A Pointwise Energy Constraint on the Positive Count Interaction

The positive contact costs are a general feature of the stated orthogonal- column preparation, rather than a result confined to aligned beams. At the maintained diagnostic scales, a pointwise inequality covers every density ratio and every relative spatial orientation for all maintained distinct same-sector pairs. It includes the actual smooth source through every transition band.

This excludes energy below the isolated reference within that preparation. It does not calculate a physical radial force, exclude local metastable wells or cover coherent fields and time-evolved shape changes.

1. Reduce the Energy to One Scalar Ratio

Retain the complete pair preparation and notation of the positive contact calculation. For two balanced shell fields in orthogonal column blocks, initial velocities retain their own isolated clocks. Let rho_i be any nonnegative local densities, and set

$$x=\frac{x_1\rho_1+x_2\rho_2}{\rho_1+\rho_2},\qquad x_1>x_2.$$

The source coefficients are m=f_sigma prime(x) and eta=f_sigma(x)-xm. For each branch define

$$d_i(m,\eta)=(m-m_i)^2+(\eta-\eta_i)^2+ 2x_i(m-m_i)(\eta-\eta_i) +2\omega_i[(m-m_i)x_i+\eta-\eta_i]. \tag{1}$$

The complete energy difference is

$$\Delta e_+=\rho_1d_1+\rho_2d_2 =(\rho_1+\rho_2)R(x),$$ $$R(x)=\frac{x-x_2}{x_1-x_2}d_1+ \frac{x_1-x}{x_1-x_2}d_2. \tag{2}$$

The original actual-field source norm identity establishes (1). No beam profile, unit norm or orientation is used in (2). It therefore applies to arbitrary balanced isolated-shell shapes under the same preparation.

On an unsmoothed affine chart, interval arithmetic verifies d_i>0 at every other eligible branch and d_i=0 on its own branch by exact identity. Nonnegative density weights then give the desired inequality outside the smoothing bands.

2. Certify Every Smooth Transition

Sort the source lines by increasing m. At a boundary x_b between a left line and a right line, let Delta m=m_right-m_left>0. Write x=x_b+sigma(2t-1), t in[0,1], using the same common smoothing width sigma fixed by all eligible count nodes. The kernel integral and its first moment simplify to

$$C(t)=10t^3-15t^4+6t^5,\qquad M(t)=-10t^3(1-t)^3.$$

The actual smooth source is exactly

$$m(t)=m_{\rm left}+\Delta m C(t),\qquad \eta(t)=\eta_{\rm left}-\Delta m[x_bC(t)+\sigma M(t)]. \tag{3}$$

Substitution into (2) gives a polynomial of degree at most thirteen. For power coefficients a_k, its degree-n Bernstein representation is

$$R(t)=\sum_{j=0}^n b_j\binom nj t^j(1-t)^{n-j},\qquad b_j=\sum_{k=0}^j a_k\frac{\binom jk}{\binom nk}. \tag{4}$$

Every Bernstein basis term is nonnegative and their sum is one. Positive lower enclosures of all b_j therefore prove R(t)>0 on the entire band, without sampling between its endpoints. The calculation starts from exact rational count coefficients and encloses their square roots, line boundaries and common minimum-defined width with 80-digit outward interval arithmetic.

All 226 transition bands across the twelve pairs have strictly positive Bernstein coefficient enclosures without subdivision. The full enclosures are retained in the final report. An independent 100-digit evaluation of the original smooth source at five points per band lies inside the interval polynomial's enclosures, checking the polynomial construction against the separately implemented source.

3. All Maintained Pair Comparisons

The maintained tau same-branch comparison remains exactly zero and is checked separately, as are all other maintained same-branch controls. The second depth-ten branch is not identified with an observed particle. Neutrino labels remain assigned spectral comparisons.

Combining the affine-chart signs and the band certificates gives

$$\boxed{\Delta e_+\ge0,\qquad \Delta E_+\ge0} \tag{5}$$

for all nonnegative densities of each listed pair in the stated preparation. It covers arbitrary individual amplitude normalization and spatial orientation. It does not depend on the finite spatial quadratures or their convergence. Distinct branches have a strict positive pointwise cost wherever both densities are nonzero.

4. Physical Scope

The positive action retains exact isolated count clocks and its separate persistence theorem. Its orthogonal-column prepared contact nevertheless does not supply energetic binding below the isolated reference at these scales. A positive energy function can still have gradients of either sign or local wells above its asymptotic reference. Neither repulsive force at every distance nor absence of finite-lived composites follows from (5).

Coherent fields change the local interference and are not covered by this inequality. Their existing finite quadratures retain their own limits. Actual encounters can also leave the isolated-shell preparation. The result therefore constrains this specific effective interaction rather than excluding IDWT or the proposed competing-range mechanism.

Reproducibility

positive-count-contact-bound.py and its final report pass 1,362 checks, including 226 interval certificates, 1,130 independent point evaluations and six common-width comparisons. The earlier 226-check report is retained with its exact executed source. Two implementation failures are preserved: an unsupported direct Fraction-to-interval conversion and a reused variable in the added report check. Neither changes the energy formulas. The final source explicitly converts rational factors and retains separate names for scalar and list values.