IDOS Energy, Coherence, and Supporting Dynamics — Part 25

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have finite total charge. Define $H_Q=Q\log(Q/n_0)-Q+n_0$ and $\mathcal P=Q[m+\gamma V]+bH_Q+W$. At fixed $Q>0$ the Routhian, reconstruction and conserved energy are

$$\mathcal L_Q=\frac I2\left[\frac{\dot z^2}{4f}+f\dot\phi^2\right] +Qz\dot\phi-\mathcal P, \qquad \dot\theta=m+\gamma V+b\log(Q/n_0)-z\dot\phi,$$ $$p_z=\frac{I\dot z}{4f},\quad p_\phi=If\dot\phi+Qz,\qquad E=\frac{2f p_z^2}{I}+\frac{(p_\phi-Qz)^2}{2If}+\mathcal P.$$

At an interior shape point both horizontal velocities remain free initial data. The common phase rate adjusts to the charge constraint. Constant $b$ and $b=m/128$ give identical restricted dynamics at $Q=n_0$, because both $H_Q$ and $\log(Q/n_0)$ vanish. At other $Q$, variable $b$ adds a shape force $H_Q\,\partial m/128$; constant $b$ shifts the energy only. Prescribing the isolated-frequency time jets instead infers a different, possibly enormous charge. One cannot retain those arbitrary phase rates, the raw-sum magnitude and an independently fixed charge simultaneously. Spatial charge transport can still change local density; this homogeneous result is not an encounter theorem.

The Two-Group Restriction Is Not Generally Invariant

For $d=2,N_a=1,N_b=2$, the extra labels $(2,0),(1,1),(0,2)$ have one, two and one phase edges. If their common amplitude is $b_0$, the inner amplitude is $a$, and the relative phase is $\phi$, their phase torques are proportional to

$$(1,2,1)\,2a^3b_0^3\sin\phi.$$

At zero horizontal velocity this gives the exact acceleration difference $-2Q\gamma\eta a^3b_0\sin\phi/I$. Equal outer phases therefore split. Even with $\eta=0$, the variance potential has unequal radial gradients at $(2,0)$ and $(1,1)$. A two-group evolution averages these forces and is a restricted variational model, not an invariant subsystem of the full label action.

Within that restricted model, smooth positive kinetic geometry and conserved energy bound physical horizontal velocities globally on its compact shape space. Coordinate poles need regular patches. The report two-band-charge-dynamics/report.md supplies all eleven actual family-count relations, six-depth controls, 45 exact checks and four bounded mathematical integrations with energy drift below $8.4\times10^{-12}$. The integrations do not model actual W/Z or electron/muon encounters. Localization, full-label response and supporting-field backreaction remain required.

The charge and boundedness argument extends beyond this non-invariant two-group truncation. For the complete finite homogeneous carrier, eliminate only the common phase at fixed $Q$. Its remaining horizontal metric is positive, and its smooth shape space is compact. This is a mathematical space of real field components, not physical position space. The energy is

$$E=\frac I2\|K\|^2+Q[m(e)+\gamma V(e)]+b(e)H_Q+W_R(e), \qquad \|K\|^2\le\frac{2(E-Q\mu_{\rm floor})}{I}.$$

The positive lower bound $m\ge\mu_{\rm floor}$ and nonnegative remaining potentials establish this inequality. Smooth forces on the compact finite carrier and bounded velocity give global continuation in time; phase reconstruction is bounded as well. This homogeneous theorem does not require the two-group manifold to be invariant. It assumes frozen positive $n_0$, flow and frame, finite cutoff, and finite energy in the homogeneous-cell description. It supplies no bound uniform in cutoff, no spatial PDE theorem and no localized particle. Local density transport and supporting-field backreaction remain outside it.

All supporting fields and their explicit time dependence are frozen in this control, so their omitted terms are carrier-independent constants. Adjusting the common phase rate does not remove the relative kinetic cost. For the normalized coherent sum with real overlap $C$, envelope ratio $x$ and $R=1+x^2+2Cx\cos\Delta$, direct real-metric projection gives

$$\|K\|^2=\frac{(1-C^2)(\dot x^2+x^2\dot\Delta^2)}{R^2}.$$

The temporal energy contribution is $I\|K\|^2/2$. It is unchanged by a common phase-rate shift. With stationary envelopes and their isolated frequency gap, the equal-envelope constructive/destructive coefficients $\|K\|^2/(\omega_b-\omega_a)^2$ are $(1-C)/[4(1+C)]$ and $(1+C)/[4(1-C)]$. The latter is the maximum over envelope ratio and phase. These are kinematic energy contributions, not a complete encounter energy or an observed coupling. Principal's pair-horizontal-energy script proves the identity and records all eleven actual-family comparisons.

An Exact Permutation-Invariant Carrier Subsystem

The failed two-group closure can be replaced by an exact symmetry reduction of the full finite homogeneous carrier. First retain a complete cumulative prefix $|k|\le N_{\max}$ in one active depth-$d$ coordinate face, with zero coefficients and velocities outside. Occupation terms have zero real first derivatives in omitted coordinates. For the phase gates, an occupied upper label references only the zero label, its active unit and its predecessor; all lie inside the same prefix. Omitted upper gates have zero first derivative. Thus this face-prefix subspace is invariant, including its horizontal kinetic equations. Arbitrary sparse supports are not protected: an occupied phase gate can force a missing predecessor.

Axis permutations preserve every remaining term of the homogeneous action. Their fixed set is therefore invariant as well. Its labels are integer

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