Shell-Preserving Potentials with Full Spatial Persistence

Preserving a stationary mass-frequency does not require retaining every derivative of the earlier constitutive law. This distinction supplies a concrete inverse construction beyond the pair-preserving repair obstruction. One common formula retains all integer-shell backgrounds and their energy while changing their response to spatial disturbances. It uses no fitted field amplitude and no separate adjustment for an individual particle.

The successful persistence result below uses the preferred-time density $\rho$, rather than a Lorentz proper-current norm. It is a conditional mathematical construction on the earlier ordinary sphere, not a completed physical particle model. Known mass targets are legitimate inputs to this inverse investigation; preserving them is distinct from deriving the action or demonstrating observational agreement beyond those targets.

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.

1. Common Constitutive Coordinates

At physical depth $d$, use

$$P_d(h)=\frac1{d!}\prod_{j=1}^d(h+j-d/2),\qquad c(h)=\epsilon_dP_d'(h),\qquad x(h)=\sqrt{1-c(h)^2}.$$

Work on an open interval with $0<c<1$, $x>0$, and $P_d''>0$. The inverse coordinate $h=h_d(x)$ is then smooth. The original potential is specified parametrically by

$$f_d(x(h))=\epsilon_dP_d(h)-h c(h), \quad f_d'(x)=\frac{hx}{c}, \quad f_d''(x)=\frac h{c^3}-\frac{x^2}{c^2\epsilon_dP_d''(h)}. \tag{1}$$

Its maintained integer spatial shells have $h_l=l+d/2$, $l=n-1$. The clocks are $\omega_lR=\epsilon_dP_d(h_l)$. The same formula applies at every occupied depth and, within its constitutive chart, every integer shell.

2. A Smooth Correction Preserving Every Shell Source

Define

$$A_d(h)=\frac1{2\pi^2} \left[\epsilon_dP_d''(h) -\frac{h\epsilon_d^2P_d''(h)^2}{c(h)x(h)^2}\right],$$

and set

$$\widehat f_d(x)=f_d(x)+A_d(h_d(x)) \sin^2\!\left[\pi(h_d(x)-d/2)\right]. \tag{2}$$

Since $h'=-x/(c\epsilon_dP_d'')$, this coefficient is $-f_d''/[2\pi^2(h')^2]$. At every integer shell the sine and its first derivative vanish, while its second derivative with respect to $x$ is $2\pi^2(h')^2$. Consequently

$$\boxed{\widehat f_d(x_l)=f_d(x_l),\qquad \widehat f_d'(x_l)=f_d'(x_l),\qquad \widehat f_d''(x_l)=0.} \tag{3}$$

The correction changes the law between shells. It does not preserve the earlier complete pair dynamics, so the obstruction's stronger hypothesis does not apply. Smoothness is asserted on the stated open chart; neither its endpoints nor the zero-field vacuum receive an unstated extension. The integer sine is a constructed shell-locking prescription, not a derived physical selection mechanism.

3. Complete Spatial Linearization

Use the canonical first-order action with potential

$$U_d(\Phi)=\frac\rho R\widehat f_d(J/\rho),\qquad \rho=\operatorname{tr}\Phi^\dagger\Phi, \quad J=\operatorname{tr}\Phi^\dagger\beta\Phi. \tag{4}$$

The ordinary sphere's spatial Dirac operator $D_R$ is self-adjoint, $\beta$ is Hermitian and $\{D_R,\beta\}=0$. Its selected positive matrix shell $E_l$ has eigenvalue $h_l/R$. The earlier exact state is

$$\Phi_l(t,q)=e^{-i\omega_lt}(a_lI+b_l\beta)E_l(q), \qquad a_l^2=\frac{1+c_l}2,\quad b_l^2=\frac{1-c_l}2.$$

It has $J/\rho=x_l$, including where its density varies spatially. The full nonlinear source of (4), before restricting any disturbance, is

$$G(\Phi)=\frac1R\left[ \widehat f_d'(x)\beta+ (\widehat f_d(x)-x\widehat f_d'(x))I\right]\Phi.$$

Equation (3) preserves this source and hence every original stationary clock and background. Its zero second derivative removes both coefficient variations at the shell, for every complex matrix disturbance. In the co-rotating frame the complete linearized equation is

$$i\partial_t\delta\Phi=\mathcal A_l\delta\Phi, \qquad \boxed{\mathcal A_l=D_R+ \frac{h_lx_l}{Rc_l}\beta-\frac{h_l}{Rc_l}I.} \tag{5}$$

This is a complex-linear self-adjoint operator on its full Dirac domain. It includes every spatial angular representation and column polarization, not just the earlier radial family. Its full spectrum is

$$\frac1R\left[ \pm\sqrt{(m+d/2)^2+(h_lx_l/c_l)^2}-h_l/c_l\right], \qquad m=0,1,2,\ldots. \tag{6}$$

Every value is real. The linearized evolution is unitary and conserves its complete norm; zero eigenvalues are semisimple. This proves full linear norm persistence for all fourteen maintained assignments under (4). It does not prove nonlinear stability for the sine correction alone.

The potential remains homogeneous of degree two in $\Phi$. Stationary complete energy therefore retains $E_l=\omega_l Q$, with the original canonical action $Q$. No amplitude has been selected to fit that clock.

4. Smooth Tangent Patches Give a Nonlinear Persistence Variant

A slightly different common prescription makes the same conclusion exact in an open neighborhood. Choose one fixed smooth cutoff $\chi(u)$, equal to one for $|u|\le1/8$ and zero for $|u|\ge1/4$. Define

$$f_d^{\rm flat}(x)=f_d(x)+\sum_{l\ge0}\chi(h_d(x)-h_l) \left[f_d(x_l)+f_d'(x_l)(x-x_l)-f_d(x)\right]. \tag{7}$$

The supports are disjoint and locally finite; terms are included where their shell lies in the chart. This is one fixed rule, with the same cutoff widths at every depth. It has exactly the retained shell values and slopes. On each plateau it is affine in $x$, so (4) becomes the exact quadratic expression

$$R U_d=f_d'(x_l)J+ [f_d(x_l)-x_lf_d'(x_l)]\rho. \tag{8}$$

The field equation there is exactly linear, with (5), for disturbances of finite amplitude that remain inside the plateau.

For the selected zonal matrix shell, the scalar Gegenbauer polynomial and its derivative do not vanish simultaneously. Their second-order ODE would otherwise force the polynomial to vanish identically; its endpoint values are nonzero. Thus $E_lE_l^\dagger$ is positive at every point. Also $a_lI+b_l\beta$ is invertible for $c_l>0$. The complete background density consequently has a positive minimum on the sphere.

Take an integer Sobolev index $s>d/2$. The norm $\sum_{j=0}^s\|\mathcal A_l^j\delta\Phi\|_2^2$ is equivalent to the squared ordinary $H^s$ norm by elliptic estimates on this compact spatial manifold. Equation (5) preserves it. Sobolev embedding therefore keeps a sufficiently small initial disturbance uniformly small at every later time. The positive background minimum makes $J/\rho$ uniformly continuous in that neighborhood. Choose the initial norm so this ratio stays inside the plateau; the resulting unitary solution then remains there for all time and solves the exact nonlinear equation. This closes the persistence argument without a linearization approximation.

The standard embedding and elliptic estimates used here are given in Freed, Geometry of Dirac Operators, lemma 3.1.21, proposition 3.2.15 and exercise 3.2.20. The bounded smooth zeroth-order terms in $\mathcal A_l$ retain those estimates. Their application to the constructed plateau law is the argument above, not a claim supplied by that reference.

Equation (7) therefore supplies conditional nonlinear Lyapunov persistence near each retained state in these norms. It is expressly an inverse, smooth, generally nonanalytic law. It does not establish that nature uses flat constitutive patches, nor derive realistic interactions between different particles. The sine and flat-patch laws are alternatives, not one action to which both nonlinear conclusions can be transferred.

5. Physical and Evidential Scope

The successful action uses $\rho$, a density in a chosen time frame. It lacks the original potential's Lorentz spinor-boost invariance. Replacing $\rho$ by the proper vector norm restores additional current derivatives, so (5) cannot be carried over to that action. A covariant same-field time structure would need its own action and complete disturbance analysis; its fluctuations cannot be frozen silently.

These states occupy the conditional ordinary sphere. Persistence on that carrier does not establish a localized finite-energy object in unbounded space, complete translational inertia, observed spin statistics, electric charge, or observed lifetimes. This construction demonstrates that an inverse count clock can coexist mathematically with full spatial persistence. The recipe also works for other admissible monotone spectral functions, so it does not independently select IDWT's assigned dimensions.

Reproducibility

The common checker and report retain all fourteen assignments, one $\epsilon_d$ per sector, 140-digit constitutive derivatives, independent generic full complex matrix action variations, and the complete spectral formula. The reported numerical checks support equations (2)–(6); the nonlinear flat-patch conclusion also uses the analytical Sobolev and plateau argument given above.

The responding same-field clock calculation provides a Lorentz-scalar continuation with complete background variation and energy. Its full ground current spectrum includes a coexact channel missed by radial tests; one common small load has a conditional complete linear orbital persistence proof. This does not transfer the earlier nonlinear plateau proof or establish persistence of assigned excited states.