IDOS Energy, Coherence, and Supporting Dynamics — Part 81

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$J_i=\langle k_i,M_{\rm kin}\dot{\mathbf q}+G\mathbf q\rangle$. Static states $K_i=(k_i,\mathcal Bk_i)$ and generalized states $W_i=(z_{1,i},M_{\rm kin}k_i+\mathcal Bz_{1,i})$ obey $\Omega_{\rm can}(K_i,W_j)=(M_{\rm eff})_{ij}$ and $\Omega_{\rm can}(W_i,W_j)=0$. The latter follows from their occupied blocks: correctors have only $u,\chi,v$ entries, and eligible static modes only support, horizontal-phase and frame entries. The actual complete gyro has zero pairing between two such correctors. Thus this finite chain space has an invariant nondegenerate symplectic projection, with drift rates $M_{\rm eff}^{-1}J$. It gives no decay estimate on its invariant complement.

For $\eta>0$, the static/chain/generalized triples at the maintained depths are $(58,57,115)$ at $d=2$, $(59,58,117)$ at $3$, $(60,59,119)$ at $4$, $(61,60,121)$ at $5$, $(62,61,123)$ at $6$, and $(66,65,131)$ at $10$. For $\eta=0$ retain the actual count $D$. No isolated depth-five effect or additional preference for the observed depths follows from these formulas.

Centroid, Regularity And Physical Scope

For an exact solution, assume finite conserved energy $E\ne0$, integrable momentum density, finite absolute first moment $\int|x||T^{00}|dx$, vanishing total energy and momentum boundary fluxes, and, for smooth cutoffs $\zeta_R$,

$$\lim_{R\to\infty}\int x_i(\partial_j\zeta_R)T^{j0}dx=0.$$

Take these limits locally uniformly in time or in their corresponding integrated-in-time form. Symmetry and conservation of the full stress then give

$$X_E=E^{-1}\int xT^{00}dx,\qquad \dot X_E=P/E,\qquad\ddot X_E=0.$$

Indeed the cutoff moment derivative equals $\int\zeta_RT^{i0}dx+\int x_i(\partial_j\zeta_R)T^{j0}dx$. The actual tail and controlled relative first jets supply these assumptions on the exact branch and its stated regular tube; small physical norm alone does not. The exact boost has $E=Q\Omega\gamma_v$, $P=Q\Omega\gamma_v v$. The complete energy centroid need not track a recognizable perturbed core indefinitely and is not the electron's internal deterministic orbit.

Since $\int w|x|^{2j}dx<\infty$ exactly for $j<3$, active rotation and boost parameter orbits are physically $C^2$ but generically not $C^3$. The finite tangent Gram supports local geometric alignment in an admissible neighborhood, not an analytic modulation expansion of arbitrary order. Removing the finite chain space leaves the zero essential threshold. There is no new nonlinear uniformly rotating branch, global orbital stability, continuum decay, core-persistence theorem or observed lifetime. Constants are for one fixed finite carrier and need not stay bounded as $\lambda\downarrow1$, $\varepsilon$ or $g\downarrow0$, or the carrier and unit ratios grow.

The count clock, cumulative flat occupied band, equal participating energy and $Q\Omega$ rest energy remain properties of the reconstructed action.

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