Polarization Contact and Reciprocal Spatial Response

The hypothesis that coupling is higher-dimensional polarization becomes testable when a stated field energy supplies both objects' response. This note takes one simple local term and calculates its consequences. It gives a conditional mechanism with explicit limitations, not the completed IDWT interaction law.

The calculation retains the full field coefficients, including their relative phases. A two-form describing average circulation is not always enough: the existing complete polarization calculation and equal-current example show that ten-dimensional patterns can share their current tensor while differing in higher-order polarization and response. Those older results retain their own field equations; the contact below is a new, separately specified benchmark.

All fields live on the single physical ten-space plus time. The two contributions below are a trial decomposition of Psi_infinity, not two mediator fields. Their coefficients are classical field data and introduce no position-probability law. Amplitudes remain symbolic and uncalibrated.

Both trial contributions use a common component carrier so their overlap is defined. Applying this benchmark to two actual particle depths requires their compatible embeddings in the same field; assigning different support dimensions alone does not derive those embeddings.

1. One Local Term and Its Complete Cross Contribution

Consider the constitutive premise

$$V(\Psi)=\frac{\lambda}{2}(\Psi^\dagger\Psi)^2, \qquad \lambda>0,\qquad \Psi=u+v.$$

This is a rotationally invariant density term in the specified time frame. Its full spacetime covariance and compatibility with a sustaining kinetic action are additional requirements. No claim that it is the already derived master action is made.

Write

$$p=u^\dagger u,\quad q=v^\dagger v,\quad c=\operatorname{Re}(u^\dagger v).$$

Subtracting the two isolated contact densities gives the exact local identity

$$V(u+v)-V(u)-V(v) =\lambda\bigl[pq+2(p+q)c+2c^2\bigr]. \tag{P1}$$

The first term depends on intensities. The other terms retain the coherent polarization overlap. They are contributions from one field term, not separately postulated forces. The full local variation is $\partial V/\partial\Psi^*=\lambda(\Psi^\dagger\Psi)\Psi$. Equation (P1) must not be reduced to an overlap alone while keeping the same claimed action.

For real nonnegative spatial envelopes f and g and unit component vectors $\chi_A,\chi_B$, take the one-harmonic trial preparations

$$u=f\chi_A e^{-i\omega_A t},\qquad v=g\chi_B e^{-i\omega_B t+i\delta_0},\qquad z=\chi_A^\dagger\chi_B, \quad \Delta=\omega_A-\omega_B.$$

Then $c=fg r(t)$, with $r(t)=\operatorname{Re}[z e^{i(\Delta t+\delta_0)}]$. Define the spatial overlaps $I_{rs}=\int f^r g^s\,dx$, where the integration support is specified below. The contact cross energy is

$$U(t)=\lambda\bigl[(1+2r(t)^2)I_{22} +2r(t)(I_{31}+I_{13})\bigr]. \tag{P2}$$

This trial uses one harmonic of each particle's proposed fundamental, not its C-times registered mass-frequency. It is not a reconstruction of C spatial portions or the complete sawtooth waveform. A full nonsinusoidal calculation retains the corresponding harmonic sums and possible resonances instead of applying (P2) without change.

2. Unequal Clocks Do Not Remove All Polarization Dependence

For an interval of length T centered at $t_c$, define $\operatorname{sinc}x=\sin x/x$, with value one at zero. Direct integration gives

$$\overline r=\operatorname{Re}[z e^{i(\Delta t_c+\delta_0)}] \operatorname{sinc}(\Delta T/2),$$

$$\overline{r^2}=\frac{|z|^2}{2} +\frac12\operatorname{Re}[z^2e^{2i(\Delta t_c+\delta_0)}] \operatorname{sinc}(\Delta T). \tag{P3}$$

For fixed nonzero detuning and long observation time this yields

$$\overline U\longrightarrow\lambda I_{22}(1+|z|^2). \tag{P4}$$

Rapid beating removes the terms sensitive to a common relative phase, but not the dependence on the full polarization overlap. Orthogonal component patterns give $\lambda I_{22}$; identical polarizations give $2\lambda I_{22}$. This specific contact therefore cannot explain complete interaction suppression by orthogonality alone.

The equal-clock case is different. At $\Delta=0$, r is constant and all terms in (P2) remain. Taking the long-time limit before letting detuning go to zero changes the answer. The maintained electron, muon and tau share a fundamental under the kind-specific premise, so their comparison belongs to this coherent case unless additional physical averaging is justified. Their different counts must not be substituted for different fundamentals.

The sector prescription gives the following fundamental ratios relative to the charged-lepton unit:

$$\omega_2/\omega_6=18564\sqrt{2890},\qquad \omega_3/\omega_6=18564\sqrt{32\sqrt7},\qquad \omega_4/\omega_6=\frac{18564}{15}\sqrt{48/\sqrt7}, \quad \omega_{10}/\omega_6=1.$$

These specify detuning inputs for this diagnostic; they do not supply actual particle profiles. The separate depth-five absolute fundamental remains uncalibrated here. For a full periodic waveform, near-resonant combinations of its harmonics need their own check.

3. Nested Spatial Support Restricts the Contact Force

Let a shallower pattern use a shared a-dimensional coordinate x, and a deeper pattern use those directions plus b−a coordinates y, with $a\le b\le10$. This is an explicitly nested, aligned support comparison in the same physical space. Choose Gaussian controls

$$f=A\exp[-|x-R_A|^2/(2w_A^2)],$$

$$g=B\exp[-(|x-R_B|^2+|y-Y_B|^2)/(2w_B^2)].$$

The shallower pattern is uniform along y. Let $R=R_A-R_B$, and for positive powers r,s put $\alpha=r/w_A^2$, $\beta=s/w_B^2$. Completing squares gives

$$I_{rs}=A^rB^s(2\pi)^{b/2} (\alpha+\beta)^{-a/2}\beta^{-(b-a)/2} \exp\left[-\frac{\alpha\beta}{2(\alpha+\beta)}|R|^2\right]. \tag{P5}$$

This is an integral over the union of their bound directions, or contact energy per unit volume of any common uniformly extended directions. It does not establish a finite complete ten-dimensional particle energy. Each isolated pattern's kinetic normalization and support completion remain separate questions. The Gaussian shape and widths are specified controls, not fitted or derived particle geometries.

Crucially, $I_{rs}$ does not depend on $Y_B$. The contact energy cannot supply a force moving the deeper pattern along those extra directions in this uniform-extension control. It responds to separation in the shared bound directions. A finite envelope, spatial texture or altered support in the formerly uniform directions can change this conclusion.

Set $\kappa_{rs}=\alpha\beta/(\alpha+\beta)$. Since $\nabla_R I_{rs}=-\kappa_{rs}R I_{rs}$, the contact contributions to the two spatial forces are

$$F_A=-\nabla_{R_A}U =\lambda R\bigl[(1+2r(t)^2)\kappa_{22}I_{22} +2r(t)(\kappa_{31}I_{31}+\kappa_{13}I_{13})\bigr], \qquad F_B=-F_A. \tag{P6}$$

The use of r for an overlap power in (P5) is distinct from the time-dependent real overlap r(t). The displayed force is only this contact term's contribution. The complete force also includes the kinetic and sustaining terms of the action, with all profiles responding.

For unequal clocks after full beat averaging, (P6) becomes $F_A=\lambda(1+|z|^2)\kappa_{22}RI_{22}$: this positive density contact contributes repulsion in the shared directions. A coherent negative real overlap can instead give attraction over part of the separation range. Both signs arise from (P1), without introducing separate exchanged objects.

4. Reciprocal Polarization Torques Follow from the Same Energy

For one common physical rotation generator S on the field components, rotate A by $\chi_A\mapsto e^{-i\epsilon S}\chi_A$, while B is held fixed. The relative overlap changes by

$$\partial_{\epsilon_A}z=i\chi_A^\dagger S\chi_B, \qquad \partial_{\epsilon_B}z=-i\chi_A^\dagger S\chi_B.$$

For the isotropic envelopes in their fixed bound subspaces, rotations preserving those subspaces leave the I factors unchanged. The contact torques are therefore

$$\tau_A=-\lambda[4r(t)I_{22}+2(I_{31}+I_{13})] \operatorname{Re}[i\chi_A^\dagger S\chi_B e^{i(\Delta t+\delta_0)}], \qquad \tau_B=-\tau_A. \tag{P7}$$

Common physical rotation of all positions, support subspaces and components also leaves the complete scalar contact invariant. Tilting one bound subspace relative to the other introduces envelope-orientation derivatives, which (P7) alone does not include. The displayed cancellation establishes reciprocal virtual-work contributions in the specified comparison, not an angular acceleration without a kinetic coefficient.

5. A Radial Balance Is Not Yet a Stable Polarized Pair

An exact example exposes the extra requirement constructively. At equal depths and widths, with equal amplitudes and fixed coherent overlap $r(t)=-s$, write $x=|R|/w$ and $K=A^4(\pi w^2/2)^{d/2}$. Then

$$U/(\lambda K)=(1+2s^2)e^{-x^2/2}-4s e^{-3x^2/8}.$$

For $0<s<1/2$, the contact has a radial minimum at

$$x_*^2=8\log\frac{1+2s^2}{3s}. \tag{P8}$$

For example, s=1/4 gives $x_*^2=8\log(3/2)$. But this is a minimum only with the polarization overlap held fixed. At nonzero separation,

$$\frac{\partial U}{\partial r} =4\lambda K[r e^{-x^2/2}+e^{-3x^2/8}]>0 \quad\hbox{for }-1\le r\le1.$$

Thus this contact generally favors decreasing the real overlap, and the radial balance is not a full stationary configuration when that change is accessible. A repair must derive a restriction or restoring dynamics for polarization from the same field, then test the complete constrained action. Freezing the polarization by declaration would not finish that step. Relative phase, spin orientation and fixed total field norm are distinct restrictions.

In particular, independent amplitudes on overlapping trial profiles do not have independent canonical kinetic terms automatically. If $\Psi=\sum_j f_j c_j$ with fixed profiles and finite norm, a first-order field kinetic term pulls back to a Gram matrix $G_{ij}=\int f_i^*f_j\,dx$. Its equations are $i\kappa G\dot c=\partial H/\partial c^*$, and its conserved total norm is $c^\dagger Gc$. Moving profiles add derivative terms. For uniformly extended controls, that kinetic integral needs the specified support or finite-volume completion. The forces and torques above do not silently supply this missing kinetic reduction.

6. Why the Full Polarization Matters at Ten

The current-matched pair $z_\pm=(|0\rangle\pm|1234\rangle)/\sqrt2$ has the same two-form J but opposite higher-grade polarization. In the fixed chiral component system of the earlier calculation, $z_+^\dagger z_-=0$. With a comparison partner of polarization $z_+$, the averaged factor $1+|z_A^\dagger z_B|^2$ is 2 for the plus preparation and 1 for the minus preparation. A model using only their identical J tensors would miss this distinction.

The complete normalized coefficient matrix $\varrho=zz^\dagger$ gives $|z_A^\dagger z_B|^2=\operatorname{Tr}(\varrho_A\varrho_B)$. Within a selected chiral depth-six multiplet, scalar and two-form coefficients already reconstruct that matrix. At ten, scalar, two-form and four-form coefficients are required. This is a difference in spatial polarization information, not evidence for more observed spin projections or another physical manifold. A common relative temporal phase must additionally be retained in the coherent case.

7. Consequences for the Sector Programme

This contact gives a concrete conditional route from full polarization to reciprocal spatial derivatives. It preserves the separation between the count-scaled mass label and the directional response, but its scalar baseline prevents identifying all coupling with orientation matching. More selective interactions must follow from an actual current, derivative structure or dynamics of the same field.

The nested-support result is useful independently of that limitation: in the aligned uniform-extension control, the force contribution lies in shared bound directions. It does not grow new directional components merely because the deeper object has more dimensions. The role of tilted subspaces and finite support is a subsequent calculation, not already covered by the nested Gaussian result.

Combining this contact with the critical first-order bubble action has a concrete compatibility condition. For any regular finite-norm stationary solution with admissible dilation variations, the clock-balance derivation gives

$$\omega N=(2-d)K,\qquad K=\frac{\lambda}{2}\int\rho^2\,d^dx.$$

The coefficients at depths two, three, four, five, six and ten are 0, −1, −2, −3, −4 and −8; five is not exceptional. For positive contact coefficient and d>2 the signed frequency would be negative in this convention, without assigning that sign a particle interpretation. The unchanged bubble cannot satisfy the new stationary equation because its density is not constant. This is a necessary condition for a changed action, not proof that a new localized solution exists or that the kind fundamental has been selected. Its independent clock review and corrected domain are recorded in the linked current note.

No absolute force, interaction rate, count-selection rule, stable particle pair or observed charge normalization is derived here. All 184 author checks passed. Claude independently reviewed the frozen mathematical text, SHA-256 0fcf32ba96cd5ec59e650368a647ee4be28bb3ed71af499bfd8fb7f5c5fd8b42, and confirmed equations (P1)–(P8) and the equal-current polarization example. This version changes status, provenance and the explicit common-carrier qualification, retaining those equations unchanged. The fundamental ratios are prescription inputs and were not part of Claude's independent verification.

The governing review is agents/claude/polarization-20261102/review-reciprocal-contact.md, read with review-reciprocal-contact-correction-1.md. Its long-window numerical control reached an integration subdivision limit and is not claimed as converged evidence; the exact finite-window sinc expressions give the long-time limit analytically. The original slow review script was stopped and replaced with bounded checks. The review also confirms that the fixed-overlap radial minimum is not a stationary point when overlap can vary. For identical envelopes at coincidence and opposite phase, the two trial fields cancel and the Gram matrix is singular: that endpoint is not a bound pair. No full dynamics is inferred from decreasing contact energy alone.

The reproducible principal program and results are stored in the database under the original exact source identities, including principal checker SHA-256 7269becc7476fa6a7c862e6619e7a63b1685e6ace1b28635ce05d413f3798942 and results SHA-256 b09961856052a695fc8a32bb3972800345f818736818ae18c5b92f0f3fd20b3f. Claude's independent bounded checker, kinetic control and corrected review are stored under agents/claude/polarization-20261102/. Their additional Gram-matrix examples retain their deliberately specified energy matrix and do not supply predicted IDWT frequency shifts.

8. Density-Law Contact Between Localized Profiles

A later control uses the focusing density law itself rather than the quartic comparison above. For local densities $x,y\geq0$, $q=1+p=10/9$, and positive rescaled coefficient $\delta$, its incoherent contribution is

$$U_{\rm inc}=-\frac{\delta}{q}\int \bigl[(x+y)^q-x^q-y^q\bigr]\leq0.$$

The inequality follows from convexity and is strict when both densities are nonzero on a set of positive measure. It is an energy sign, not by itself a force direction: radial attraction requires $U'(R)>0$. The computed separations show decreasing magnitude and attractive secant slopes for this frozen contribution; a complete reciprocal encounter has not been solved.

If $c$ is the modulus of the local spinor overlap, then $0\leq c\leq\sqrt{xy}$. A stipulated uniform relative-phase average gives $\langle(x+y+2c\cos\theta)^q\rangle\geq(x+y)^q$, and this average increases with $c$. Thus

$$U_{\rm maximal\ phase\ average} \leq U_{\rm actual\ phase\ average}\leq U_{\rm inc}.$$

The left envelope uses maximal pointwise overlap and need not be attainable by one globally specified pair of spinor profiles. The inequalities are not bounds for every fixed phase; destructive equal fields give a positive nonlinear cross contribution. Both profiles in this control have the same specified frequency, so its phase loop is an imposed averaging prescription, not a derived unequal-clock beat.

For a common quadratic operator $H_0$, the full cross energy also contains $2\operatorname{Re}[e^{i\theta}\langle\phi_A,H_0\phi_B\rangle]$. A justified uniform phase average removes that term. Without it, the quadratic kinetic and mass cross terms remain. This connects the diagnostic to the existing total-field-minus-isolated-energies definition without turning one contribution into the complete physical force law.

9. A Finite-Range Slope and an Asymptotic Tail Are Different Tests

The author's contact run passed 20 of 30 criteria and returned exit one: all ten stated finite-radius exponential-rate bounds failed. Its rising finite-range local-slope proxy does not prove an asymptotic law. It also has no contact-quadrature or tail-domain convergence study.

There is nevertheless an independent conditional far-tail derivation. On an infinite-domain localized positive-gap branch, with $\kappa=\sqrt{m_0^2-\omega^2}>0$ and nonlinear potential tending to zero, the radial equations give

$$F,G\sim r^{-(D-1)/2}e^{-\kappa r},\qquad \rho\sim A r^{-(D-1)}e^{-2\kappa r}.$$

For equal $\kappa$ and $0<p<1$, the nonlinear cross integrand is bounded by positive constants times $\min(x,y)\max(x,y)^p$. Along the line between centers its smallest exponential suppression occurs where the leading densities have equal exponential size. This gives logarithmic decay rate $(1+p)\kappa$, with algebraic prefactors depending on geometry. The phase-averaged envelopes retain that leading rate. A finite wall or spliced numerical tail does not prove the required infinite-domain branch.

This conditional limit supplies no rate lower bound at the tested finite radius. In particular, the potential's slow $\rho^{1/9}$ decay can delay the linear-tail regime; discretization and interpolation have not been excluded as additional contributors. The control does not derive observed electric far fields or exclude other same-field responses.

10. The Symmetry Condition Uses the Complete Patterns and Boundaries

Let $S_A,S_B$ be the actual translation-invariance subspaces of the two complete patterns, including their boundaries. Reciprocity and that symmetry give

$$\nabla_R U\in(S_A+S_B)^\perp=S_A^\perp\cap S_B^\perp.$$

Only when each complete free extension is uniform, so $S_i=B_i^\perp$, does this become the intersection of the bound spaces $B_A\cap B_B$. A finite actual-contact-ended span, spatial texture or bending can break the symmetry. Endpoint stresses and variations can then contribute in directions that were free in the ideal bulk. The nested Gaussian result is retained with its premises, rather than imposed as a restriction on every physical encounter.

These added identities and source/output checks have independent analytical assessment SHA-256 14a9b274021e209211432cdc80013986af8679689d1392514f5d7499c0440daa. The numerical values remain author execution; no independent replay, absolute amplitude calibration or new physical degree of freedom is claimed.