IDOS Energy, Coherence, and Supporting Dynamics — Part 51

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$e_t=(H+b\log r)Je+K$; continuity fixes $n_t$ and physical reconstruction retains its density connection. The unique density optimum in this family has $r=\exp[-(H-m_H+\lambda t+\nu)/b]$, with $\lambda=C\int rt\,d\mu$ and normalization fixing $\nu$.

For $N_H=2,N_L=1$, the 72 controls at all six maintained depths, three lengths and both coefficient and phase-edge choices give 48 lower numerical energies: 2 of 24 at $\ell=1$, 22 of 24 at $\ell=10$, and 24 of 24 at $\ell=100$. Four quadrature orders resolve the actual layers; the final energy refinement is at most $9.592\times10^{-13}$ per $Q$. Sixteen independently adaptive evaluations agree within $4.69\times10^{-13}$ per $Q$. These are numerical accuracy diagnostics, not rigorous continuum error enclosures. Density remains analytically positive even when its logarithm is below $-3500$ and floating output underflows. In the depth-ten, $\ell=100$, positive-edge constant-$b$ example, about $80.45\%$ of charge lies in the joining layers; the low plateau's charge is extremely depleted.

The separate actual-prefix batch covers eleven maintained same-depth pairs and thirteen prefix-one controls. It compares the same five bounded trials $r_\alpha=e^{\alpha f}/Z_\alpha$, $f=t/(E_c/D)$, $\alpha\in\{-4,-1,0,1,4\}$, so $e^{-4}\le r_\alpha\le e^4$. Across 288 comparisons it finds 225 lower-energy selections: 51 of 96, 78 of 96 and 96 of 96 at the three lengths. Twenty-two independent adaptive controls agree to relative error at most $2.191\times10^{-14}$. The selected negative margins exceed $0.3022$ per $Q$; the largest selected quadrature refinement is $1.582\times10^{-5}$ per $Q$. This batch selects among five trials; it does not compute the density optimum. Common $\mu_d=1$ units are not the calibrated unequal physical sector units. Bottom remains unassigned, and tau has no invented same-depth observed partner.

Compact relative recovery also works for these nonoptimal trials. On the unchanged high core and exterior their original density is the same $c_\infty=1/Z_\alpha$, with the same $H=m_H$, $b=b_H$ and charge features. Use the preceding cutoff and normalized core correction. All linear energy terms have the same coefficient $m_H+b_H\log c_\infty$ on the changed regions and cancel upon integration. The excess is exactly $b_H\int\rho\,k(r_R,c_\infty)$ on those regions, of order $R^{-6}$ when $c_\infty\ne1$. Hence every genuinely negative continuum margin survives sufficiently remote recovery. The proof does not itself certify floating signs, a uniform evolution time or a trajectory to these data.

The independently reviewed sources smooth-radial-density-batch and actual-prefix-radial retain all trial choices, source versions and quadrature ledgers. No special depth-five effect, observed decay or physical mass-operator selection follows from these comparisons.

Joint Scalar And Vector Support Recoil

The scalar recoil endpoint admits a larger, exactly solvable velocity family. Keep rational support positions, rest $U$, constant frame and

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