persistence. Encoding the target in that action is not an independent derivation, and a full spatial stability test remains necessary for an electron candidate.