Exact Beam Contact and a Strong-Overlap Energy Constraint

The count envelope predicts an explicit interaction between the closed-circle fields without adding a separately fitted pair coefficient. For the preparation below it is attractive, and alignment maximizes its magnitude. This is a mathematical consequence of the specified effective action. Its strong overlap also reveals a negative complete energy preparation, so the interaction cannot be promoted into a satisfactory physical encounter law.

1. Two Branches of One Field

Use the same round sector sphere, scale and finite envelope as before:

$$V(s,\rho)=\max_k(m_ks+\eta_k\rho),\qquad f_i=m_ix_i+\eta_i,\qquad x_i=\sqrt{1-c_i^2}.$$

Put two independently oriented massive beam fields in orthogonal column blocks of the same classical spinor field. Each separately has density rho_i, scalar density x_i rho_i and zero tangential current. The complete summed invariants are

$$\rho=\rho_1+\rho_2,\qquad s=x_1\rho_1+x_2\rho_2,\qquad j=0. \tag{1}$$

This is a substantive field-component preparation. It does not assert that all pairs occupy orthogonal columns. In this preparation the action has no relative-phase dependence: each column's phase cancels from (1). The particles retain their own isolated frequencies; synchronization is not imposed. Cross-depth extensions and physical detection are not supplied.

Define the endpoint deficits

$$A_k=f_1-(m_kx_1+\eta_k),\qquad B_k=f_2-(m_kx_2+\eta_k). \tag{2}$$

The unique isolated winner conditions give A_k,B_k nonnegative. The complete interaction-energy difference is exactly

$$\boxed{\Delta V=-K(\rho_1,\rho_2),\qquad K(a,b)=\min_k(A_ka+B_kb)\ge0.} \tag{3}$$

The kinetic energy adds across the column blocks; no kinetic cross term has been dropped. The proper-current envelope gives the same instantaneous potential and source when j=0, but its subsequent response differs.

For distinct branches and a,b>0, K is strictly positive: no single chart wins at both isolated endpoints. For the same branch, K=0 identically, for every relative orientation. Thus this particular action and column preparation do not supply a universal same-species force.

2. A Positive Layer-Overlap Formula

Order the active charts from branch 1 to branch 2 so that A_k increases and B_k decreases. Their positive adjacent crossing ratios and slope drops are

$$r_j=\frac{A_{j+1}-A_j}{B_j-B_{j+1}},\qquad \sigma_j=B_j-B_{j+1}>0.$$

The concave piecewise-linear function min_k(A_k+B_k r) begins with slope B_1 and ends with slope zero. Hence

$$\boxed{K(a,b)=\sum_j\sigma_j\min(b,r_ja),} \tag{4}$$

with telescoping identities sum sigma_j=B_1 and sum sigma_j r_j=A_2. This is an exact decomposition of the actual interaction, not an assumed density-overlap proxy.

The elementary layer identity

$$\int\min(b,r_ja)\,dV =\int_0^\infty \operatorname{Vol}\{b>u,\ r_ja>u\}\,du \tag{5}$$

proves a global orientation statement. Each beam density is a monotone function of squared distance from its great-circle plane. When the circles are aligned, all their superlevel sets are nested. Their intersections then attain the largest possible volume, the smaller of the two individual volumes. Every positive term in (4) is maximized simultaneously.

Alignment therefore maximizes attractive contact over all relative circle orientations, with no sampled-angle argument required. A uniform branch gives orientation-independent contact; the maintained depth-three down comparison has l=0 and provides that control. The statement supplies an energy gradient and reciprocal generalized orientation torque under the common rotation-invariant action. It does not derive the resulting encounter trajectory or identify an empirical interaction.

3. Exact Spatial Integration and Bounds

For aligned normalized beams, rho_i=N_i t^(l_i). Each term in (4) is integrated by splitting at

$$t_j=\left(\frac{r_jN_1}{N_2}\right)^{1/(l_2-l_1)}$$

when that number belongs to (0,1). The two integrals are regularized incomplete beta functions with parameters l_i+1 and (d-1)/2. This gives the displayed aligned values at high precision without a spatial simulation.

For orthogonal circle planes, available when d>=3, the exact classical shape overlap is

$$\mathcal O_{12}=\int\sqrt{\rho_1\rho_2}\,dV =\sqrt{N_1N_2} \frac{\Gamma(1+l_1/2)\Gamma(1+l_2/2)} {((d+1)/2)_{(l_1+l_2)/2}}. \tag{6}$$

Since K<=min(A_2 rho_1,B_1 rho_2),

$$\boxed{|\Delta E|\le\sqrt{A_2B_1}\,\mathcal O_{12}.} \tag{7}$$

For the depth-six electron/muon comparison, aligned contact magnitude is 2.09130667135 in inverse-radius units, while the orthogonal-circle bound is 0.000198774882. The strong orientation dependence is therefore proved by the bound, rather than relying on rare-overlap numerical samples. Depth five and the depth-ten adjacent-shell control also have strong proved aligned-versus-orthogonal separation. No uniquely small depth-five interaction follows. At depth two, any two great circles intersect and the orthogonal-plane comparison is unavailable.

4. Complete Energy Changes the Interpretation

The same calculation retains the complete pair energy. At unit norm for each isolated shape,

$$E_{\rm pair}=\omega_1+\omega_2-|\Delta E|. \tag{8}$$

An independent piecewise integration uses the actual winning potential and the kinetic term h_1 c_1+h_2 c_2. It agrees with (8) at more than 70 decimal places.

The depth-ten second shell is an unassigned mathematical control, rather than an additional observed particle. Same-branch zero-contact controls include every maintained assignment.

At six the attraction exceeds the sum of the isolated positive energies: the aligned electron/muon preparation has negative complete energy. Amplitude homogeneity then permits an arbitrarily negative energy under overall scaling if norm is unrestricted. This does not invalidate the local nonlinear persistence theorem near a nodeless isolated state. It does constrain interpreting this first-order envelope as a global physical encounter action. A positive completion must be varied and checked afresh; the attraction and alignment results cannot be transferred automatically.

5. Reproducibility and Scope

count-envelope-beam-contact.py and its report pass 445 checks. At 100-digit precision they verify every endpoint deficit, the complete finite envelope, the layer decomposition, aligned beta integrals and the independent complete-energy ledger. The final report retains the negative energy classification explicitly.

Bounded importance quadrature at 2,048 and 8,192 samples illustrates nonzero angles. Independent variation against one fixed proposal checks the orientation-energy derivative. Rare orthogonal overlap is not resolved at these sample sizes; equation (7), rather than a small numerical sample mean, is the reliable claim. Exact earlier source and report bytes remain preserved.

This test connects the count action to real spatial contact and also exposes its strong-overlap energy limit. It does not supply observed charge, color, cross-depth coupling, phase-sensitive matching or a neutrino-specific suppression. Those are additional constraints on any improved action.