A stationary collective source for the spatial mass operator

The bounded normal carrier supplies an actual spatial operator and a nonzero force on its proposed geometry. A collective completion can now make that geometry stationary without shifting the operator's frequencies. Its full finite-dimensional linear response also satisfies a common stability criterion at every maintained particle index, including depths three and four.

The completion is an explicit inverse choice. It retains two geometric mode coordinates of the same proposed field and includes their energy and canonical response. It does not derive their stiffness or inertia from the master field, prove stability in the full spatial complement, or make the carrier's finite-hopping spectrum exactly proportional to the simplex count. Mass remains the selected particle frequency in equivalent units.

The spatial operator and its geometric source

Use the complete projected pair of the linked normal carrier, with normal winding $d$, maintained grade $N$, and

$$H_{N,d}^{(0)}=\frac1{C_d} \begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix},\qquad T_{ll}=\beta+l-\frac{d-1}{2},\quad T_{l+1,l}=\gamma h_l, \quad C_d=\frac{d!}{\gamma^{d-1}\prod_l h_l}. \tag{1}$$

For the positive surface branch, $\beta=N+1+(d-1)/2$. The negative branch is retained as well. The $h_l>0$ are integrals of the actual localized normal modes. The factor $C_d$ defines one common frequency unit at a depth; it is not adjusted for each particle.

The smallest positive frequency of (1) is $\lambda_{N,d}=\binom{N+d}{d}[1+O(\gamma^{-2})]$. The exact finite correction is retained in every calculation below. The normal winding equal to physical depth, the surface assignment and the controlled spatial projection remain the carrier's declared premises. All physical coordinates are ordinary and uncompactified.

The reciprocal spatial source shows that a constant-polynomial deformation produces the opposite endpoint entry $\delta T=\gamma\kappa_d\epsilon|0\rangle\langle d-1|$. In the common dimensionless coordinate $a=\gamma\kappa_d\epsilon/C_d$, its two real source operators are

$$E=|\text{lower},0\rangle\langle\text{upper},d-1|,\qquad O_1=E+E^\dagger,\quad O_2=i(E-E^\dagger). \tag{2}$$

Thus $H(a)=H^{(0)}+a_1O_1+a_2O_2$ is the retained linear geometric mode. The physical stiffness in $\epsilon$ must be transformed by this coordinate conversion; a coefficient in $a$ is not directly a measured spatial stiffness. The full nonlinear deformed normal operator also has higher-order terms not included in (2).

One complete square supplies both reciprocal sources

Let $\psi$ contain every canonical component of the finite pair, and define

$$Q=\psi^\dagger\psi>0,\qquad g_j=\frac{\psi^\dagger O_j\psi}{Q},\qquad r_j=K a_j+g_j.$$

Retain $a_j$ and its canonical momentum $P_j$, with common positive coefficients $K,I$. The Hamiltonian is

$$\boxed{ H_{\rm int}=\psi^\dagger H^{(0)}\psi +\frac{Q}{2K}\sum_{j=1}^2 r_j^2 +\frac{|P|^2}{2I Q}.} \tag{3}$$

Its canonical action is $\int [\tfrac i2\psi^\dagger\overleftrightarrow{\partial_t}\psi +P\cdot\dot a-H_{\rm int}]dt$. Eliminating $P$ gives the positive kinetic term $QI|\dot a|^2/2$. Carrying the geometric energy by the same canonical norm is a stated degree-two collective premise. It differs from adding a static geometric energy independent of the wave norm.

The square contains all three required terms:

$$\frac{Q}{2K}|Ka+g|^2 =\frac{KQ}{2}|a|^2+\sum_j a_j\psi^\dagger O_j\psi +\frac{1}{2KQ}\sum_j(\psi^\dagger O_j\psi)^2. \tag{4}$$

The last self-current term is part of this proposed kernel. It is not discarded when computing the frequency or the response. Direct variation gives

$$\begin{aligned} i\dot\psi={}&H^{(0)}\psi+ \sum_j\left[\frac{r_j}{K}O_j\psi+ \left(\frac{r_j^2}{2K}-\frac{r_jg_j}{K}\right)\psi\right] -\frac{|P|^2}{2IQ^2}\psi,\\ \dot a={}&\frac P{IQ},\qquad \dot P=-Q r. \end{aligned} \tag{5}$$

The normalization derivatives in the first line are required. These equations conserve $Q$ and the complete Hamiltonian. At fixed nonzero $Q$, finite-dimensional energy bounds keep $\psi,a,P$ bounded: the wave sphere is compact, $|g_j|\le1$, and the square and kinetic terms are positive. Smooth local solutions consequently extend for all finite times. This is global existence for the collective ordinary differential equations, not nonlinear orbital stability or a spatial field theorem.

The geometry is stationary and the frequency is unchanged

For any normalized positive eigenvector $H^{(0)}\psi_n=\lambda_n\psi_n$, set

$$\boxed{\psi(t)=e^{-i\lambda_nt}\psi_n,\qquad a_j=-g_{j,n}/K,\qquad P=0.} \tag{6}$$

Then $r=0$. The complete square, its wave source and its geometric source all vanish. The same action gives

$$H_{\rm int}=\lambda_n Q,\qquad \omega_n=\lambda_n. \tag{7}$$

The geometric equilibrium is allowed to differ between occupied branches; the action and its coefficients are unchanged. This resolves the earlier nonzero source within the retained linear geometric mode. It does not assert that the original radial geometry remains undeformed or that the full spatial source has been solved.

Under $\psi\mapsto b\psi$, $P\mapsto b^2P$, with $a$ fixed, the Hamiltonian scales as $b^2$. The frequency in (6) is independent of this normalization. No field magnitude is being used as mass.

Complete finite-band linear response

Remove the decoupled norm and common phase directions of $\psi_n$. Keep every other positive and negative eigenvector $\psi_l$, and put

$$\nu_l=\lambda_l-\lambda_n,\qquad b_{lj}=\langle\psi_l,(O_j-g_{j,n})\psi_n\rangle.$$

Write $\delta\psi=\sum_l\psi_l(q_l+ip_l)/\sqrt2$. Let $C$ have columns $\sqrt2(\operatorname{Re}b_j,\operatorname{Im}b_j)^T$. Rescale the geometric canonical coordinates by $a'=\sqrt{I\Omega_f}\,\delta a$, $p'=\delta P/\sqrt{I\Omega_f}$, where $\Omega_f=\sqrt{K/I}$. The complete real Hessian is

$$\mathscr H= \begin{pmatrix} \operatorname{diag}(\nu,\nu)+CC^T/K&C/(KI)^{1/4}&0\\ C^T/(KI)^{1/4}&\Omega_f I_2&0\\ 0&0&\Omega_f I_2 \end{pmatrix}. \tag{8}$$

The generator is $J\mathscr H$, with the canonical real symplectic matrix $J$. Equation (8) includes the feedback in both directions. Holding the wave fixed or omitting the self-current Hessian gives a different spectrum.

Eliminating the geometric position block in the Hessian gives exactly

$$\operatorname{diag}(\nu,\nu)+CC^T/K -\frac{CC^T}{\sqrt{KI}\,\Omega_f} =\operatorname{diag}(\nu,\nu). \tag{9}$$

Thus the full Hessian remains nondegenerate on the reduced space and has the same inertia as the uncoupled wave and positive geometric oscillators. Here “inertia” of a matrix means its numbers of positive and negative eigenvalues; it is not a particle's physical inertial mass.

A finite-dimensional spectral separation criterion

Turn on the coupling continuously: replace $C$ in the mixed blocks by $tC$ and $CC^T$ by $t^2CC^T$, for $0\le t\le1$. Equation (9) holds throughout. The uncoupled generator is normal in these canonical coordinates. Since $\|O_j\|\le1$, $\sum_j\operatorname{Var}_{\psi_n}O_j\le2$, and hence

$$\boxed{\|J(\mathscr H(t)-\mathscr H(0))\| \le\varepsilon=\frac4K+\frac2{(KI)^{1/4}}.} \tag{10}$$

The normal resolvent bound puts every perturbed eigenvalue inside the union of radius-$\varepsilon$ disks around the uncoupled imaginary frequencies. Let $\delta$ be the minimum distance between such frequencies whose uncoupled modes have opposite signs of quadratic energy. This includes the positive geometric oscillators and all positive and negative wave modes.

If $\delta>2\varepsilon$, every connected disk cluster contains only one energy sign. Its invariant spectral subspace retains a nondegenerate restriction of $\mathscr H(t)$: the conserved-form identity

$$(J\mathscr H)^\dagger\mathscr H+ \mathscr H(J\mathscr H)=0$$

makes separated clusters orthogonal in that form, and each disk cluster is invariant under reflection $z\mapsto-\overline z$. Degeneracy of the restricted form would therefore imply degeneracy of the full form, contradicting (9). Its sign cannot change continuously. It remains definite on each cluster, so the generator there is skew-adjoint in that definite norm. Its eigenvalues are imaginary and semisimple.

This proves bounded linear evolution on the reduced finite space under the stated criterion. It does not replace a spatial stability analysis of the discarded normal complement or higher surface disturbances.

For the single common choice

$$K=16,\qquad I=256,\qquad \Omega_f=1/4,\qquad \varepsilon=1/2, \tag{11}$$

90-digit spectra at $\gamma=10000$ satisfy the criterion for every maintained index at depths 3, 4, 5, 6 and 10, on both surface signs. The smallest separation is approximately $1.74999978779$, at the depth-three down control. All other separations are larger. The bottom entries remain its two component controls; a physical geometric-mean bottom branch has not been constructed by this calculation.

Equation (11) is an illustrative common region of stability, not a derived physical value or a predicted new internal particle frequency. Its units and the physical geometric stiffness remain to be fixed.

The surviving vertex is a transverse response

At equilibrium the complete linear wave vertex of a geometric displacement is

$$\boxed{b_j=(O_j-g_{j,n})\psi_n,\qquad \langle\psi_n,b_j\rangle=0.} \tag{12}$$

The old diagonal expectation $g_{j,n}$ sets the equilibrium deformation, but cancels from the linear phase vertex once the square's normalization terms are retained. It must not be reported as an unchanged coupling.

On the large-hopping endpoint branch, $g_1$ approaches $(-1)^{d-1}$ on the positive surface sign. Its variance tends to zero. The second operator has zero mean and variance approaching one. It couples mainly to the opposite small-frequency partner, whose rotating frequency is $-2\lambda_n$. Thus a nearly branch-independent transverse matrix element survives, while the associated transition frequencies differ.

In this two-endpoint limit the coupled squared frequencies satisfy

$$s^2-\left(4D^2+K/I-4D/K\right)s+4D^2K/I=0, \qquad D=\binom{N+d}{d}. \tag{13}$$

They are positive and real in the control region (11). This exact limit helps interpret the complete calculation; it does not substitute for the full finite-band test.

Quasistatic elimination of a free collective $a$ cancels the square's interaction. A time-dependent source instead gives a nonzero delayed response. Physical coupling between two spatial particles would require the reciprocal source and spatial gradient energy of that same geometric mode. No observed force, interaction rate or neutrino suppression is derived here.

Positive inertia in a stated collective Lorentz completion

There is a direct collective action for the center $X(t)$:

$$S=\int dt\left[ \frac i2\psi^\dagger\overleftrightarrow{\partial_t}\psi +P\cdot\dot a-H_{\rm int}\sqrt{1-|\dot X|^2}\right]. \tag{14}$$

The internal canonical one-form is unchanged; the internal Hamiltonian advances in proper time. On (6), canonical differentiation with respect to $\dot X$ gives

$$E=\gamma\hbar\lambda_nQ,\qquad \mathbf P_X=\gamma\frac{\hbar\lambda_nQ}{c^2}\mathbf v, \qquad M=\frac{\hbar\lambda_nQ}{c^2}>0. \tag{15}$$

For $Q=1$, this is the mass-frequency value. It includes the retained collective geometry's energy; no additional static energy was omitted. Equation (14) is an explicitly stated collective Lorentz completion. It is not yet the stress tensor of a derived local master-field solution on the full curved spatial carrier.

Reproducible calculations

completed_geometry_source.py contains the full canonical gradient and Hessian. The 142-check report verifies independent full variations, stationary sources, joint scaling, the exact Schur cancellation and the two-endpoint limit.

completed_geometry_scan.py recomputes the actual normal-mode hopping integrals at two quadrature resolutions and evaluates the entire finite Dirac spectrum at 90 digits. Its 83-check report retains every maintained control and both surface signs. Raw double diagonalization would lose the small frequency in the deepest cases; the reported factorization and separation use the full high-precision matrix. Neither report changes the physical scope stated above.