IDOS Energy, Coherence, and Supporting Dynamics — Part 1

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A Conservative Equal-Energy Family

Expand two real fields in a finite orthonormal spatial band. With canonical coefficients $Q_j,P_j$, write $I_j=(Q_j^2+P_j^2)/2$, and choose phases so that $\dot\theta_j=\partial H/\partial I_j$. Let $e_j=\nu_j I_j$, where the bare rates $\nu_j>0$. Define

$$H_0=\sum_j e_j,\quad \bar e=H_0/D,\quad C=\frac1{2\varepsilon}\sum_j(e_j-\bar e)^2,\quad H=H_0+\kappa C F(\theta),\qquad \kappa>0.$$

For a finite integer resonance set $\mathcal R$, take

$$F=1+\frac\eta{|\mathcal R|}\sum_{r\in\mathcal R}\cos(r\cdot\theta), \quad |\eta|<1,\quad \sum_jr_j=0,\quad \sum_jr_j\nu_j=0.$$

Set $F=1$ when the set is empty. This is an explicitly nonlocal spectral interaction of the same field. Its modal frame and finite band are additional premises. The complete variation gives

$$\dot I_j=\frac{\kappa\eta C}{|\mathcal R|} \sum_r r_j\sin(r\cdot\theta),\qquad \dot\theta_j=\nu_j[1+\kappa(e_j-\bar e)F/\varepsilon].$$

Both $H_0$ and total action $\sum I_j$ are conserved. Every uniform preparation $e_j=e_0>0$ is an exact invariant family, with $H=De_0$ and unchanged phase rates $\nu_j$. At $H_0=D\varepsilon$,

$$\sum_j(e_j-\varepsilon)^2 \le\frac{2\varepsilon(H-H_0)}{\kappa(1-|\eta|)}.$$

Small energy deviations therefore stay small for all time. Below the bound that keeps all energies positive, the finite-dimensional smooth flow is global. There is reciprocal energy exchange away from the uniform family: for $\nu=(1,2,3)$, resonance $(1,-2,1)$, and $e=(11/10,3/5,13/10)$, choosing the resonant phase $\pi/2$ and $\varepsilon=\kappa=1,\eta=1/2$ gives $\dot e=(13/200,-13/50,39/200)$.

This sustains equal energy but does not attract arbitrary initial conditions or select $e_0=\varepsilon$. The mismatch energy is conserved. The action also requires a physically justified modal frame within degenerate shells. The same rule works at all six depths; at $D=1$ the mismatch vanishes. A positive arithmetic comparison spectrum $\nu_k=\omega_d(k+1)$ permits the rule, but is not the actual disturbance spectrum of the entropy core. The latter includes neutral modes that cannot carry positive oscillator energy through $e_j=\nu_j I_j$.

Coherence Does Not Require a Single Degenerate Shell

For real spatial modes $u_j$, a paired field $z=\sum a_j u_j e^{-i\nu_jt}$ has stationary squared magnitude precisely when every nonzero beat coefficient vanishes pointwise:

$$\sum_{\nu_j-\nu_k=\Delta}a_j a_k^*u_j u_k=0\qquad(\Delta\ne0).$$

Generic overlapping scalar modes beat. Bounded beats, orthogonal real component planes and nonlinear phase locking are distinct possible patterns; stationary density is not a universal persistence requirement. Integral orthogonality alone does not remove local beats.

An explicit projected quartic example uses normalized weighted-ball modes $u_0=1$, $u_1=\sqrt{d+3}\,x_1$, with $C_d=\int u_1^4=3(d+3)/(d+5)$. For

$$H=\nu_0 I_0+\nu_1 I_1+ \frac g2[I_0^2+C_dI_1^2+4I_0I_1+2I_0I_1\cos(2\delta)],$$

let $J=I_0+I_1$, $\Delta=\nu_1-\nu_0>0$, $g>0$. There is an exact mixed common-clock state at $\delta=0$ with

$$I_1=\frac{\Delta+2gJ}{g(5-C_d)},\qquad J>\frac{\Delta(d+5)}{6g},\qquad \nu_{\rm relative}^2=4g^2(5-C_d)I_0I_1>0.$$

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