Source frequencies in spatial coupling

Summary: The full field source, spatial observables and changing mode patterns carry different temporal frequencies, whose relation to normalized response determines whether a slow bound oscillation weakens coupling.

Identify the clock carried by the interacting spatial pattern

A low bound oscillation frequency can affect coupling only through the source actually sampled by the other object. In particular, the frequency in a spatial susceptibility need not equal the phase frequency of the unperturbed bound field. The distinction can be established directly from the written source, without assigning a weak coefficient to depth five.

All fields below live on the one nested manifold of ordinary macroscopic spatial dimensions. Mode indices and amplitude components describe fields over that space. They introduce neither further spatial directions nor interaction particles. A finite expansion illustrates the source algebra; it is an exact solution of the nonlinear field equation only when its profiles and amplitudes also satisfy that equation.

Bilinear sources carry transition frequencies

For normalized spatial modes, write

$$\Psi(t,x,y)=\sum_j c_j\Phi_j(x) e^{i(k_jy-E_jt)}. \tag{1}$$

The normal coordinate $y$ is an existing containing spatial direction. For any time-independent Hermitian source matrix $A(x)$,

$$\boxed{\rho_A=\Psi^\dagger A\Psi =\sum_{i,j}\bar c_i c_j\Phi_i^\dagger A\Phi_j e^{i[(k_j-k_i)y-(E_j-E_i)t]}.} \tag{2}$$

The transition density includes both spatial polarizations, their relative phase and the overlap required by the adopted nested inclusion. Integrating it against a spatial kernel preserves its temporal frequency. Its contribution can vanish geometrically even when that frequency is allowed. The complete spatial vertex is derived in Transition-density coupling.

For a single stationary field $e^{-iMt}\phi(x)$, every such bilinear is stationary. This applies to its scalar density and ordinary Dirac current. For a superposition, off-diagonal terms can describe actual moving spatial patterns. Their frequencies are differences, rather than a common phase.

A pure occupied mode can still have nonzero transition matrix elements to other modes. The absence of a pre-existing off-diagonal mean density is not the absence of a response to an applied source.

The complete nonlinear field source retains a carrier

A bilinear density is only one factor in the scalar Dirac source. For $\mathcal N[\Psi]=(K\rho_\beta)\beta\Psi$, the complete term has

$$\boxed{\Omega_{i;j\ell}=E_j+E_\ell-E_i, \qquad q_{i;j\ell}=k_j+k_\ell-k_i.} \tag{3}$$

Here $K$ is a time-local spatial kernel; a specified kernel with temporal memory would add its own response. The source coefficient includes the full cubic spatial overlap, as derived in Dirac source linearization. For a single frequency,

$$\mathcal N[e^{-iMt}\phi]=e^{-iMt}\mathcal N[\phi]. \tag{4}$$

Thus cancellation of the phase from $\Psi^\dagger\beta\Psi$ does not make the full nonlinear field source time independent.

A linear output-mode projection of $\Psi$ samples (3); a photon realized as a bilinear collective observable samples (2) and its tangent response. The physical output map must decide which construction applies. They cannot be identified solely because each contains a small frequency.

The principal clocks can be recorded without conflating them:

The last row is an exact nonlinear construction in Two-frequency spatial Dirac currents. It supplies a real spatial clock tied to the bound frequency; its amplitude and coupling still need to be calculated.

Slow inter-mode clocks can survive large normal momentum

For the normal Dirac branch, let $E_i(k)=\sqrt{M_i^2+k^2}$ in units $\hbar=c=1$. At the same normal momentum,

$$\boxed{E_j(k)-E_i(k) =\frac{M_j^2-M_i^2}{E_j(k)+E_i(k)}.} \tag{5}$$

It equals $M_j-M_i$ at rest and approaches $(M_j^2-M_i^2)/(2|k|)$ at large $|k|$. Therefore a hierarchy of small bound frequencies can provide slow transition clocks even for energetic propagation. The frequency relation alone does not supply the required spatial transition matrix element or a physical neutrino assignment.

Other transitions have different clocks. Within one branch,

$$E(k+q)-E(k) =\frac{2kq+q^2}{E(k+q)+E(k)}. \tag{6}$$

At fixed nonzero forward momentum transfer, this need not vanish as $M$ tends to zero. The complete interaction must retain such channels when its source permits them.

The already derived polarization wave has $\omega(k)=\sqrt{M^2+k^2}-M$. Consequently

$$\frac{\partial\omega}{\partial M}\bigg|_k =\frac{M}{\sqrt{M^2+k^2}}-1<0\quad(k\ne0). \tag{7}$$

Reducing the bound frequency at fixed normal wavelength increases this particular fluctuation frequency. Scaling the wavelength together with $M$ is a different comparison. The scalar moment source of these orthogonal polarization waves has zero first variation at every depth where the stated construction applies.

A source phase $\Omega t-qy$ sampled by a target moving with speed $v$ in that direction has proper-time frequency $\gamma_v(\Omega-vq)$. Both the temporal and spatial phase are needed when comparing clocks in different preparations.

Include the response scale and normalization

The frequency barriers are expansions about zero forcing frequency with fixed profiles and regular response functions. An amplitude proportional to $\omega$ or $\omega^2$ is small relative to the corresponding spectral and source scales. Those coefficients can change when the particle's bound scale changes.

The zero-bare-mass scalar moment branch gives an exact control. Let $D_\lambda\phi(x)=\lambda^{d/2}\phi(\lambda x)$. For unit spatial norm, dilating a complete bound solution gives

$$\phi_\lambda=D_\lambda\phi, \quad M_\lambda=\lambda M, \quad a_\lambda=\lambda^3a, \quad g_\lambda=\lambda^5g. \tag{8}$$

The scalar second moment scales as $\lambda^{-2}$. A quadrupole source column $b_A=\beta(x^TAx)\phi$ therefore scales as $\lambda^{-2}D_\lambda b_A$, while the full fluctuation operator scales as $\lambda D_\lambda L D_\lambda^\dagger$. Both ordinary and conjugate resolvents yield

$$\boxed{\chi_\lambda(\omega) =\lambda^{-5}\chi_1(\omega/\lambda), \qquad g_\lambda\chi_\lambda(\lambda z)=g\chi_1(z).} \tag{9}$$

This holds for the complete self-consistent shape response, not just a bare overlap. A common change of bound scale does not suppress this dimensionless response at a fixed frequency ratio.

There is a corresponding fixed-$g$ statement for the single-frequency family of proper density $s=\lambda^5$. Its field has the additional factor $\sqrt{s}$. Each source column gains $\sqrt{s}$, cancelling the $\lambda^{-5}$ factor in the susceptibility. At fixed $g$,

$$\chi_s(\omega)=\chi_1(\omega/\lambda). \tag{10}$$

This is a density-family identity, not permission to vary an isolated particle's norm to fit its mass. A two-frequency background has additional periodic response coefficients and must be treated separately. Different physical particle scales could still provide a small frequency ratio; their selected profiles and normalization must establish it.

Conditions for the proposed neutrino mechanism

The proposed five-dimensional application requires a selected spatial mode whose actual source has no unsuppressed static contribution, followed by a derived slow source clock in the frame of its interaction partner. The full response must include both polarizations and all accessible ordinary, conjugate and spatial channels. An incoming mode also has to satisfy its own field equation; Incoming spatial shape modes checks that distinction explicitly.

A physical photon additionally has a massless propagating branch. Its outgoing Green function can have poles where a regular static susceptibility does not. The low-frequency power of a gapped shape response cannot by itself determine the normalized photon amplitude or its emitted or scattered flux.

V2's neutrino spectrum provides proposed mass indices and a conditional scale reconstruction. The projected mass-operator construction still leaves the depth-five normalization to be derived. Neither the frequency rules nor a chosen small normalization complete that task.

Finite Harmonic Occupation Gives a Fundamental Density Beat

The sawtooth sector calculation takes the maintained mass-frequency $K\omega_{0,d}$ as its supplied upper harmonic, with $K=S(n,d)$. For the real local preparation $f_K(\theta)=\sum_{r=1}^K\sin(r\theta)/r$, adjacent harmonics give the exact density coefficient $c_1=1-1/K$ at the sector fundamental. A complex positive-frequency local evaluation gives the same coefficient under equal mean-source normalization while retaining a constant global mode norm.

Its spatial source still has to contain the corresponding matrix elements in (2). A point evaluation can retain these beats; an integrated identity on orthogonal modes can remove them. The temporal coefficient alone is not the physical vertex.

The calculation distinguishes a direct density action from a velocity-source action and applies one common shape receiver across depths two, three, four, five, six and ten. Direct density response is not small at depth five. The derivative source has no static term and, below the receiver gap, gives

$$Q_K^{(1)}(r)\simeq\frac{13}{30}r^2\left(2K-H_K-\frac12H_K^{(2)}\right), \qquad r=\omega_{0,d}/\omega_{0,6}.$$

Here $H_K$ and $H_K^{(2)}$ are the first and second harmonic sums. The formula retains the upper-cutoff dependence as well as the fundamental. Its small depth-five value is conditional on the derivative action, common receiver scale and stated neutrino calibration. The same rule suppresses any sufficiently slow source, so it supplies no independent theorem selecting five spatial dimensions.

Verification

source_frequency_checks.py compares full complex spatial bilinears and a nonlocal cubic source with independent temporal Fourier transforms. It checks normal transition clocks, observer phases and complete scalar matrix elements. Independent spatial radial solves and both signs of the fluctuation resolvent verify (8)–(10). Reports preserve previous runs with timestamped filenames; all conditions remain active under Python optimization.

Related

• Dirac source linearization — fixes the complete field-source and rotating-background conventions.
• Transition-density coupling — supplies the two-sided spatial vertices accompanying the clocks.
• Two-frequency spatial Dirac currents — gives an exact changing current with a bound-frequency clock.
• Incoming spatial shape modes — distinguishes a free pattern from a forcing coefficient.