Pair geometry and reciprocal encounters

Physical interaction premise

Particles interact through extended tails with changes in mutual influence propagating no faster than light over their separation. Keep both full spatial patterns, their multiple orientations, shape, phase and reciprocal response. Within the ten-dimensional matter domain, seven directions are unresolved by a three-dimensional observer; infinitely many additional directions are unbound for every particle. Electromagnetic dimensional reduction remains a derivation target. Conventional antisymmetric amplitudes remain external comparisons.

An encounter is a calculation for two polarized, spatially responding patterns of the same field $\Psi_\infty$ on one infinite-dimensional physical spacetime, using a ten-spatial-dimensional matter domain plus time. Each begins at the isolated clock specified by its action. In the current portion proposal that is its kind's fundamental: each portion shares that frequency, while the mass-frequency is the count-multiplied registration. The older mass-frequency carrier examples below are separate models. Unlike fundamentals are not synchronized by assumption; same-kind fundamentals can agree without equal mass. Frequency shifts, sidebands or capture must follow from the equations.

The intended comparison is compatibility of actual spatial shapes, orientations and phases in one ordinary higher-dimensional manifold. Shared dimensional support alone is not a nonzero interaction vertex. Both particles must be allowed to respond: a fixed target, an isotropic density or a prescribed overlap number gives only a restricted snapshot. The attractive force between an electron and a nucleus and the electron's literal external trajectory are distinct from internal count-field patterns. The electron has no inherent physical three-plus-three split.

No mediator particle or exchange mechanism is introduced. Any response variable must be a retained or eliminated mode of the same field with a stated action. A local operator comparison, a prepared contact energy, a canonically normalized vertex, a force, an encounter rate and a complete trajectory answer different questions. The evidence below preserves those differences.

Reviewed Connections as of 5 November 2026

The new isolated translating constructions supply more concrete candidates for eventual encounter tests. Their global inverse moment and variance kernel have no proved separated-cluster limit, so two distant copies are not yet independent physical particles under that law.

The commuting-tetrahedron contact has I+Swap eigenvalues zero and two. Its lower antisymmetric cost is not attraction or binding. The separate focusing-density profiles do not inherit that contact's preference or the centered-window motion theorem. A physical encounter must still evolve both actual preparations at their own clocks with complete energy, momentum, phase and orientation response.

Counts, source strength and cluster separation

The current pair comparison must retain evenly spaced and uneven internal phases, different kind fundamentals, and the actual assigned counts. Additive long-time event counting survives arbitrary fixed offsets, but a coherent force can retain phase-dependent cross terms. A detector's average count alone therefore cannot establish the pair force or remove its frequency dependence.

Source strength is independent data until the field equations connect it to count. The newer cluster candidate supplies a shared phase-action leaf $P=\hbar$ and $B=\Omega[P+\hbar(C-1)]+U$. It can then increase its mechanical mass with $C$ while leaving the reference total phase action fixed. Whether electric charge follows that source requires its actual coupling; neither equality of charge nor multiplication of charge by $C$ follows just from counting portions.

For two disconnected clusters, the proposed relativistic energy is

$$H=\sqrt{c^2|p_1|^2+B_1^2}+\sqrt{c^2|p_2|^2+B_2^2}.$$

Replacing it by $\sqrt{c^2|p_1+p_2|^2+(B_1+B_2)^2}$ would omit relative kinetic energy. A rule that joins clusters must retain that energy and the derivatives of its joining function. The connectivity proposal gives a conditional spectral interpolation, but has not yet derived physical joining, reciprocal capture or the selected count.

The uniform-probe shape result provides a separate useful ingredient: a translation-compatible first spatial deformation and positive susceptibility of one compact portion. Its probe is prescribed, so it is not yet a reciprocal encounter or a loaded many-portion solution. A comparison must use the same complete action for both sides before combining these ingredients.

Normalize the same field law before comparing response

The canonical normalization calculation starts with spatial maps $S_d$, positive kinetic Gram matrices $Z_d=S_d^\dagger S_d$, and a raw coherent vertex $K_{ab}$. Independent basis coordinates become canonical under $z_d=Z_d^{1/2}c_d$, giving

$$\widehat K_{ab}=Z_a^{-1/2}K_{ab}Z_b^{-1/2}.$$

If its retained same-field response amplitude has kinetic coefficient $Z_A>0$, the complete vertex also contains $Z_A^{-1/2}$. Cross-family kinetic terms require normalizing the full combined Gram matrix; a singular redundant basis cannot be inverted. A current has two matter legs, so a current-current coefficient cannot be normalized by applying the displayed two-leg vertex formula just once.

A conserved generator already equal to the same scalar $q$ on two families gives unnormalized charge matrices $qZ_d$ and normalized matrices $qI$. This proves preservation of an assumed common charge, not its electric identification or value. Its zero-momentum form factor $F_d(0)=q$ can agree while resolved form factors and encounters differ.

For actual particle comparisons keep the supplied count ratios and the maintained occupied depths two, three, four, five, six and ten under one action. A neutral insertion or special depth-five coefficient cannot establish rare neutrino encounters. A photon transverse control and higher-depth polarization need their own full spatial fields; photon transversality is not imposed universally.

Positive-wave contact has its own energy

The positive count action has $E_+=\|\dot\Phi\|^2+\|F[\Phi]\|^2$. Its interaction must be varied from that action. Here $D_R$ is the specified spatial Dirac parent, $\beta$ is its Hermitian scalar-current matrix, and $f_\sigma$ is the common smoothed count constitutive function. Define $s=\operatorname{tr}(\Phi^\dagger\beta\Phi)$. For $\rho=\operatorname{tr}\Phi^\dagger\Phi$, $x=s/\rho$, $m=f_\sigma'(x)$, $\eta=f_\sigma-xm$, and $v=(\beta-xI)\Phi$, the complete real derivative is

$$DF[\Phi]u=(D_R+m\beta+\eta I)u +\frac{2f_\sigma''(x)}\rho v\operatorname{Re}\operatorname{tr}(v^\dagger u).$$

The final term remains in smooth chart transitions. Omitting it freezes a responding source coefficient and changes the model.

For two balanced branches in orthogonal column blocks, retain $\dot\Phi_i=-i\omega_i\Phi_i$ and $x=(x_1\rho_1+x_2\rho_2)/(\rho_1+\rho_2)$. The isolated source coefficients are $m_i=f_\sigma'(x_i)$, $\eta_i=f_\sigma(x_i)-x_im_i$. Put $\delta m_i=m-m_i$, $\delta\eta_i=\eta-\eta_i$, $g_i=-(x_i\delta m_i+\delta\eta_i)$. Then the exact prepared energy difference is

$$\Delta e_+=\sum_i\rho_i\{(\delta m_i)^2+(\delta\eta_i)^2 +2x_i\delta m_i\delta\eta_i-2\omega_i g_i\}.$$

The positive mismatch square competes with the first-order deficit. The pointwise interval proof retains the actual smooth source and proves $\Delta e_+\ge0$ for all density ratios and relative orientations in this preparation at its stated diagnostic scales. The certified shell comparisons are:

Distinct branches have strictly positive cost where both densities are nonzero; every maintained same-branch orthogonal-column control has zero cost. The proof covers 226 smooth transition bands with positive interval Bernstein coefficient enclosures. It is independent of beam quadrature convergence. It excludes energy below the isolated reference in that preparation, not every force, metastable well, coherent encounter or time-evolved shape.

Coherent fields restore relative-phase response

The coherent positive-count calculation uses actual matrix fields $\Phi=\Phi_1+e^{i\varphi}\Phi_2$ and both isolated clocks. Writing $\delta=(l_2-l_1)\theta$, its full Clifford traces give a density interference term proportional to $\cos\varphi\cos\delta$. Depth two also retains a three-generator trace term proportional to $-\sin\varphi X_0\sin\delta$, absent in the retained full representations at three, four, five, six and ten. At depth at least three the $\varphi=\pi/2$ preparation reduces to the orthogonal-column energy comparison; the depth-two term prevents that identification.

Prepared contact costs vary with phase, but the finite quadratures are not uniformly certified through every sharp transition. For unit individual norms, the depth-six finest estimates are about 49.60, 54.19 and 60.73 times isolated complete energy at phases 0, $\pi/4$,$\pi/2$; the independently resolved orthogonal-column value is 60.8189. Depth-two phase zero remains unresolved with quadrature order. These large positive prepared costs do not supply the desired binding mechanism. Equal isolated phase-action preparation is a separate normalization and gives different values. Five's phase response is shared with other depths; it is not neutrino-specific suppression.

The following unbound packet and current-law comparisons supply complementary dynamical and geometric controls. Their equations and numerical values retain their own actions, spatial preparations and units. No union of those controls is presented as one complete particle-pair solution.

Spatial packet compatibility

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Compare both patterns using the same spatial rule

Use the local scalar Dirac equation

$$i\partial_t\Psi=D\Psi+g\sigma\beta\Psi, \quad D=-i\alpha\cdot\nabla, \quad \sigma=\Psi^\dagger\beta\Psi, \tag{1}$$

with its full complex source and reciprocal variation as in Dirac source linearization. The incoming field is a sum of two opposite complete kinetic chiralities, $\Psi=\Psi_u+\Psi_v$. Their individual scalar self-currents vanish, while their cross pattern supplies the entire scalar. The positive-frequency vertices retain its polarization, phase and spatial interference.

For a fixed comparison duration $T>0$, let $U(t)=e^{-iDt}$ and define

$$\boxed{\mathcal C_T[\psi]= \int_0^Tdt\int d^Dx\, \left[(U(t)\psi)^\dagger\beta U(t)\psi\right]^2.} \tag{2}$$

Both complete fields enter (2). A fixed target, a count of components, or an isotropic density replacement would define a different functional. It is nonnegative and uses exactly the scalar entering (1). With dimensionless positive norm and $\hbar=c=1$, $[\mathcal C_T]=\mathrm{length}^{1-D}$, so $g\mathcal C_T$ is dimensionless. It is not a measured interaction probability or a particle reaction rate.

The free propagation in (2) specifies a controlled encounter for the chosen preparation and duration. It permits reproducible comparisons of spatial patterns without declaring every overlap measure physically interchangeable. It is the spatially localized counterpart of the traveling-pulse compatibility comparison.

The first scattering variation preserves this functional

Use the positive spatial inner product and the complex gradient convention

$$\delta\mathcal C_T=2\operatorname{Re} \langle\nabla_*\mathcal C_T,\delta\psi\rangle. \tag{5}$$

Vary the local scalar before integrating. Its exact gradient is

$$\boxed{ \nabla_*\mathcal C_T[\psi] =2\int_0^Tdt\,U(-t) \left[\sigma(U(t)\psi)\,\beta U(t)\psi\right].} \tag{6}$$

The finite spatial regulator uses the orthogonal Galerkin projection of the displayed source and the same kinetic norm; no additional interaction coefficient appears.

Let $\Psi_g(T)$ be the full nonlinear solution at time $T$ and define its outgoing interaction-picture field by $\psi_{\rm out}^I=U(-T)\Psi_g(T)$. The same inverse free evolution removes free motion from both patterns. The comparison is $\Delta\mathcal C_T=\mathcal C_T[\psi_{\rm out}^I]-\mathcal C_T[\psi_{\rm in}]$, retaining every generated component.

Duhamel's formula for (1), at fixed $T$ and in a regime where the small-$g$ expansion is valid, gives

$$\boxed{\psi_{\rm out}^{I} =\psi_{\rm in}-\frac{ig}{2} \nabla_*\mathcal C_T[\psi_{\rm in}]+O(g^2).} \tag{7}$$

Consequently the first derivative of $\mathcal C_T[\psi_{\rm out}^I]$ at $g=0$ vanishes:

$$2\operatorname{Re}\left\langle \nabla_*\mathcal C_T,-\frac{i}{2}\nabla_*\mathcal C_T \right\rangle=0, \qquad \boxed{\Delta\mathcal C_T=O(g^2).} \tag{8}$$

The leading field deformation can therefore be nonzero while the leading change in this particular compatibility measure vanishes. Its later change depends on the evolving kinetic and source response. Equation (8) does not fix that change's sign or impose alignment.

Fully localized five-dimensional preparations

The axial spatial family reconstructs all five ordinary spatial directions while retaining both complete complex polarization vectors. It is a declared angular family of unbound fields, not a selected neutrino or another bound particle geometry.

Let $R_b(p,z_0,w_z,\xi)$ denote the unit-norm positive-frequency projection of the following raw field in chiral block $b$:

$$\phi_0^0(r)\,W\xi\, (\pi w_z^2)^{-1/4} \exp\left[-\frac{(z-z_0)^2}{2w_z^2}+ip(z-z_0)\right], \tag{9}$$

with radial scale $w=2$. Here $\phi_0^0$ is the normalized transverse radial ground profile, and $W$ is the specified chiral angular/polarization lift in the axial Dirac packet construction. They are the pinned source's spatial basis maps, not new particle support assumptions. The projection uses the complete free spatial Dirac operator. Every $R_b$ is spatially localized and lies in its full positive free band. Its transverse profile is a Gaussian preparation, not a confining potential.

Set $\eta=0.6$, $\vartheta=0.7$, $b=(\cos\vartheta,e^{0.31i}\sin\vartheta)^T$, and let $S$ swap the two polarization coordinates. Prepare

$$\begin{aligned} u_0&=R_u(2,-4,2,e_1) +\eta e^{0.6i}R_u(2.35,-3,1.5,e_2),\\ v_0&=e^{i\varphi}R_v(-2,4,2,b) +\eta e^{-0.4i}R_v(-1.7,3,2.3,Sb),\\ \psi_{\rm in}&=u_0/\|u_0\|+v_0/\|v_0\|. \end{aligned} \tag{10}$$

Thus both full incoming patterns have positive norm one. The parameter $\varphi$ changes the phase of the main mode of the second pattern relative to its additional texture mode. It is not a global phase change of that entire pattern. The first pattern also has a nonconstant complex polarization texture; neither side is a fixed target.

Evolve (1) with the same $g=50$ and $T=10$ in both cases below. These are specified numerical units. Since $[g]=\mathrm{length}^{D-1}$, comparing this number with a coefficient in another spatial dimension would require its physical normalization; no common observed hierarchy coupling is inferred.

The complete nonlinear equation produces both signs at the same coupling. The effects are small. They show changing compatibility under the defined control, not attraction toward an automatically preferred orientation. The actual outgoing fields also differ from their free controls, with full positive-norm distances approximately $0.03223$ and $0.02698$.

Separate pattern change from redistribution of norm

For nonzero chiral norms $Q_u,Q_v$, define the same encounter with each complete pattern rescaled by its positive real norm:

$$\boxed{\mathcal S_T[\psi] =\mathcal C_T\left[\frac{\psi_u}{\sqrt{Q_u}} +\frac{\psi_v}{\sqrt{Q_v}}\right] =\frac{\mathcal C_T[\psi]}{Q_uQ_v}.} \tag{11}$$

The equality follows because each individual chiral scalar vanishes, free evolution preserves both chiral norms, and the cross scalar scales by $1/\sqrt{Q_uQ_v}$. Every relative phase and every spatial component is retained. This removes an amplitude product from the comparison.

For the two refined preparations in the table, both final chiral norms differ from one by less than $1.2\times10^{-8}$. Recomputing the comparison with (11) changes their reported $\Delta\mathcal C_T/ \mathcal C_T$ values by less than $6\times10^{-12}$ in absolute value. Both signs and their displayed sizes survive. The decrease is therefore not explained by a loss of the product of the two chiral norms.

More generally, start with equal norms $Q_u=Q_v=q>0$, and put $q_1=\partial_g Q_u|_{g=0}$. Conservation of their sum gives $Q_uQ_v=q^2-g^2q_1^2+O(g^3)$. If $\Delta\mathcal C_T=g^2c_2+O(g^3)$, then

$$\boxed{\frac{\Delta\mathcal S_T}{\mathcal S_{T,\rm in}} =g^2\left[\frac{c_2}{\mathcal C_{T,\rm in}} +\left(\frac{q_1}{q}\right)^2\right]+O(g^3).} \tag{12}$$

The direct coupling variations calculate $c_2$ and both norm variations from the complete field equation. The normalized comparison still uses the prescribed encounter protocol; it is not a universal definition of a measured coupling.

Current pair coupling

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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

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

For the principal rotation angle $0\le\alpha\le\pi$, combining (5) and (7) excludes the sign-changing channel 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. 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 strength $gR/c=100$, the electron/muon comparison has a threshold about $115.9^\circ$ under this specific spatial-current preparation. The depth-ten index 23/24 control has a threshold about $10.4^\circ$; index 24 is unassigned. These are model polarization thresholds, not measured cross sections. Their coefficient normalization and actual current law must be retained when comparing another depth or particle family.

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.

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.

Conditional frame matching without a physical three-plus-three premise

The scattering-frame variational calculation retains a useful mathematical result. Given two supplied oriented orthonormal frames $\widehat O,\widehat H$, the positive Procrustes functional

$$V(Q)=\frac\kappa2\|\widehat O-Q\widehat H\|_F^2, \qquad Q\in SO(3),\qquad\kappa>0,$$

has minimizer $Q=\widehat O\widehat H^T$ and tangent Hessian $2\kappa I_3$. For a mixed tensor with positive singular values $a,b,c$, the corresponding rotational curvatures are $\kappa(b+c),\kappa(a+c),\kappa(a+b)$; rank two with $a,b>0,c=0$ also gives a strict local orientation minimum.

This is conditional variational geometry for supplied frame data and positive coefficient. The field action has not derived those scattering frames, the sign or magnitude of $\kappa$, or their reciprocal motion. It does not establish an inherent physical split of the six-dimensional electron, select its orbit around the nucleus or determine a particle frequency. The newer literal-trajectory requirement keeps the electron itself six-dimensional and treats the proposed four-direction external motion as a separate force-and-recoil problem.

Two patterns side by side

Two solved depth-two patterns of the density law, evolved under one coefficient of the law, give direct readings of how each moves and how each clock changes. These patterns are uniform along their free directions; amplitudes are uncalibrated.

Equal clocks. Patterns in step pull together and their clocks slow. Patterns out of step push apart and their clocks quicken. This holds in all six cases, at separations 10, 16 and 20. In step they do not stay merged: each stretches toward the other, the two merge and compress to about three times their peak density, and they come back out. Out of step they separate with no visible change of shape. The frozen energy slope over the inertia gives the right sign and the size within a factor of about two.

Unlike clocks. At fixed coefficient of the law, unequal clocks make one pattern much denser: $\omega=0.8$ beside $0.7$ makes the slower-clocked pattern 38 times denser and about 26 times heavier. With a beat of 0.1, the lighter pattern shows no net approach over four beats, where the frozen average pull predicts a clear fall. The actual average pull is at least ten times smaller than the frozen one. With a beat of 0.05, the lighter pattern falls into the heavier within one beat and is absorbed. The two cases differ in density ratio as well as beat, so the beat is not isolated. The steady push of a rigid jog at the beat rate accounts for only 2 to 3% of the frozen pull, so the patterns' own response removes the pull.

The lighter clock quickens. Beside a heavier pattern the lighter pattern's clock runs quicker, by $9.9\times10^{-3}$ at separation 16 and $2.8\times10^{-3}$ at 20. Along the family of patterns the norm goes as $\delta^8$, so a pattern holding a fraction $x$ of its norm has gap $\delta\,x^{1/8}$. For the measured norm fractions that gives $8.9\times10^{-3}$ and $2.8\times10^{-3}$. The quickening is what a lighter member of the same family does, and the heavier clock slows by a tenth as much.

Depth two in a plane is the only case run; depths 3 and 4 and nested unlike depths have not been. Pushing one pattern is in volume 12, and the frozen contact range in volume 14.

Contact and what a detector registers

Hidden offsets. Two electrons in a three-direction apparatus have three unsteered offsets in the hidden directions. Take the normalized pair-energy cross term as the contact measure, whose tail falls as $r^{-13.4}$, with offsets uniform in a three-ball of radius $L$ and $a$ equal to two reduced Compton wavelengths. The effective contact radius is then

$$r_{\rm eff}=a\sqrt{2K_5(a/L)^3},\qquad K_5=0.256\text{–}0.422 .$$

A contact radius below the $10^{-17}$ cm of high-energy electron scattering needs $L/a$ above $3.1$ to $3.7\times10^4$, a hidden spread of 24 to 29 nm. Under that condition the apparent smallness of the electron comes from the spread of the offsets, not from the size of the pattern. $L$ is a condition here, not a derived length.

The neutrino. A neutrino bound in two of the three observed directions is spread along a line in ordinary space, so contact falls by about $a/L$ per unbound observed direction, the scaling being $(a/L)^{3-b}$ with $b$ the number of bound observed directions. If the unbound direction is the line of flight, helicity along the flight is exact (volume 16).

Detector weights. Take a detector to register the increase of the self-term $\rho^{1/9}$ when the pattern's density is added to a uniform ambient density, with $u$ the ratio of the two. The weight is $(1+u)^{1/9}-1$, proportional to density for small $u$. The total-variation distance between the registered distribution and the density-proportional one is 0.016 at core value $u_c=1$ and 0.0885 at $u_c=10$. The density and energy-density weights converge over all six directions of an electron, while the self-term rate $\rho^{2/9}$ converges only over its three unresolved directions.

Electron size. An extended charge of mean-square radius $\langle r^2\rangle$ shifts an $n$S level. The Dirac term and measured Lamb shift together equal $\langle r^2\rangle=0.786\lambda^2$, with $\lambda$ the reduced Compton wavelength. The exact branch at depth six gives 1.2, 1.0, 1.6 and 0.4 $\lambda^2$ for the four candidate densities that might carry the charge. These bracket the measured value with no tuning, but each misses the 2S level by 7 to 23 Lamb shifts. Which density couples to the nucleus is open.

What a complete encounter must establish

A complete pair calculation must start from actual isolated spatial solutions and each pattern's own frequency, evolve both through the same action, and retain full energy, momentum, phase charges, supporting stress, orientation and phase. Derive any force from the position dependence of that complete interaction; derive capture or separation from the reciprocal evolution. Loss of energy from a selected pair must be accounted for in other modes of the same field. No automatic alignment, imposed binary compatibility gate or common frequency is assumed.

The unbound five-dimensional packet comparison above demonstrates both signs of change in one defined compatibility functional, not rare neutrino interactions. Prepared positive contact energies constrain one action and preparation; they do not provide encounter trajectories. Current-law thresholds and sideband energies are useful restricted controls, with their optimization residuals retained. Canonical normalization preserves assumed common charges without making finite-momentum response universal. These results identify reusable mechanisms and remaining tests without claiming an observed force, rate or complete mass-and-coupling operator.