Relative Phase in Coherent Positive Count Fields

The orthogonal-column pair has no relative-phase dependence. Coherent spatial fields restore it without adding another coupling coefficient. This calculation uses the same positive count action and an explicit full matrix carrier, rather than claiming that the minimum one-column fields have been classified. The results are prepared energies, without inferred radial forces or encounter trajectories.

1. Actual Matrix Fields Give the Interference

Choose the same circle plane (1,2) for two different shell levels. Write E_i=sqrt(rho_i) U_i C, U_i=cos(l_i theta)I+sin(l_i theta)Gamma_1 Gamma_2, with C=I/sqrt(r), the identity on the actual spinor representation of dimension r. The massive fields are Phi_i=(a_i+b_i beta)E_i, with the usual count coefficients. Take Phi=Phi_1+exp(i phi)Phi_2 and retain each component's own isolated clock in its initial velocity.

Set delta=(l_2-l_1)theta, A=a_1a_2+b_1b_2 and B=a_1b_2+b_1a_2. Direct Clifford traces give

$$\frac1r\operatorname{tr}(U_1^\dagger U_2)=\cos\delta,\qquad \frac1r\operatorname{tr}(U_1^\dagger\beta U_2)= \begin{cases}iX_0\sin\delta,&d=2,\\0,&d=3,4,5,6,10.\end{cases} \tag{1}$$

The depth-two term follows from its nonzero three-generator volume trace; the same triple trace vanishes in the retained full representations at the other occupied depths. This is verified using actual matrices, including the physical position dependence of beta=Gamma_X.

Let p=cos(phi)cos(delta), q=-sin(phi)X_0 sin(delta) at depth two and q=0 otherwise. Then

$$\boxed{\rho=\rho_1+\rho_2+2\sqrt{\rho_1\rho_2}(Ap+Bq),}$$ $$\boxed{s=x_1\rho_1+x_2\rho_2+2\sqrt{\rho_1\rho_2}(Bp+Aq).} \tag{2}$$

The actual common source uses x=s/rho and the same m=f_sigma prime(x), eta=f_sigma(x)-xm. With u_i,v_i defined in the positive-contact note,

$$|F|^2=\sum_i(u_i^2+v_i^2)\rho_i+ 2\sqrt{\rho_1\rho_2} \big[(u_1u_2+v_1v_2)p+(u_1v_2+v_1u_2)q\big], \tag{3}$$

and

$$|\dot\Phi|^2=\sum_i\omega_i^2\rho_i+ 2\omega_1\omega_2\sqrt{\rho_1\rho_2}(Ap+Bq). \tag{4}$$

Equations (2) through (4) preserve both the field interference and its source. For distinct levels the azimuthal integral of the kinetic cross term is zero. At d>=3 and phi=pi/2 the formulas reduce to the orthogonal- column energy preparation, providing an independent comparison. At depth two its q term remains and prevents that identification. The two signs of X_0 give equal angular integrals by delta to minus delta symmetry.

The relative temporal phase evolves with omega_2-omega_1. No common clock, phase lock or externally imposed synchronization is introduced.

2. Finite Spatial Integration

Gaussian Jacobi quadrature integrates t=|w| squared with its actual sphere volume weight. A periodic midpoint rule integrates one complete difference- harmonic period. Orders 96, 192 and 384 are retained separately. Logarithmic local density scaling avoids underflow before forming x; the discarded underflow energy weights are recorded. Finite sampled positive density does not prove a global density floor: the pure beams have nodes at t=0.

The following values illustrate contact energy divided by isolated complete energy at the finest rule, using unit individual norms:

These are finite-quadrature illustrations, not certified precision results. In particular the depth-two phase-zero unit-norm sequence changes from 0.0000645 to 0.0001138 to 0.0001575 and is unresolved. The orthogonal-column comparison has an independently resolved exact piecewise integration and smooth-band correction; its depth-six ratio is 60.8189. The coherent phase-pi/2 sequence approaches that value imperfectly, retaining the need for transition-aware integration. Large positive costs are not the desired binding mechanism.

Equal isolated phase-action preparations are also retained. Their phase response differs from unit-norm preparations; neither normalization can be silently substituted for the other. At depth ten the second shell is the unassigned adjacent control. Five's phase response is not unique: six and other depths also respond under the same law. No neutrino-specific suppression or color selection follows.

3. A Useful Exact Integration Reduction

For d>=3 write z=sqrt(rho_2/rho_1)=C t^((l_2-l_1)/2). At fixed p the crossing of any scalar-ratio chart boundary x_b obeys the quadratic

$$(x_2-x_b)z^2+2p(B-x_bA)z+(x_1-x_b)=0. \tag{5}$$

Positive roots in the physical z interval split the t integral into exact unsmoothed source charts. Every term in (3) and (4) then has an incomplete beta integral, including the cross power (l_1+l_2)/2. This reduction provides a way to resolve moving sharp transitions before angular integration. The present finite quadratures do not execute that reduction or resolve every narrow smoothing band. Depth two retains an additional sqrt(1-t) term and requires its own crossing calculation.

Verification

coherent-positive-count-contact.py and its report pass 504 algebraic, trace, positivity and kinetic-integration checks, while retaining 108 spatial estimates for 36 phase-and-normalization preparations. The report includes the actual source hashes and convergence sequences. A failed raw-density pilot is preserved; its high powers underflowed before the invariant ratio was calculated. Logarithmic scaling fixes that numerical issue without changing the field.

For every maintained assignment, a separate 100-digit comparison finds positive source-norm changes on every other unsmoothed count chart. This finite chart diagnostic is not a proof for smooth transitions or every coherent field. The exact two-component identity in the positive-contact note supplies its interpretation.