IDOS Energy, Coherence, and Supporting Dynamics — Part 23

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$c_A=\prod_{B\ne A}\|z_A-z_B\|^{-6}$. It can strongly weight unrelated clocks between targets: the constructive equal-envelope electron/muon preparation gives about $0.739664$ weight to up and $0.259410$ to strange. These are geometric weights, not particle conversion probabilities or numerical comparisons of sector units.

Removing target-distance normalizers gives another positive interpolant with the same exact values and flat jets but weights proportional to $\|z-z_A\|^{-6}$. The same preparation then weights the third neutrino target by about $0.998578$. Neither Euclidean distance rule was derived as an interaction law. This comparison exposes a genuine gap between reproducing isolated clocks and determining encounter response.

One separate common-function coefficient choice is $\kappa_a(z)=8/m(z)$ and $b(z)=m(z)/128$. Their product stays $1/16$, preserving the saturated phase cone bound. At a selected branch, the static Schur term becomes $7n\omega I/8-S_{\rm grad}\ge n\omega I/8$. Algebraic coefficient derivatives introduce lower-order interactions, not new principal cross terms. This changed law removes the global $\omega_{\min}$ from fractional compression near each target, but retains the intrinsic slope $128(r-1)/\sqrt r$, $r=D_b/D_a$. For electron/muon this is approximately $1831.650671$; a log-density change of $0.1$ has leading envelope scale about $5.46\times10^{-5}$. That is an asymptotic preparation scale, not a finite bound or observed coupling. Off-branch interpolation sensitivity remains.

The reproducible calculation in causal-pair-density passes 767 checks and records 165 preparations across all eleven actual same-family pairs, plus mathematical controls at all six maintained depths. It keeps sector units, lengths, amplitude normalization and interaction strengths explicit. It supplies constrained local data and density diagnostics, not a pair solution, scattering law or force obtained by differentiating an incompatible prescribed overlap.

A Common Flattened Count Constitutive Law

An explicit alternative removes unrelated-sector clock mixing on the same-depth preparations above. Let the nonnegative mean labels be $q_i=\langle e,T_i e\rangle_H$, and set

$$z_1=\sum_iq_i,\quad z_2=\sum_iq_i^2,\quad a=\frac{z_1^2}{z_2},\quad N= z_1+\frac{z_2}{z_1}\qquad(z_1>0).$$

The physical domain obeys $1\le a\le10$ and $z_1\le N\le2z_1$. On a uniform depth-$d$ prefix of cutoff $n>0$, these recover $a=d,N=n$. For general anisotropic fields, $a$ is a participation ratio, not support rank. For example $q=(1,1,4,0,\ldots)$ gives $a=2,N=9$ despite three nonzero axes. Occupation selection remains a separate part of the action.

Define one flattened count coordinate and six positive count functions:

$$s(N)=N-\frac{3\sin(2\pi N)}{4\pi} +\frac{3\sin(4\pi N)}{20\pi}-\frac{\sin(6\pi N)}{60\pi}, \qquad s'(N)=\frac{16}{5}\sin^6(\pi N),$$ $$f_d(N)=\frac{\mu_d}{d!}\prod_{j=1}^d[s(N)+j],\qquad \mathcal D=\{2,3,4,5,6,10\}.$$

At every nonnegative integer $n$, $s(n)=n$ and derivatives one through six vanish. Thus $f_d(n)=\mu_d\binom{n+d}{d}$ with flat derivatives. Set $L_d(a)=\prod_{j\in\mathcal D,j\ne d}(a-j)^6$, $W_d=L_d/\sum_jL_j$, and $F(a,N)=\sum_dW_d(a)f_d(N)$. The denominator is everywhere positive; at a maintained depth only its own weight remains, with derivatives one through five flat.

All zero-label states coincide. They cannot carry six distinct clocks through these descriptors. Choose the common value $\omega_0=\mu_3$

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