IDOS Energy, Coherence, and Supporting Dynamics — Part 7

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The zero label uses only $u_0$. This gives $|e_0|=1$, $a_0=0$ and a constant active matrix with diagonal $N(2N+d)/[(d+1)(d+2)]$ and off-diagonal $N(N-1)/[(d+1)(d+2)]$. Its nonzero eigenvalues are

$$\lambda_\perp=\frac{N(N+d+1)}{(d+1)(d+2)},\qquad \lambda_\parallel=\frac{N[(d+1)N+1]}{(d+1)(d+2)},$$

with multiplicities $d-1$ and one. For $N>0$ the rank is $d$; for $N=0$ the matrix vanishes and cannot determine a depth. The trace is $t=dN(2N+d)/[(d+1)(d+2)]$.

One explicit bounded completion uses the smooth step $s(x)=e(x)/[e(x)+e(1-x)]$, where $e(x)=e^{-1/x}$ for $x>0$ and zero otherwise. Set $\eta=1/22$ and

$$r(C)=\operatorname{tr}s(2C/\eta-1),\qquad \chi_d(r)=1-s([16(r-d)^2-1]/3).$$

The nonnegative gates have disjoint positive supports. At all maintained nonzero targets, $r=d$ on a neighborhood, including repeated eigenvalues. For maximum cutoffs $(M_2,M_3,M_4,M_5,M_6,M_{10})=(94,3,71,21,34,22)$, define

$$F_d(n)=1+\sum_{j=1}^{M_d}\binom{j+d-1}{d-1}s(n-j+1),\qquad n_d(t)=\frac{-d+\sqrt{d^2+8t(d+1)(d+2)/d}}4.$$

The Hockey-Stick identity gives $F_d(N)=\binom{N+d}{d}$, with every positive-order derivative zero at the retained integers. Put $m_d(t)=\kappa_dF_d(n_d(t))$, $m_0=\kappa_3$, $K=8933/3$ and $\zeta(t)=1-s((t-K)/K)$. One common local law is

$$m(C)=m_0+\zeta(t)\sum_{d\in\{2,3,4,5,6,10\}} \chi_d(r(C))[m_d(t)-m_0],\qquad t=\operatorname{tr}C.$$

It is positive, bounded and invariant under a common physical spatial rotation. At all fourteen assigned targets, $m=\kappa_dD$ and $Dm=0$. The zero-matrix down target uses the supplied baseline $\kappa_3$, not an inferred depth. The common formula still contains prescribed sector units, rank gates and a finite ladder. Its flat integer jets lose the positive count-direction curvature of the earlier oscillatory polynomial flattening. It supplies no special depth-five coefficient or exceptional response.

The trace cutoff is needed for the stated global regularity estimate. Bounded eigenvalue gates alone allow one transition eigenvalue while another grows without bound. Here $Dm$ vanishes for $t\ge2K$, so both $Dm$ and $tDm$ are bounded on the positive-semidefinite cone. The local energy and variation coefficients obey, with $z=\psi$, $v=\nabla\psi$,

$$|h|\le C(|z|^2+|z||v|),\qquad |h_z|\le C(|z|+|v|),\qquad |h_v|\le C|z|.$$

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