A reciprocal matrix-current source with real uniform frequencies

One conditional quartic action produces the cumulative frequency on a coherent uniform branch and admits a complete analytic test of its spatial linearization. A separate localized calculation is required before that branch can represent a particle. The retained matrix fields in this note are proposed modes of the one field, not additional particles. Their physical spatial origin, occupation and canonical normalization remain inputs to this inverse construction.

All spatial derivatives act on ordinary flat R^d, with even d. The coefficient spaces below do not replace these spatial dimensions. Set hbar = c = 1. The full local Dirac principal matrices are Hermitian, and beta anticommutes with every spatial principal matrix.

One action and its canonical weights

Let the partner rank be r = 2^(d/2), and define

$$F_q(C)^2=qr\operatorname{Tr}(C^2) +(1-q)(\operatorname{Tr}C)^2, \qquad \mathcal L_q(C)=qrC+(1-q)\operatorname{Tr}(C)I, \quad 0<q\leq1.$$

The identity has the full retained matrix size. The default comparison q = 1/(r+1) gives equal exchange and trace contractions, but invariance alone does not select it. Let R_k be the retained scalar current formed by the existing oscillator splitting isometries, summing target grades N >= k and tracing their discarded spectators.

Choose one fixed finite ceiling K. For each k <= K retain a canonical matrix field B_k carrying the same coefficient representation as R_k. Write

$$h_k=\binom{k+d-1}{d-1},\qquad C_k=R_k+\frac{B_k^\dagger\beta B_k}{h_k},$$

$$\boxed{U=\sum_{k=0}^{K} \left[\frac{g h_k}{2}F_q(C_k)^2 +m_B\operatorname{Tr}(B_k^\dagger\beta B_k)\right]}, \qquad g>0,\quad m_B>0. \tag{1}$$

Every target and B_k has a unit canonical kinetic coefficient. Equivalently, b_k = B_k/sqrt(h_k) has kinetic weight h_k; that weight must be retained in both its norm and field equation. The m_B term is a common proposed source gap, not a separately chosen particle mass. All interactions in (1) are local matrix-current terms. No independent scalar exchange field or Yukawa coupling has been supplied.

Variation of (1) gives the target source by lifting g h_k beta Psi L_q(C_k) through its original splitting map. The normal source is

$$\mathcal N_{B_k}=g\beta B_k\mathcal L_q(C_k)+m_B\beta B_k. \tag{2}$$

The same action therefore determines both responses, including the target self term. Its field source is polynomial and smooth at zero current. This does not prove global regularity of its spatial solutions.

The uniform coherent branch

Let z be a unit coefficient ray, and let P_z,k project onto z^k. Prepare an upper target at one grade N with scalar density a > 0, and each per-copy normal mode with density b > 0. Then

$$R_k=\frac a r I_r\otimes P_{z,k},\qquad \frac{B_k^\dagger\beta B_k}{h_k} =\frac b r I_r\otimes P_{z,k} \quad (k\leq N).$$

On the occupied ray L_q(C_k) acts as a+b, independently of q. The full phase equations give

$$\boxed{\omega_N=g(a+b)D_N},\qquad D_N=\sum_{k=0}^Nh_k=\binom{N+d}{d}, \qquad \omega_{B_k}=m_B+g(a+b). \tag{3}$$

For k > N, the normal phase is m_B+gb. The unoccupied target equations vanish. Thus the different occupied grades belong to one finite action and the same initial normal preparation. The weighted normal density is b sum_{k=0}^K h_k for all the compared grades.

These a and b are uniform local densities, not finite global particle norms. On infinite spatial volume this background has infinite norm. Taking equal positive occupation to infinite coefficient order also has infinite norm independently of spatial volume. Equation (3) does not select the background, its physical carrier, its ceiling or its weights. It proves the Hockey-Stick frequency conditional on them.

Complete oscillator sectors outside the occupied ray

Choose a coefficient basis with z = e_0. This is a change of mathematical components, not a choice of compact spatial geometry. Give an oscillator monomial of degree n the nonnegative charge j = n - occupation(e_0). Simultaneous phase rotation of every coefficient direction orthogonal to e_0 leaves the action and the prepared state invariant.

For the ordinary Hessian A and the conjugate Hessian C, invariance gives

$$[J,A]=0,\qquad JC+CJ=0. \tag{4}$$

All charges are nonnegative, so C has support only at j = 0. The spatial Dirac operator acts on the spinor index and preserves j. Every positive-charge block is consequently an ordinary Hermitian evolution operator, with real Fourier frequencies. This retains the oscillator components outside the coherent ray; it does not discard them.

The full partner block has a positive squared-frequency form

At j = 0 retain all partner matrices, upper and lower spinors, and both ordinary and conjugate variations. There is one trace block and r^2-1 traceless blocks. Label the canonical species A,B_0,...,B_N, with

$$Q=(a,h_0b,\ldots,h_Nb),\qquad G_{AA}=gD_N,\quad G_{Ak}=G_{kA}=g,\quad G_{kk}=g/h_k, \tag{5}$$

and all other normal cross entries zero. This symmetric matrix is positive semidefinite because

$$x^\dagger Gx=g\sum_k h_k|x_A+x_k/h_k|^2.$$

Define W = diag(m_i + sum_j G_ij Q_j), with m_A = 0 and m_k = m_B, and K = sqrt(Q) G sqrt(Q). In a traceless block replace K by qK. For the trace block the exact identity

$$x^\dagger(W-K)x =\sum_i m_i|x_i|^2 +\frac12\sum_{ij}G_{ij} |\sqrt{Q_j}x_i-\sqrt{Q_i}x_j|^2 \tag{6}$$

shows that V = W-K is strictly positive: every normal gap is positive and every normal species connects to the target. The same conclusion holds for qK with 0 < q <= 1.

After a unitary lower-spinor transformation, the squared Fourier frequencies are the eigenvalues of

$$M(p)=\begin{pmatrix} p^2I&-2pW\\ -2pV&p^2I+4W^2 \end{pmatrix}. \tag{7}$$

H = diag(V,W) is positive and HM is Hermitian. More precisely,

$$\begin{pmatrix}x\\y\end{pmatrix}^{\!\dagger} HM\begin{pmatrix}x\\y\end{pmatrix} =(px-2Wy)^\dagger V(px-2Wy) +y^\dagger(p^2W+4WKW)y. \tag{8}$$

For p != 0 this is strictly positive. All squared frequencies are therefore positive and all Fourier frequencies are real. At p = 0, the phase and density block can have secular phase drift; no exponentially growing mode appears. Equation (8) is not a proof of nonlinear stability, uniform-in-time perturbation bounds, or stability of a nonconstant core. It also works for arbitrary positive h_k, so stability does not derive their binomial values.

The complete spatial density response

Consider the normal background by itself, driven by a small target scalar density delta sigma_A. Put

$$\omega_0=gb,\qquad \mu=m_B+\omega_0,\qquad \kappa^2=4\mu\omega_0.$$

Eliminating both lower Dirac partners and both conjugate frequency components gives

$$\Delta(\Omega,p)=(\Omega^2-p^2)^2-4\mu^2\Omega^2+\kappa^2p^2,$$

$$\frac{\delta\sigma_B}{\delta\sigma_A} =-\frac{\kappa^2p^2}{\Delta(\Omega,p)},\qquad \boxed{\frac{\delta(\sigma_A+\sigma_B)}{\delta\sigma_A} =\frac{(\Omega^2-p^2)^2-4\mu^2\Omega^2}{\Delta(\Omega,p)}}. \tag{9}$$

The actual normal poles are real:

$$\Omega_\pm^2=p^2+2\mu^2 \pm2\mu\sqrt{\mu^2+(1-\omega_0/\mu)p^2}. \tag{10}$$

At zero drive frequency and nonzero momentum, (9) reduces to

$$\delta(\sigma_A+\sigma_B) =\frac{p^2}{p^2+\kappa^2}\delta\sigma_A. \tag{11}$$

In position space this is the screened spatial response (-Laplacian)(-Laplacian+kappa^2)^(-1). This operator was derived from the declared Dirac equations. Its static use does not replace the causal time-dependent response (9), introduce particle exchange, or establish a physical IDWT coupling. A local Dirac equation with these smooth sources has finite propagation during smooth existence.

At nonzero p the complete source in (9) has zeros at Omega = sqrt(p^2+mu^2) +/- mu. They are cancellations in this specified density channel, not electric neutrality or a neutrino prediction. At p = 0, a conserved uniform normal norm cannot acquire the static density change obtained by first taking Omega = 0 and then p -> 0. These limits do not commute. The unforced lower-state singularity at Omega = 2mu must also be handled using the full equations.

The small-momentum response speed is sqrt(omega_0/mu), while the static inverse length is kappa. At fixed omega_0 and m_B, increasing b and decreasing g leaves both unchanged. It reduces the local target correction relative to the background, but does not derive the background or prove that finite-particle coupling survives that limit.

Verification and remaining spatial work

matrix_hierarchy_checks.py verifies every finite oscillator component, both Hessians, their charge identities, the complete partner decomposition, the positive form, and independent energy and source variations. Its 20260913T204251Z report passes 168 checks through depth-two degree three. matrix_higher_depth_checks.py retains all 8192 complex components at depth six, degree two, without a dense Hessian. All 1041 checks pass in its 20260913T210031Z report, including every zero-charge column.

matrix_screening_checks.py verifies the full driven spinor equations and complex susceptibility at depths two and four. Its final 20260913T204516Z report passes 208 checks. These numerical controls support the stated analytic identities; they do not supply the missing physical premises.

A nonconstant target makes the local total current vary. Its background must solve its own spatial equation, and its complete energy, canonical norm and inertial response must be retained. The current localized calculation uses the same smooth binding law as the earlier root action and solves for the responding background. That is a separate conditional construction, still requiring independent matrix checks and full stability.

Related research

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