IDOS Energy, Coherence, and Supporting Dynamics — Part 65

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do not add canonical momentum terms because $\epsilon$ has no derivative dependence; its spacetime derivative is nevertheless present in current conservation.

For a nonnegative compact radial $f$, set $p=a^2f^2<1$ and prepare the occupied/empty pair as $(\sqrt{1-p},af\exp[J(l-k)\cdot x/\ell])$. Choose relative phase rate $-1/[\epsilon_a(1-p)]$ and total clock $U\cdot a=h+\Xi_q$, where $\epsilon_a=\epsilon(p(x))$. Then $\rho=\rho_0$ and the empty charge vanishes pointwise. With radial amplitude rate $\gamma$ the remaining canonical momentum density is $\epsilon_a\rho_0\gamma\,\partial_i p/[2(1-p)]$; its integral vanishes, as does angular momentum. Define

$$A_e=\int\rho_0\epsilon_a\frac p{1-p}\,dx,\quad A_{\rm inv}=\int\rho_0\frac p{\epsilon_a(1-p)}\,dx,$$ $$P(a)=\int\rho_0[h+\Xi_q-\Omega]dx+ \frac{a^2}{8}\int\rho_0\epsilon_a\frac{|\nabla f|^2}{1-p}\,dx.$$

The exact energy difference is $P+\gamma^2A_e/2+A_{\rm inv}/2$. Taking $\gamma^2=-(2P+A_{\rm inv})/A_e$ restores energy exactly when its smooth limiting value is positive. That limit is $\Gamma^2-\lambda_f/4$, where $\Gamma^2=-\epsilon_k^{-2}-2Q_0/\epsilon_k>0$ and $\lambda_f=\int\rho_0|\nabla f|^2/\int\rho_0f^2$. Broad fixed cutoffs make $\lambda_f$ small. The radial initial rate $\gamma(0)$ differs from the ground-mode exponent $\Gamma$.

The independent review checks the full constrained coframe Hessian: only its horizontal block changes to $\rho_0\epsilon(p)(UU-P/4)$. The phase block remains $(\rho/b)(U-h_A)(U-h_A)-\rho h_{AA}$; support, frame and flow blocks remain present. Derivatives of $\epsilon$ enter lower-order forces. The phase speed bound below $3/4$ is in the $U$ rest frame; it is not a laboratory-frame bound under arbitrary boosts. The physical unit cone is retained. Every fixed regular positive chart and finite collar supports the local symmetric-hyperbolic theorem and comparison with the full linearized system on each prescribed finite horizon.

The compact exact-constraint data therefore have a growing linear projection with an $O(a^2)$ nonlinear error on each fixed horizon after making $a$ sufficiently small for that horizon. Constants and allowable amplitude can depend on time and collar. This does not give fixed-size nonlinear escape, global control through the exterior, vacuum evolution or an observed decay.

Verification And Scope Of The Construction

The amended participation report has 4,023 exact author assertions and an independent full review plus a targeted OB1–OB4 amendment review. The kinetic report has 1,324 assertions over 110 controls; its independent review has 5,754 assertions, including 154 directly enumerated non-origin singletons, two exact-matrix and two Arb certificates and an unchanged author replay. All six maintained depths and both selector, phase and entropy alternatives are represented. The displayed inequalities and operator proofs establish the general claims; finite controls check their implementation.

The derivations retain the passive-covariance, unit-knot and positive-floor qualifications stated above.

The construction advances a common action compatible with the nested count clocks and complete inertia. It does not establish that nature realizes this action, derive distinct sector units, favor the assigned depths over all alternatives, or supply a fundamental finite-component realization. Classical field coefficients and detector averages do not replace literal deterministic electron trajectories in six ordinary spatial dimensions.

Polynomial Support, A Uniform Spatial Chart And Nonlinear Departure

Two explicit changes of the participation action improve its exterior analysis while preserving its exact count-clock branches. First replace the reference density by the common polynomial-tail law below. Then, in a separate action, change the vector-support gradient square. These choices have different evolution theorems and must remain distinguishable. Both are inverse constructions on ordinary $\mathbb R^{10}$ with a fixed finite carrier; neither derives the sector units or observed occupied cutoffs.

A Common Positive Tail And Its Normalization

Write $R=|x|$, $t_R=R^2/L^2$, and choose common positive $L,\sigma,\varepsilon_p$. The exponent $p$ describes the spatial tail and is unrelated to modal intensities $p_k$ or the selector exponent $q$. Set

$$\psi(s)= \begin{cases} s e^{-1/s},&s>0,\\ 0,&s\le0, \end{cases} \qquad F_p(s)=[1+\psi(s)]^{-p}, \qquad w(W)=\sigma F_p\!\left(\frac{W^{-1/4}-2}{\varepsilon_p}\right).$$

Replace $\rho_0(W)$ everywhere in the full action by $w(W)$, including its support, horizontal, frame and flow coefficients. The retained exact support and its kinetic coefficient are

$$W_0=(1+t_R)^{-4},\qquad Z_0=(x/L)(1+t_R)^{-5},\qquad w_0=\sigma F_p((t_R-1)/\varepsilon_p),\qquad \mu_s(W)=C_s\frac{w(W)}{W^2},\quad C_s=\frac{L^2\omega_*}{212}.$$

The density is smooth and positive at every finite spatial point, exactly $\sigma$ for $R\le L$, and asymptotic to

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