A complete finite normal band

A finite sequence of normal modes can be a complete invariant band of an ordinary spatial operator. It need not be obtained by throwing away the end of an oscillator tower. The construction below produces the curvature simplex product in such a band and gives it a local, first-order, canonically normalized completion.

This is an inverse construction. Its oscillator background, component couplings and selected normal degree are inputs. The calculation does not derive the supporting field, choose the particle bands, or establish their complete inertia. Mass remains the selected frequency.

Use both ordinary normal coordinates

Let a proposed surface feature have two ordinary normal coordinates y_1 and y_2, with one common profile width a. These coordinates are part of the containing physical space, not compactified directions. Set x_i=y_i/a and define

$$a_i=\frac{\partial_{x_i}+x_i}{\sqrt2},\qquad N_i=a_i^\dagger a_i,\qquad N=N_1+N_2. \tag{1}$$

Here the symbol a_i denotes an operator; the unindexed a denotes a length. The normal fields are the complete normalized Hermite functions in both coordinates. The eigenspace N=q is

$$\mathcal K_q=\operatorname{span} \{\phi_n(x_1)\phi_{q-n}(x_2):0\le n\le q\}. \tag{2}$$

It has dimension q+1. Both J_z=(N_1-N_2)/2 and J_+=a_1^\dagger a_2 preserve it, as do their adjoints. These are operators on spatial oscillator functions. The representation dimension is not a number of additional physical dimensions.

For the surface spinor operator B_K already derived in A spatial spinor parent for the curvature frequency, use a common frequency unit nu=c_t/R. Its dimensionless positive eigenvalue is beta=L+(d-3)/2. On the selected normal band consider

$$T_q=\beta+J_z+\gamma J_+, \qquad H_q=\begin{pmatrix}0&T_q\\T_q^\dagger&0\end{pmatrix}. \tag{3}$$

All components have the same canonical time coefficient. In the basis (2), T_q has diagonal beta+n-q/2 and lower diagonal gamma sqrt(n(q-n+1)). The whole Hermitian operator preserves (2). This supplies the adjoint closure missing in the single-oscillator cutoff test in Restoring the normal oscillator tower.

The frequency product and its normalization

The smallest positive frequency is the smallest singular value of T_q. At fixed q and beta, its exact triangular inverse gives

$$\omega_q=\frac{\nu}{\gamma^q q!} \prod_{n=0}^{q}\left(\beta+n-\frac q2\right) \left[1-\frac{(\beta-q/2)^2+(\beta+q/2)^2} {2q\gamma^2}+O(\gamma^{-4})\right]. \tag{4}$$

This formula assumes positive diagonal entries, as in the assigned controls. It follows by scaling the inverse by its largest corner entry; the two adjacent entries supply the first correction to its norm. The full singular vector, rather than only an end component, supplies the kinetic normalization.

Choose q=d-1. Then beta+n-q/2=L-1+n, and

$$\boxed{\omega_L\sim\frac{d\nu}{\gamma^{d-1}} \binom{L+d-2}{d}.} \tag{5}$$

The factorial q! in (4) is fixed by ordinary Hermite normalization. It changes the frequency unit from that of an equal-link chain. It does not change the simplex polynomial. The choice q=d-1 is still a physical selection requirement, rather than a consequence of (1).

Every vector in K_q has integral (x_1^2+x_2^2)|psi|^2=q+1 at unit norm. Thus its root-mean-square normal extent is a sqrt(q+1), independent of its surface harmonic. The finite completion below has a small additional occupation in K_(q-1); that correction is retained.

Complete the second-order normal operator

Equation (3) contains second normal derivatives. To test a local first-order completion, retain a target two-component field psi and two additional two-component mode fields eta_1, eta_2. Each component also carries the complete surface spinor. They are proposed components of the same field, not exchanged particles.

Write A psi=(a_1 psi,a_2 psi), and order the four additional components as (1x,1y,2x,2y). Define the Hermitian matrix

$$C=\begin{pmatrix} 0&1/2&0&\gamma\\ 1/2&0&0&0\\ 0&0&0&-1/2\\ \gamma&0&-1/2&0 \end{pmatrix}. \tag{6}$$

In dimensionless frequency units, the complete linear equation is

$$i\partial_t\begin{pmatrix}\psi\\\eta\end{pmatrix} =\nu\begin{pmatrix} \tau_x RB_K&\sqrt M A^\dagger\\ \sqrt M A&-M C^{-1} \end{pmatrix} \begin{pmatrix}\psi\\\eta\end{pmatrix}. \tag{7}$$

The dimensionless M is one common additional-component frequency scale divided by nu. The canonical norm includes both psi and eta. Eliminating eta at dimensionless frequency w gives the exact target operator and its kinetic metric

$$\begin{aligned} H_{\rm red}(w)&=\tau_x RB_K+ A^\dagger C(I+wC/M)^{-1}A,\\ Z(w)&=I+\frac1M A^\dagger C^2(I+wC/M)^{-2}A. \end{aligned} \tag{8}$$

The second expression is positive and equals the full eliminated field norm. Its correction must not be discarded when comparing frequencies or interaction coefficients. At large M, the first expression reduces to (3).

The conserved graded normal number is N on psi and N+1 on eta. Consequently the complete band is psi in K_q and eta_i in K_(q-1). Its dimension is 2(q+1)+4q=6q+2, multiplied by the actual surface spinor rank. A and A^dagger map these entire spaces into each other. No unretained normal component is generated by either equation.

Check all spatial propagation directions

The normal derivatives in (7) are first order. Their symbol has norm at most nu a sqrt(M/2)|p_normal|. The tangential symbol has norm c_t|p_tangent|. With epsilon=a/R, a sufficient total speed bound is

$$v_{\max}\le c_t\sqrt{1+M\epsilon^2/2}. \tag{9}$$

Choosing c_t no larger than c divided by the square root in (9) therefore bounds every spatial direction by c. All spatial components, including the additional ones, enter this estimate.

This is a Hermitian first-order parent with a finite propagation bound. Its principal matrices are not a full irreducible ambient Dirac Clifford representation. Equation (7) is therefore not yet a derivation from the master Dirac equation, nor a proof of Lorentz covariance. The oscillator coordinate terms are a specified background profile requiring its own same-field origin and stress calculation.

Numerical controls and the remaining selection

schwinger-normal-parent-20260914T033434_202537Z.json passes 128 checks. Direct Hermite derivatives verify both normal equations and both adjoints. Complete spatial quadrature verifies their canonical norm and extent. Independent 100-digit singular values check the full first-order parent and its complete eigenfields.

At gamma=10000 and M=10^18, the assigned dimensionless frequencies are approximately 1.113839450877e-15 at q=5, L=14; 2.303021493953e-13 at q=5, L=36; and 6.451218434029e-28 at q=9, L=24. These numbers use one declared dimensionless normal hierarchy. A physical frequency unit remains to be supplied.

All normal bands remain present in the complete equation. In particular, when q has the parity of 2 beta and q>=2 beta, a diagonal entry of T_q vanishes and the complete Hermitian band has zero modes. Thus conservation of a selected q is not a derivation of its occupation or isolation under arbitrary perturbations.

The ordinary Euclidean tube also changes B_K with the radial coordinate. It therefore mixes q bands. Curvature of the complete normal band tests that specific geometric correction and retains its complementary modes. The origin of the normal background, selection q=d-1, relation of frequency units across depths and complete particle inertia remain open.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.