A canonical mode-field realization of the collective source

The collective geometric source has an exact realization in a single canonical coefficient field. The realization specifies the interaction through mode averages. This assumption makes its coherent family invariant and keeps all the additional oscillator modes under control at linear order.

The oscillator indices describe candidate modes of the same field. They are not extra physical dimensions or additional particles. Identifying these modes with the actual spatial carrier remains a separate derivation. In particular, a moment of a whole mode is not automatically a local interaction on the physical manifold.

The complete action

Let the finite carrier Hilbert space be $V$, with Hermitian frequency operator $H_0$ and two bounded Hermitian source matrices $O_j$. Retain one field

$$\Xi\in V\otimes\mathcal F_1\otimes\mathcal F_2, \qquad Q=\langle\Xi,\Xi\rangle>0, \qquad g_j=\frac{\langle\Xi,O_j\Xi\rangle}{Q}. \tag{1}$$

Here $\mathcal F_j$ is the usual oscillator mode space and $b_j$ its lowering operator. No occupation statistics or particle exchange is being assigned to this notation. The matrices $O_j$ act on $V$ and the $b_j$ on the oscillator factors.

For common positive $K,\Omega$, define

$$u=\frac1{\sqrt{2K\Omega}},\qquad A_j=b_j+u g_j, \qquad \boxed{E[\Xi]=\langle\Xi,H_0\Xi\rangle +\Omega\sum_j\|A_j\Xi\|^2.} \tag{2}$$

Use the canonical kinetic term $\tfrac i2\langle\Xi,\overleftrightarrow{\partial_t}\Xi\rangle$. The current-dependent displacement in (2) is a stated inverse kernel. It is homogeneous of degree two in $\Xi$; it is not a primitive quartic interaction with a discarded source norm.

Write $h_j=\operatorname{Re}\langle\Xi,A_j\Xi\rangle/Q$. The full field equation is

$$\boxed{i\dot\Xi=H_0\Xi+\Omega\sum_j \left[A_j^\dagger A_j\Xi+2u h_j(O_j-g_j)\Xi\right].} \tag{3}$$

The second term inside the brackets is required by variation of the current in $A_j$. Both $Q$ and the complete energy are conserved. Homogeneity gives $\operatorname{Re}\langle\Xi,i\dot\Xi\rangle=E$.

Exact frequency preservation

For a normalized carrier eigenvector $H_0\psi_n=\lambda_n\psi_n$, take the oscillator coherent states with amplitudes $\alpha_j=-u g_{j,n}$, where $g_{j,n}=\langle\psi_n,O_j\psi_n\rangle$. Then

$$\Xi_n=\psi_n\otimes|\alpha_1,\alpha_2\rangle, \qquad A_j\Xi_n=0. \tag{4}$$

Every added term in the energy and its first variation vanishes. Consequently

$$\Xi_n(t)=e^{-i\lambda_nt}\Xi_n(0),\qquad E_n=\lambda_n Q. \tag{5}$$

Mass is this selected frequency in equivalent units. Its value does not come from adjusting $Q$. The construction preserves whichever spectrum the stated $H_0$ actually has. Applied to the linked carrier it therefore preserves its finite-hopping corrections as well; it does not turn them into an exact simplex law or derive the carrier's mode assignments.

The collective kinetic term follows from the same field

On a product coherent state write

$$\alpha_j=\sqrt{I\Omega/2}\,a_j +\frac{i p_j}{\sqrt{2I\Omega}},\qquad I=K/\Omega^2. \tag{6}$$

Choose the coherent-state phase convention whose Berry one-form is $p\cdot da$. The product field's canonical one-form is then

$$\frac i2\psi^\dagger\overleftrightarrow{d}\psi +Qp\cdot da.$$

Thus $P=Qp$ is the geometric canonical momentum, including when the field normalization is varied. Equation (2) restricts exactly to

$$E=\psi^\dagger H_0\psi+ \frac{Q}{2K}|Ka+g|^2+\frac{|P|^2}{2IQ}. \tag{7}$$

This supplies the earlier collective norm factors from an explicit canonical field ansatz. The coherent family is also dynamically invariant: the complete generator obtained by expanding (3) is a sum of a state-dependent operator on $V$, a forced harmonic oscillator operator, and a scalar. Such evolution preserves product states, and a forced harmonic oscillator preserves coherent states.

More generally, even an entangled initial coefficient field evolves by a product of time-dependent unitary operators on its factors. The eigenvalues of its reduced density matrices, or Schmidt weights, are conserved. This strong restriction follows from the chosen moment interaction. It is a physical modeling assumption to assess, not a generic property of local field interactions.

All additional modes have a known linear response

Displace the oscillator basis so that (4) is $\psi_n\otimes|0,0\rangle$. Let $v_j=(O_j-g_{j,n})\psi_n$. The variation of each completed square is

$$\delta(A_j\Xi)=b_j\eta+ 2u\,\Xi_n\operatorname{Re}\langle v_j\otimes0,\eta\rangle \quad(Q=1). \tag{8}$$

Its active subspace consists of every other carrier vector tensored with the oscillator vacuum, together with the occupied carrier vector and one quantum in either oscillator. Its complete real Hessian is exactly the collective Hessian already derived from (7).

Every remaining mode is a spectator at linear order. Its rotating frequency is

$$\boxed{\nu_{l,k}=\lambda_l-\lambda_n+\Omega(k_1+k_2).} \tag{9}$$

It evolves by multiplication by a phase. Coincident frequencies, including zero, do not create a coupling between these spectators. Thus the proven finite collective stability result extends to the entire oscillator complement of this particular moment model. This statement includes both signs of every retained carrier frequency. It does not establish stability in the physical normal complement that has not been identified with these oscillator modes.

Equal restricted energies do not determine the full source

A superficially similar action replaces the product of moments by an operator product. Put $c=\sqrt{\Omega/(2K)}$, $X_j=b_j+b_j^\dagger$ and $N_j=b_j^\dagger b_j$. That comparison is

$$E_{\rm op}=\left\langle H_0+\Omega\sum_jN_j +c\sum_jO_jX_j\right\rangle Q +\frac Q{2K}\sum_jg_j^2. \tag{10}$$

Angle brackets in (10) are normalized field expectations. Its difference from (2) is precisely the covariance

$$E_{\rm op}-E=cQ\sum_j \left(\langle O_jX_j\rangle-\langle O_j\rangle\langle X_j\rangle\right). \tag{11}$$

It vanishes on product fields, but its first variation need not vanish there. At the proposed stationary product (4), the omitted full-field source is

$$R=c\sum_j (O_j-g_j)\psi_n\otimes|1_j;\alpha\rangle, \qquad \boxed{\|R\|^2=c^2\sum_j\operatorname{Var}_{\psi_n}(O_j).} \tag{12}$$

For the two-endpoint limit $H_0=D\sigma_z$, with source operators $\sigma_z,\sigma_x$, the variance sum is one. The product is therefore not an exact stationary state of (10). Solving that action's full nonlinear stationary equation at $K=16,\Omega=1/4$ gives a frequency increase $0.00442969434406$ at $D=1$, and $0.000262336796110$ at $D=15$. Refining the oscillator cutoff preserves these shifts.

An independent small-transverse-coupling expansion includes the complete longitudinal oscillator displacement. Its second-order coefficient is

$$\delta\omega^{(2)}=c^2\sum_{m=0}^{\infty} \frac{e^{-\mu}\mu^m/m!}{2D+2/K-\Omega(m+1)}, \qquad \mu=\frac2{K\Omega}, \tag{13}$$

away from a denominator resonance. The checked small-coupling solutions approach (13). The comparison demonstrates why restricting an energy to a coherent ansatz is insufficient: the source normal to that ansatz must also vanish. A local spatial current requires its own calculation, as developed in the locality tests.

Verification

moment_square_checks.py checks arbitrary complex variations, normalization, Schmidt invariance, the exact collective block, every retained spectator through two quanta per oscillator, and maintained simplex endpoint counts across depths 3, 4, 5, 6 and 10. Its 141-check report passes. The infinite spectator statement follows from (8)–(9), rather than from extrapolating that finite cutoff.

oscillator_parent_checks.py independently verifies (10), its full real-linear Hessian, stationary energy accounting and the expansion (13). Its 21-check report also records the complete signed response at three oscillator cutoffs for $D=1$. Those finite spectra do not supply an infinite-cutoff theorem for (10).