Full Variation and Pair Energy of the Positive Count Action

The positive count wave action retains exact isolated frequencies and has nonnegative complete energy. Its interaction must be calculated from that action. This calculation verifies its full source variation and evaluates actual aligned two-branch preparations under the same smoothed law. Every displayed aligned contact energy is positive relative to the isolated reference. This is an energy result, without an inferred physical radial force or encounter trajectory.

1. The Complete Real Source Derivative

For rho=tr Phi dagger Phi, s=tr Phi dagger beta Phi and x=s/rho, keep

$$G=(m\beta+\eta I)\Phi,\quad m=f_\sigma'(x),\quad \eta=f_\sigma(x)-xf_\sigma'(x),\quad F=D_R\Phi+G.$$

Let v=(beta-xI)Phi. The full real derivative is

$$DF[\Phi]u=(D_R+m\beta+\eta I)u+ \frac{2f_\sigma''(x)}\rho v\operatorname{Re}\operatorname{tr}(v^\dagger u). \tag{1}$$

The last term remains at a smoothed transition even though it vanishes near every isolated count node. Independent differentiation in four real component coordinates checks the gradient, source Hessian, real adjoint and variation of norm(F) squared across a transition at all six occupied depths. The algebraic carrier is a finite Hermitian matrix solely for this variation check. It is not substituted for the sphere's spatial operator. The relative positive-square variation error is below 1.4e-87; the independently differentiated source derivative error is below 3.5e-77. Spatial domain and persistence statements use the analytical proof in the linked positive action note.

2. Recalculate the Prepared Pair Energy

Put two balanced count branches in orthogonal column blocks of the one field. Each retains its own isolated clock in the preparation:

$$\Phi=[\Phi_1,\Phi_2],\quad \dot\Phi_i=-i\omega_i\Phi_i,\quad \rho=\rho_1+\rho_2,\quad s=x_1\rho_1+x_2\rho_2.$$

The local source uses m=f_sigma prime(x) and eta=f_sigma(x)-xm. Write Phi_i=(a_i+b_i beta)E_i, where D_R E_i=h_i E_i in radius units R=1. Then its actual source is

$$F_i=(u_i+v_i\beta)E_i,\quad u_i=h_i a_i+m b_i+\eta a_i,\quad v_i=-h_i b_i+m a_i+\eta b_i. \tag{2}$$

Balance gives norm(F_i) squared=(u_i squared+v_i squared)rho_i. Consequently the complete energy density is

$$\boxed{e_+=\sum_i[\omega_i^2+u_i^2+v_i^2]\rho_i\ge0.} \tag{3}$$

For delta m_i=m-m_i, delta eta_i=eta-eta_i and the tangent deficit g_i=-(delta m_i x_i+delta eta_i), the change from the isolated energy is

$$\boxed{\Delta e_+=\sum_i\rho_i \left[(\delta m_i)^2+(\delta\eta_i)^2+ 2x_i\delta m_i\delta\eta_i-2\omega_i g_i\right].} \tag{4}$$

The source-mismatch square competes with the negative term. Under an unsmoothed winner, g_i is nonnegative; in a smoothed band its sign follows the actual source and must not be presumed. Formula (4) is verified independently against the full two-component norm (2), with relative error below 5.3e-93. The first-order attractive deficit is not the energy of this action.

At an unsmoothed chart, a useful equivalent formula is

$$\Delta e_i/\rho_i=g_i^2+c_i^2(\delta m_i)^2-2\omega_i g_i. \tag{5}$$

It displays explicitly why retaining an isolated clock does not retain its previous pair response. Two patterns in the same branch have exactly the same local affine source at every density ratio; their orthogonal-column contact remains zero for all fourteen maintained same-branch controls.

3. Integrate the Actual Smoothed Source

Aligned beam densities are rho_i=N_i C_i t^(l_i), with t the squared circle-plane projection, C_i=((d+1)/2)_(l_i)/l_i factorial and ordinary volume density b(1-t)^(b-1), b=(d-1)/2. The integrated norm N_i is free. The unsmoothed chart contributions in (4) are exact incomplete beta integrals. The smoothing correction is integrated separately through each transition band; it is included in every result below.

For ratio r=rho_2/rho_1 and L=l_2-l_1>0,

$$r=\frac{x_1-x}{x-x_2},\quad t=\left(\frac{rN_1C_1}{N_2C_2}\right)^{1/L},\quad \left|\frac{dt}{dx}\right|= \frac{t(x_1-x_2)}{L(x_1-x)(x-x_2)}. \tag{6}$$

Using x=boundary+sigma u resolves even very narrow smoothing bands at 100-digit precision. It avoids missing a transition between finite t samples. The pair energy comparison uses both N_i=1 and equal isolated phase action Qplus_i=1, which implies N_i=1/(2 omega_i).

At ten, level 23 is an unassigned adjacent-shell control; level 22 is the maintained tau. Same-branch tau contact is checked separately and is zero. The depth-three uniform level makes this preparation orientation independent. The law has no special depth-five coefficient; five's positive contact is not uniquely suppressed. These large costs expose a substantial interaction constraint despite the successful isolated frequencies and persistence.

The preparation has no relative-phase dependence because its column blocks are orthogonal. Coherent preparations require a new calculation. These are internal spatial patterns on an assumed sphere, with isolated references; there is no established interparticle separation coordinate in this model. Positive contact energy alone is not proof of radial repulsion.

4. A Competing-Range Interaction Hypothesis

A long-range attractive response partly cancelled by a shorter-range repulsive response can yield net attraction increasing with separation. Both could be contributions from the same physical field. A simple mathematical comparator is

$$U(r)=\sigma r+A a e^{-r/a},\qquad F_r=-\frac{dU}{dr}=-\sigma+A e^{-r/a},\quad 0<A<\sigma. \tag{7}$$

The inward force magnitude sigma-A exp(-r/a) increases monotonically toward sigma. This proves the cancellation mechanism is mathematically possible; it does not derive either term from the count action. With two finite-range exponentials, increasing attraction can instead hold over a finite interval before both responses decay.

The force and separation energy are distinct tests. The conventional heavy-quark Cornell comparison U=sigma r-kappa/r has inward force sigma+kappa/r squared, tending toward a constant even though the energy increases with separation. See the explicit potential and phenomenological comparison in Mateu et al., equation (1). This is an external comparison, without importing its mediator interpretation into IDWT. Increasing separation energy does not require increasing force or uniquely imply two forces.

Testing the competing-range hypothesis requires a physical separation coordinate, field profiles extending through the containing manifold and the derivative of their complete same-field energy. Neither circle alignment nor the positive prepared contact supplies that calculation. In particular, the zero same-branch contact does not explain color binding. A viable common mechanism must also explain why the analogous higher-depth channels do not generate their own strong interactions.

Verification

A pointwise interval certificate now extends the positive prepared-energy result to every maintained distinct same-sector pair, all nonnegative density ratios and arbitrary spatial orientation, including every smooth transition. It excludes energy below the isolated reference within the orthogonal-column preparation, while retaining coherent fields, evolving shapes and force gradients as separate questions.

The coherent matrix-field continuation restores relative-phase interference under the same source, including the exceptional depth-two cross grading. Finite quadratures retain positive costs and explicit convergence limits; they do not select depth five or establish a physical force.

positive-count-contact.py and its report pass 526 checks of the full variations, source norms, all maintained same-branch zero controls and twelve smoothed pair preparations. The source and dependency hashes identify the executed equations. Analytical incomplete-beta integration and explicitly resolved transition corrections are distinct from a field evolution or measured force comparison.