An Inverse Local Current Action for Exact Spatial Count Clocks

What the Construction Establishes

The positive local-current calculation proved that nonnegative potentials of the form U=n f(J/n) compress bare-shell frequency ratios on a connected stationary pair branch. An attractive contribution can evade that restriction. The following inverse construction gives exact actual spatial solutions with the supplied count frequencies, without fitting field amplitudes.

The count polynomial is an input to the constitutive action. This is therefore a consistency benchmark for an inverse mechanism, not an independent derivation of that spectrum or new evidence for the assigned sector depths. The same construction works at every occupied depth. It leaves a single parent action, localization and full spatial stability unestablished.

Measured particle masses are already known during model development and cannot be un-known. Fitting those targets is legitimate; historical ignorance of them is not a prerequisite for useful model construction. Assess a fitted construction through its common constraints, mathematical consistency and further tested consequences, while identifying which quantities were supplied and which follow from the equations. In this construction the count polynomial is explicitly supplied, whereas the exact spatial closure, complete energy and restricted persistence follow from the resulting action.

Measured masses are inputs to the upstream constrained spectral inference. The sector depths and mode indices are inferred assignments under the shared counting and scale rules, rather than additional independently supplied measurements. Those inferred assignments become inputs to this downstream dynamical realization test. Assignment uniqueness is a separate question, restricted to the documented search domain and constraints.

A Single Reconstruction Formula Across Occupied Depths

A direct auxiliary-variable formulation now gives a compact forward proof of this same conditional action. Stationarity identifies its algebraic feedback label with the actual spatial Dirac eigenvalue; the complete energy then returns the count clock. The equivalence clarifies the mechanism inside the effective model without independently deriving its counting constitutive premise.

Use the full balanced same-spinor matrix carrier, ordinary sphere Dirac operator and actual opposite-sign shell pair from the preceding note. Let physical radius be R_d, and write the dimensionless bare frequency as h=l+d/2. Define

$$P_d(h)=\frac1{d!}\prod_{j=1}^d(h+j-d/2),\qquad P_d(l+d/2)=\binom{l+d}{d}.$$

The continuous polynomial between integer shells defines a constitutive function; it does not assert fractional spatial shell occupation. Put ε_d=μ_d R_d, where μ_d is the sector frequency unit. Initially take one common diagnostic ε=2⁻³² across depths 2,3,4,5,6,10. This is an arbitrary dimensionless comparison scale, not a measured radius or a derived cross-sector scale relation.

For n=√(ρ²−|j|²)>0 and x=J/n, reconstruct f_d by

$$\boxed{ c=\sqrt{1-x^2}=\varepsilon P_d'(h),\qquad f_d(x)=\varepsilon P_d(h)-hc.} \tag{1}$$

This is minus a Legendre transform of the supplied convex polynomial. It uses the same formula at each depth, but supplies dimension-dependent constitutive functions. A reduction producing all of them from one master-field action has not been derived.

For an explicit open chart choose h>(d−1)/2, with 0<c<1. All factors in P_d are then positive. Its second derivative is strictly positive:

$$P_d''=P_d\left[ \left(\sum_j\frac1{h+j-d/2}\right)^2 -\sum_j\frac1{(h+j-d/2)^2}\right]>0.$$

Thus the inverse in (1) is unique on that interval. The half-shell extension puts the physical ground shell inside the chart, allowing arbitrary sufficiently small complex field variations there. It is a declared mathematical extension, not a new particle mode. No global continuation of the constitutive function beyond the chart is selected.

The physical local potential is

$$U_d=\frac n{R_d}f_d(J/n).$$

J and n are the summed scalar and proper Dirac currents. The interaction is locally Lorentz scalar and has amplitude degree two. Its spatial dimensions remain ordinary physical directions; the conditional sphere does not compactify them. Spinor columns are a field-component premise, not separate particles. The field is an internal classical structure, distinct from literal deterministic six-dimensional electron position.

Keep the Entire Local Source and Spatial Kinetic Term

In dimensionless time t/R_d, the complete source is

$$G=\left[f_d'(x)\beta+ \frac{f_d(x)-xf_d'(x)}n \left(\rho I-\sum_a j_a\alpha_a\right)\right]\Phi. \tag{2}$$

No current derivative is frozen. On the actual balanced shell field Φ=aE_l+bβE_l, the summed spatial current vanishes and J/n=2ab for real a,b with a²+b²=1. Let c=a²−b² and x=2ab>0. Equation (1) gives

$$f_d'(x)=\frac{hx}{c},\qquad f_d''(x)=\frac h{c^3}-\frac{x^2}{c^2c'},\qquad c'=\varepsilon P_d''(h)>0. \tag{3}$$

The complete spatial equation DΦ+G=ωΦ reduces exactly to the pair matrix hσ₃+f_d'(x)σ₁+(f_d−xf_d')I. Its positive mixed branch is

$$a=\sqrt{(1+c)/2},\qquad b=\sqrt{(1-c)/2},\qquad \boxed{\omega R_d=hc+f_d=\varepsilon P_d(h).} \tag{4}$$

Restoring a physical integrated norm Q and volume V_d multiplies the unit spatial shape by √(Q/V_d). The local ratios, source coefficient and selected frequency are unchanged. Complete energy includes the bare spatial kinetic term:

$$\boxed{E=\omega Q,\qquad \omega=\mu_d\binom{l+d}{d}.} \tag{5}$$

Both spatial Dirac signs and their full actual kernels are retained. The construction tests l=N=n−1 as the map from cumulative cutoff to individual spatial shell. This remains an additional hypothesis. Selecting R_d=ε/μ_d imports an already supplied sector unit into the radius; it does not independently reproduce its numerical value.

Attraction Is Necessary Here

At every integer shell l≥0 and occupied depth,

$$f_d=\varepsilon(P_d-hP_d')<0.$$

At l=0, P_d=1 and P_d'=H_d, the harmonic number, so P_d−hP_d'=1−dH_d/2<0 for d≥2. Its derivative is −hP_d''<0. The negative sign therefore follows on the whole tested positive-shell interval. It is not a choice to discard an unfavorable result.

The total stationary energy in (5) is positive, while the local interaction contribution is attractive. That does not prove a positive unrestricted Dirac Hamiltonian. A negative interaction energy alone also does not prove an exponential instability or decay.

What Persistence Has Actually Been Proved

On the fixed-norm Bloch sphere, the pair energy is hc+f_d(x), with x²+y²+c²=1. At the stationary real pair, its two independent shape curvatures are

$$E_{cc}/Q=-1/c',\qquad E_{yy}/Q=-h/c.$$

Both are negative: the state is a strict local energy maximum within this invariant family. Conserved pair energy controls displacement from that maximum, giving local orbital persistence modulo overall phase. The full four-real-component pair Jacobian includes two zero phase/amplitude directions and an elliptic shape pair with

$$\boxed{\Omega_{\rm shape}R_d =2\sqrt{\frac{(1-c^2)h}{cc'}}.} \tag{6}$$

Independent nonlinear-source finite differences verify that Jacobian. This theorem does not cover arbitrary spatial or column-polarization perturbations. It does not establish electron stability, an observed lifetime hierarchy or outgoing decay channels. Electron candidates need substantially stronger persistence evidence than unstable-particle candidates; this two-mode result cannot supply that distinction.

The chosen small ε creates very close scalar-current ratios. For the electron comparison c≈1.69389×10⁻⁶ and 1−x≈1.43462×10⁻¹². For the ground down comparison, 1−x≈9.11031×10⁻²⁰. Eighty-digit arithmetic retains these differences and the cancellation in (4). This is conditioning caused by the diagnostic scale, not physical field-amplitude calibration.

A Finite Chart Rather Than an Unlimited Ladder

Every smooth stationary pair of this class obeys d(ωR_d)/dh=c, with |c|≤1. Thus a convex polynomial count spectrum at fixed ε cannot continue to arbitrarily large l. The upper chart endpoint satisfies εP_d'(h)=1. For the shared comparison ε=2⁻³²:

All fourteen maintained assignments lie inside the chart. Many other integer shells also do. No sparse observed-index selection follows. The exact electron–muon slope bound alone requires ε₆≤22/(3838380−18564)=11/1909908. Our common comparison scale satisfies it; this inequality does not select that scale.

A Wider Frozen-Current Check and Its Limit

The complete local linearization at j=0 has nonlinear terms

$$\delta G=M\delta\Phi +\frac{2f_d''}\rho T\operatorname{Re}\langle T,\delta\Phi\rangle -\frac{2B}\rho\sum_a V_a\operatorname{Re}\langle V_a,\delta\Phi\rangle,$$

where M=f_d'β+BI, B=f_d−xf_d', T=(β−xI)Φ and V_a=α_aΦ. Freezing these coefficients at a point and retaining both Fourier wavevectors gives a Hamiltonian matrix with both ordinary and conjugate-linear current responses. Its coupled range is exactly finite: span the matrix source columns and their images under α_k and β. The orthogonal complement is self-adjoint and uncoupled for this frozen operator. Explicit projectors avoid discarding directions when c is tiny.

The reduced complex dimensions are 4,12,16,20,24,40 across occupied depths, compared with full matrix dimensions 4,16,16,64,64,1024. Rotation controls verify the same frozen growth spectrum.

At zero frozen wavevector, let λ=h/c and r₀=(ωR_d)/λ. For d≥3 the vector-current block has squared-frequency trace and determinant

$$\operatorname{tr}=2r_0(1+x^2),\qquad \det=c^4(2r_0-1)$$

in λ units. Thus r₀<1/2 gives a negative squared frequency in this frozen block. Depth two has overlapping Clifford vector channels and the distinct polynomial z²+2r₀xz+c²(1−2r₀); its imaginary-root magnitude is √[(1−r₀)²c²−r₀²] on these preparations. Exact symbolic determinants and projections from the complete frozen matrix verify both.

These are not instability theorems for the actual sphere solutions. The zero-wavevector growth is of order h/R_d, the same scale as the spatial background and connection terms that freezing omitted. Those terms cannot consistently be dropped to infer a physical electron instability. Conversely, bounded retained-pair motion does not prove their cancellation in the full spatial problem.

The sampled nonzero frozen wavevectors k/λ=0.01,0.03,0.1,0.3,1,3,10 show no imaginary frequency above the conservative floating resolution 3×10⁻⁷ in λ units. These finite probes are not a continuum stability proof. Tiny floating roots near multiple eigenvalues are kept separate from the exact zero-wavevector formulas.

Reproduction and Evidential Scope

inverse-current-ratio-checks.py and its report pass 558 evaluations across twenty-three conditions at eighty-digit precision. They retain actual spatial kernels, arbitrary complex local variations, exact rational clocks and complete pair Jacobians for all fourteen assignments. Maximum full spatial residual relative to the selected clock is 1.7×10⁻⁵³; the arbitrary variation discrepancy is below 2.1×10⁻³⁷. The first exact-check attempt failed a Python Fraction/mpmath comparison before spatial calculation; exact equality fixes that type error. Its source and failure record are retained.

inverse-current-frozen-symbol.py and its report pass 450 evaluations across thirteen conditions. They cover 112 assigned frozen spectra and 48 axis controls, complete source-range closure, Hamiltonian symmetry, independently projected low-wavevector blocks and their exact determinants.

The results give a concrete local inverse benchmark with exact energy and frequency, and specify the persistence calculation still needed. They do not derive the count input, sector units, occupied-index selection, one master constitutive law, particle localization, full spatial stability, observed spin/charge or complete particle inertia. No new assigned-depth preference or independent support for the sector case is claimed.

Actual Spatial Persistence Beyond the Pair

The complete rotational spatial calculation retains the sphere connection and all current derivatives. An exact ground-shell harmonic has growing oscillations; at depths four, five, six and ten that block is unstable throughout the open mixing chart. This does not establish instability of the assigned excited electron, muon or tau. Their finite spatial projections remain unconverged. The original pair persistence theorem retains its stated invariant-family scope.