IDOS Energy, Coherence, and Supporting Dynamics — Part 60

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$C_d=(j_2/j_1^2)F_{\rm phase}/\epsilon_c$ with $F_{\rm phase}=D^2/[(d+1)(D-d-1)]$ at $\eta=0$ or $(d+2)^3/[2d(d+3)]$ at $\eta=1$; $j_2=\int\chi^2d\mu_r$. Wall and core kinetic weights remain $\rho$, not $\rho r$. If $I_-$ and $j_{1,-}$ are the exact-shell and monotone-core lower bounds, then $r\le M=1+1/j_{1,-}+1/I_-$ and the complete entropy is bounded by $b_{\max}\log M$. This uses nonnegativity of $r\log r-r+1$ before normalization; it remains valid for variable $b_0$.

Whole-shell Arb bounds yield $E_\tau/Q-D\le Z_++\tau B_+$ with every $Z_+<0$, $B_+\ge0$. All 72 explicit trials are rigorously negative for $0<\tau\le10^{-5}$, with weakest common-threshold upper bound below $-4.9993\times10^{-6}$. At the unchanged $\tau=1$, this explicit family certifies 19 of 72 cases; the earlier 48 of 72 optimizer results remain numerical comparisons. No sign is certified for the other 53 explicit trials by this bound. All relevant diagonal charges, $P=0$ and $J=0$ are exact; continuity determines density velocity. Each fixed positive $\tau\le1$ has its own compact-data local evolution. There is no uniform lifespan, $\tau=0$ PDE or trajectory from the high state to a competitor.

Exact Response Of The Fixed Constrained Density Minimum

For fixed bounded $H,t,b_0$, $0<b_{\min}\le b_0\le b_{\max}$, nonconstant $t$, $C_d>0$ and normalized $\mu_r$, define $\mathcal F=\min_r[\int Hr+C_dy^2/2+\tau\int b_0\phi(r)]$, $y=\int tr$, $\int r=1$,

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