IDOS Energy, Coherence, and Supporting Dynamics — Part 36

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Both logarithmic normalization gradients are bounded at each fixed $k$. Thus the normalized time operator remains bounded and skew. Its bound grows with $k$; no uniform sharp-core dynamics follows.

One static term needs additional care. If $g_W=(9/4)W^{5/4}-(5/2)W^{3/2}$, the algebraic support square is

$$|\ell_k\cdot\zeta|^2,\qquad \ell_k=\frac{\sqrt{\kappa_G}}{\sqrt\mu}(-g_W,2Z).$$

Its vector coefficient grows as $r^{2k-17}$ and is unbounded for $k\ge9$. The exact form domain therefore consists of normalized $L^2$ coordinates, $H^1$ for all except vector support, with $W\nabla\zeta_Z\in L^2$ and $\ell_k\cdot\zeta\in L^2$. The latter condition controls the combined scalar expression; requiring each component product separately would remove allowed cancellations. It is automatic only for $4\le k\le8$. Compact cutoff and local smoothing still give a closed positive form and its Friedrichs realization. The bounded-gyro wave argument then supplies global occupied-block linear evolution, conserved quadratic energy and at most linear displacement growth.

On compact positive-density charts the principal speed ratios remain the SF-DW ratios, since kinetic and corresponding spatial coefficients share $\mu$. At vacuum, however, $\mu\sim Cn_{\rm peak}W^{k/2-2}(1+2kW^{1/4}+\cdots)$. The positive-side law has no matching $C^2$ extension for $k=4,5,6,7$; for $k\ge8$ an even $C^2$ continuation exists but its kinetic coefficient vanishes. Neither a regular nonlinear vacuum theory nor an infinite-$k$ evolution limit is established. The earlier density-flatness integrals remain valid, but the strict constant-$\sigma_0$ clock is still not met. The same law applies at all six maintained depths without a special five-dimensional term. Sources are flat-core-coupled/report.md and the independent 97-check flat-core-review/report.md.

Tail Energy Comparisons With Fixed Momentum

The nonlinear tail competitor can also preserve the complete initial translation momentum. For constant positive support inertia, allow only an extra scalar support velocity $W_t$, keeping its position and the other support/frame/flow data initially unchanged. Since SF-DW's auxiliary density equation contains no support derivatives, $n=\rho$ and the same prescribed carrier velocity still give exactly the same charge. The full canonical translation generator and extra energy are

$$P=-p_X-M_W\int W_t\nabla W\,dx,\qquad p_X=\int\rho a\,dx,\qquad E_{\rm support}=\frac{M_W}{2}\int W_t^2dx.$$

For the rational profile,

$$M_W\int\partial_iW\partial_jW\,dx=g\delta_{ij},\qquad g=\frac{M_W\pi^5L^8}{1890} =\frac{Q\omega_{\min}}{159}$$

for the stated common inertia. Set $W_t=-(p_X/g)\cdot\nabla W$. Then $P=0$ exactly and the cost is $|p_X|^2/(2g)$. The bubble momentum is bounded by a constant times its tail weight, so this cost is quadratic in that weight while the negative bulk gain is linear. Far enough out, the total energy is still lower at fixed $(Q,P=0)$, and the support velocity tends to zero in its kinetic norm.

Continuity fixes $n_t=\operatorname{div}(\rho h_A)$ on this initial slice; it need not equal the target-density derivative $(n_0)_t=2n_{\rm peak}WW_t$. The auxiliary reconstruction determines the corresponding phase acceleration. No density momentum or constraint velocity is omitted. This conclusion fixes no additional angular momentum or separate populations and remains an initial-data comparison, not a decay mechanism. tail-fixed-momentum/report.md and the independent 238-check tail-momentum-review/report.md give the full constrained Noether derivation.

Why The Directional Hessian Is Not A Uniform Nonlinear Approximation

A precise limitation follows already on the fixed-$Q$ carrier slice. Parametrize its normalized carrier by $e(v)=\cos|v|\,e_H+\operatorname{sinc}(|v|)v$, $v\perp e_H$, and let $F(e)$ be its exact reduced initial energy. For every fixed smooth compact tangent direction, $F'(0)=0$ and $F''(0)\ge0$. Nevertheless this does not give a uniform second-order Frechet expansion in the norm $\|v\|_X^2=\int\rho(|v|^2+|\nabla v|^2)dx$.

Indeed each tail competitor stays uniformly away from the antipode of $e_H$, so its sphere logarithm $v_X$ is smooth and compactly supported,

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