IDOS Energy, Coherence, and Supporting Dynamics — Part 58

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decomposition against $D_i=(\partial_iW,\partial_iZ)$ in the $\mu$-weighted kinetic space. The sharp integration limit gives

$$A_W^0=160\log2-328/3,\quad A_Z^0=1915/3-920\log2,\quad \frac{Q\omega_{\min}}g\longrightarrow\frac{212}{529-760\log2} =96.0082846335\ldots .$$

The combined/scalar cost ratio tends to $.7111023682\ldots$. These replace the constant-inertia values for this changed profile. At $\varepsilon_p=.01$, analytic moment bounds evaluated with Arb give $95.93859<Q\omega_{\min}/g<96.11209$. For fixed $p,Q,\sigma$ and $\omega_{\min}$ the cost and velocities stay bounded as the layer narrows. They are initial-jet results, and $g$ is not the complete boost inertia $mQ$. For a nonzero smooth radial cutoff $0\le\chi\le1$, define $g_{Aj}$ by inserting $\chi^j$ in each Gram integral. The fixed compact family's minimum uses $g_\chi=g_{W1}^2/g_{W2}+g_{Z1}^2/g_{Z2}\le g$ and $v_A=-(g_{A1}/g_{A2})p/g_\chi$. Arbitrary signed cutoffs are not included.

Physical rotations obey $(Bx)\cdot\nabla W=0$ and $(Bx)\cdot\nabla Z=BZ$ for antisymmetric $B$, so support orbital and intrinsic angular charges cancel pointwise. Material coordinates retain the induced frame covector even at zero frame velocity. Coefficient forces remain: on the stated slice,

$$\mu W_{tt}+\tfrac12\mu'(W_t^2-|Z_t|^2)=q_0'Q_{\rm der},\quad \mu Z_{tt}+\mu'W_tZ_t=0,\quad F_t=b[\nabla\cdot(q_0h_A)/q_0-(q_0'/q_0)W_t].$$

At $s=1$, $Wq_0'/q_0=(2+\varepsilon_p)/(2e\varepsilon_p)$ and $\mu'/\mu$ diverges like $16/(e\varepsilon_p)$. Bounded recoil cost therefore does not imply bounded acceleration or sharp-interface dynamics.

Explicit Positive-Entropy Competitors In The Rational Density Family

The following calculation uses the earlier rational density action, not the new plateau. In its 72 common- unit controls, $N_H=2,N_L=1$, $d=2,3,4,5,6,10$, $\ell=1,10,100$, $\eta=0,1$, and both $b_0$ laws, change

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