Spatial compatibility and reciprocal adjustment under the mass current

The homogeneous current law gives more than isolated frequencies. Applying that same local action to two spatial patterns produces a reciprocal response that depends on their relative phase and physical polarization. Holding both original patterns fixed can greatly overstate the cost of their overlap: a small adjustment of the faster field components reduces that cost substantially.

The later stability refinement is essential to the result: the present depth-five moving trial retains a downhill fast direction. Lowering the overlap energy has not yet produced a stable coupling reduction.

The calculations here are conditional tests on the stated geometry. They evaluate two-mode fields on one round reference geometry. Constructing two spatially separated particle structures and following their encounter requires an extension of this calculation. They do not yet give a scattering trajectory, a cross-depth vertex, observed force strengths, or neutrino neutrality. The frequencies remain the mass variables throughout; amplitudes enter through canonical phase normalization and the coupled field equations.

Use the same action for one pattern and for a pair

Let $\psi_a=\phi_a\otimes u_a$ and $\psi_b=\phi_b\otimes u_b$ be normalized individual spatial eigenfields, with $F[\psi_i]=\omega_i\psi_i$. Different spatial degrees are orthogonal, including after physical rotation of either field. Fix each isolated canonical phase charge to one:

$$\Psi_0=\frac{\psi_a}{\sqrt{2\omega_a}} +e^{i\theta}\frac{\psi_b}{\sqrt{2\omega_b}},\qquad \Pi_0=-i\sqrt{\frac{\omega_a}{2}}\psi_a -i e^{i\theta}\sqrt{\frac{\omega_b}{2}}\psi_b. \tag{1}$$

The complete initial energy and its excess are

$$E_{\rm pair}=\|\Pi_0\|^2+\|F[\Psi_0]\|^2,\qquad \Delta E=E_{\rm pair}-(\omega_a+\omega_b). \tag{2}$$

Equation (2) evaluates the full nonlinear source in physical space. It does not insert an independent pair potential or assume that a superposition remains a solution. A common rotation of the two fields and their geometry preserves this action.

The model is reciprocal because both patterns belong to the same varied field functional. This establishes a common source law; a dynamical encounter still requires their coupled evolution.

Interference can change the current channel

For normalized component rays, define

$$k_a=\phi_a^\dagger K\phi_a<0,\qquad k_b=\phi_b^\dagger K\phi_b<0,\qquad k_{ab}=\phi_a^\dagger K\phi_b. \tag{3}$$

At a point, write the spatial spinor directions as $\zeta_a$ and $\zeta_b$, with overlap $s=\zeta_a^\dagger\zeta_b$. For local complex coefficients $a,b$, the current is

$$j=k_a|a|^2+k_b|b|^2 +2\operatorname{Re}(\bar a b\,k_{ab}s). \tag{4}$$

Although both individual currents are negative, the pair current can be positive in some interference regions. The exact condition that prevents this for every $a,b$ is

$$\boxed{|s k_{ab}|^2\le k_a k_b.} \tag{5}$$

It is the negative-semidefiniteness condition for the full two-ray current matrix. A signed current is not particle inertia.

For $\eta\to0$, the source coefficient approaches $G=-g\operatorname{sign}(j)$. Thus interference can change the local component coupling substantially, even when regularization shifts each isolated frequency only slightly. The zero-regularizer limit is nonsmooth at $j=0$; it is not a globally smooth replacement for the finite-$\eta$ action.

A physical rotation gives a calculable compatibility threshold

For the actual highest-weight spinor, let $(\Gamma_1+i\Gamma_2)\zeta=0$. Rotate its oscillation plane toward the third coordinate by an angle $\alpha$, transporting the spinor with the same ordinary spatial rotation:

$$\zeta_\alpha=e^{\alpha\Gamma_3\Gamma_1/2}\zeta,\qquad \left(\cos\alpha\,\Gamma_1+ \sin\alpha\,\Gamma_3+i\Gamma_2\right)\zeta_\alpha=0. \tag{6}$$

Direct Clifford differentiation gives

$$\zeta^\dagger\zeta_\alpha=\cos(\alpha/2). \tag{7}$$

Combining (5) and (7), the sign-changing channel is excluded when

$$\boxed{ \alpha\ge\alpha_c, \qquad \alpha_c=2\arccos\!\left[ \min\left(1,\frac{\sqrt{k_a k_b}}{|k_{ab}|}\right)\right]. } \tag{8}$$

This follows from the actual polarization transport. It is not a chosen neutral projector or an inference from shared spatial support alone. Beyond a strict threshold, the saturated negative-current law is linear on the pair span. Its two-frequency superposition then has no nonlinear interaction within that channel. Finite $\eta$ retains a smaller response through the varying magnitude of the current, so (8) is not a general zero-interaction theorem.

At the illustrative common strength $gR/c=100$, using the same within-depth regularizer as the mass calculation, selected thresholds are approximately:

The depth-ten second index is an unassigned mathematical control. It is not a proposed additional particle. These angles describe this current law and this family of spatial patterns, not measured cross sections.

Depth-two forms have a different physical bound

The bosonic form fields use their own rotation law. For the electric family at index $n$, write

$$a_n=\sqrt{\frac{n+1}{2n+1}},\qquad c_n=\frac{n}{\sqrt{(2n+1)(n+1)}}. \tag{9}$$

After reversing the oscillation plane by an ordinary half-turn, the local form overlap, with scalar harmonic factors removed, is $a_na_m-c_nc_m$. Its normalized magnitude is bounded everywhere by

$$\boxed{ |s_{\rm form}(x)|\le \frac{|a_na_m-c_nc_m|} {\sqrt{(a_n^2+c_n^2)(a_m^2+c_m^2)}}. } \tag{10}$$

For the W/Z indices this bound is about 0.01267. It is below the current threshold in (5), so that actual form configuration also excludes the sign-changing channel. The scalar and vector companions are both included. The spinor half-angle in (7) is not being assigned to a boson. This remains a massive-mode polarization control, not a photon derivation.

A fixed superposition can be a poor encounter state

For coaxial patterns with the same spinor polarization, direct evaluation of (2) gives a large positive overlap cost at the higher depths. At $gR/c=100$, the down/strange example costs roughly $4.3\times10^4$ times the isolated pair energy. In the tested fixed-ratio regularizer family, the unadjusted cost grows approximately as $g^{2d-2}$ in the large-gap regime.

This is an initial-state calculation. It does not establish that an actual pair must retain its original faster components throughout an encounter. A small change in those components can cancel a large source that was exposed by the overlap.

To test that possibility, retain a spatial trial space invariant under $H_0$, while evaluating $F$ in the full physical space. Hold the slow component coefficients fixed and vary the faster component directions:

$$\boxed{\min_{\delta\Psi_f}\ \|F[\Psi_0+\delta\Psi_f]\|^2.} \tag{11}$$

The fast directions are orthogonal to the two occupied rays, so the initial phase charges in (1) remain fixed in this static adjustment test. The norm in (11) includes the part of the nonlinear source outside the trial space. Projecting that source away before minimization would artificially reduce the cost.

The actual spinor basis includes negative azimuths and its full moving spinor frame. Depth two instead uses all four differential-form types. No extra spatial dimensions or independent interaction field are added.

In the first common trial space, the down/strange fast field changes by only about $9.5\times10^{-5}$ in relative norm, while its excess energy falls to about 6.52 times the isolated pair energy. Enlarging the spatial basis reduces that ratio to 5.61 and then 3.65. The up/charm control falls from about 1.46 to 0.864 when the spatial basis is enlarged. Thus the large reduction is established, but the remaining interaction has not reached a demonstrated infinite-space limit.

The larger trial keeps axial radial cutoff three and two sidebands on each side of the occupied pair. Its refined excess-energy ratios are 3.65, 0.864, 0.688, 0.504 and 0.209 at depths three, four, five, six and ten respectively. Independent integration uses 256 radial nodes and 384 phase nodes. The depth-five optimizer reached its iteration limit, and all these values remain dependent on the spatial trial space.

These ratios describe energy costs. They are not physical coupling constants obtained by dividing an interaction by mass, and they do not yet establish the observed grouping of responses by depth.

Include the actual motion of the adjusted pattern

For coaxial indices $n,m$, nonlinear spatial response generates azimuths $n+k(m-n)$. Phase covariance and spatial rotation force the corresponding frequency to be

$$\boxed{\omega_k=\omega_a+k(\omega_b-\omega_a).} \tag{12}$$

The fast correction therefore moves. If $c_k$ are its orthonormal spatial coefficients, the full kinetic and potential energies are

$$T=\sum_k\omega_k^2\|c_k\|^2,\qquad V=\|F[\Psi]\|^2. \tag{13}$$

The fast stationary functional at these prescribed frequencies is $V-T$, rather than $V$ alone. The moving calculation varies that complete expression. Its phase actions are

$$\boxed{ J_a=2\sum_k(1-k)\omega_k\|c_k\|^2,\qquad J_b=2\sum_k k\omega_k\|c_k\|^2. } \tag{14}$$

Consequently,

$$\boxed{E=V+T,\qquad E-\omega_aJ_a-\omega_bJ_b=V-T.} \tag{15}$$

Equation (15) retains the complete positive energy and the phase normalization. Some signed contributions to (14) can oppose one another; that does not imply negative particle inertia. The faster components remain modes of the same field.

In the smaller common trial space, the extra kinetic energy is about $1.3\times10^{-7}$ of the isolated pair energy for down/strange, $1.1\times10^{-5}$ for up/charm, and $1.3\times10^{-5}$ for electron/muon. The phase-action changes are explicitly retained and reach roughly $2.7\times10^{-5}$ in these controls. They are not silently absorbed into the amplitudes or identified with new masses.

The moving and static adjustments remain close in those tests. The electron/muon moving optimization reached its iteration limit, so its reported state is a variational trial with a nonzero optimization residual. The full low-mode Euler equations and an encounter trajectory remain to be solved.

The pair Hessian has an additional term

An isolated eigenfield has the useful second-variation identity $\tfrac12\delta^2V=\|DFh\|^2$. A general pair does not satisfy that simplification. Its full moving quadratic form is

$$\boxed{ \frac12\delta^2(V-T)[h,h] =\|DFh\|^2+ \operatorname{Re}\langle F,D^2F[h,h]\rangle -\|\Lambda h\|^2, } \tag{16}$$

where $\Lambda c_k=\omega_k c_k$. For the local interaction map $N=F-H_0\Psi$, put $v=(K-kI)\Psi$ and define

$$k_h=\frac{2\operatorname{Re}(v^\dagger h)}{\rho},\qquad k_{hh}=\frac{2\operatorname{Re}[h^\dagger(K-kI)h]}{\rho} -\frac{4k_h\operatorname{Re}(\Psi^\dagger h)}{\rho}. \tag{17}$$

Then the required derivative is

$$\boxed{ D^2N[h,h]=2f''k_h(K-kI)h-f''k_h^2\Psi +f'''k_h^2v+f''v k_{hh}, } \tag{18}$$

with $f(k)=-g\sqrt{k^2+\eta^2}$. This term cancels in the quadratic form when $F=\lambda\Psi$, as it should. Retaining it for an adjusted pair is necessary before claiming a stable fast response.

Two independent implementations evaluate (16): direct application of the full derivative, and assembly from local current moments. They agree at the same spatial quadrature, including actual forms at depth two. The kinetic metric is canonical in both. At 128 radial and 192 phase nodes, every tested fast Hessian is positive. That observation alone is insufficient: its spatial integration must also be resolved.

Independent refinement gives the following smallest fast stiffnesses for the same smaller-space moving trial fields, in the dimensionless frequency units of the calculation:

Depth five reverses sign, and its derivative is still poorly resolved. The coarse-grid positivity therefore does not establish a stable pair response. Its background also ceases to be a fast stationary point on the finer quadrature. This is a failure of the claimed numerical reduction for that pair trial, not a demonstrated instability of the isolated depth-five mass branch.

An independent directional calculation holds the most negative 512-by-768 perturbation fixed and integrates its complete quadratic form directly, in bounded memory. Its curvature is approximately $-4.49\times10^6$, $-4.26\times10^6$, $-4.29\times10^6$ and $-4.46\times10^6$ on grids 512 by 768, 512 by 1536, 512 by 3072, and 1024 by 3072 respectively. The original grid agrees with the matrix calculation independently. The persistent negative direction strengthens the diagnosis for this trial field, while the remaining variation prevents calling the exact continuum curvature numerically certified.

Depth six remains positive, but the full matrix still changes much more than its smallest eigenvalue even at the last refinement. Its moving optimizer had already reached its iteration limit. A converged small eigenvalue alone is not a norm bound for omitted integration or spatial channels. At depths two and four, the reported matrix integration is much better resolved. The depth-three and depth-ten matrices still have larger relative changes despite their positive minima.

For a positive frozen fast stiffness with minimum $\kappa_f$, a bound on its gyroscopic coefficient gives the conditional frequency gap

$$\Omega_f\ge\sqrt{\Omega_{\max}^2+\kappa_f}-\Omega_{\max}, \qquad \Omega_{\max}=\max_k|\omega_k|. \tag{19}$$

Even at the coarse positive depth-two trial, this bound is only about twice the larger particle frequency. It does not justify a strong separation of timescales there. At other depths the coarse bound is larger, but it must not be used to bypass the integration and stationarity failures just described. Full coupled stability and encounter dynamics remain open.

What is established and what remains

One explicit local law now supplies isolated spatial frequencies, an orientation-sensitive current channel, and a reciprocal fast adjustment of overlapping patterns. The adjustment can be small in field norm and large in its effect on the overlap energy. Its forced motion and phase normalization are calculable from the same action.

The residual interaction still depends on the retained spatial space. The supporting geometry, the low-mode encounter dynamics, cross-depth maps, physical sector scales, observed vertices and complete translational inertia remain unresolved. In particular, a same-depth neutrino-pair test does not derive its interaction with ordinary matter.

Verification

current_pair_response_checks.py passes 88 checks of initial pair energy and local current compatibility. current_polarization_filter_checks.py passes 92 checks of actual spinor rotation, the separate form bound, and the finite-regularizer orthogonal-pair response. current_fast_relaxation_checks.py checks the full spatial gradient, fixed phase normalization and energy decrease of the static trial. current_fast_refinement_checks.py enlarges its physical spatial basis and independently refines integration. current_pair_motion_checks.py passes 30 checks of the moving action and complete energy bookkeeping. current_fast_hessian_moments_checks.py checks the complete real second variation against an independent derivative and verifies canonical symmetry and angular normalization. current_fast_hessian_quadrature_checks.py retains the independent integration refinements, including the depth-five sign reversal. Passing its algebraic checks does not mean the reported quadrature convergence or pair stability has passed. current_pair_directional_quadrature_checks.py independently follows the fixed depth-five downhill direction through finer spatial integration without constructing another full matrix. Optimizer termination, spatial basis dependence and quadrature error are recorded separately from those algebraic checks.