IDOS Energy, Coherence, and Supporting Dynamics — Part 10

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For arbitrary real balanced wavevectors, moments through degree $4N$ are necessary and sufficient to force at least $D$ classes. The even-polynomial Gram through polynomial degree $2N$ has rank $D$, proving sufficiency. For necessity, replace the axis submeasure on $0,\pm1,\ldots,\pm N$ by its positive symmetric $2N$-node Gaussian quadrature. It matches all moments through degree $4N-1$ with $N$ linked pairs instead of $N+1$ classes. Leaving other classes untouched gives $D-1$ classes with every retained mixed moment unchanged. These replacement roots are generally nonintegral. Complete positive-measure uniqueness follows safely by degree $4N+2$ using squared polynomial annihilators; no minimality is claimed for that stronger statement.

At any fixed degree $2r$, positive weight elimination on the original $K_N$ gives a matching subset with at most $\sum_{j\text{ even},\,0\le j\le2r}\binom{d+j-1}{j}$ classes. For the tau target, second and fourth moments therefore admit at most 56 and 771 classes, respectively, despite the full count 64,512,240. The exact thresholds and fourteen-label table are preserved in finite-moment-occupation/report.md, supported by 715 exact checks.

These are sharp information bounds within the stated balanced positive spectral class. They do not require every selecting field equation to have high derivative order: global boundary, regularity or other dynamical constraints can distinguish spectra through a lower-order equation. Nor do they establish independent canonical multiplicity, equal participating energy, nested inheritance or an unbroadened localized Fourier band.

Energy Sharing, Actual Coherence And Spectral Width

A finite conservative quartic comparison provides a useful preparation test, but does not by itself establish that flat bare frequencies are necessary for equilibration. For

$$H=\frac12p^Tp+\frac12q^TKq+ \frac\lambda4\sum_m(a_m\cdot q)^4,$$

the persistent linear dark subspace is $W=\bigcap_{j=0}^{D-1}\ker(AK^j)$, with rows $a_m^T$ in $A$. It equals $\ker A$ when $K=\omega^2 1$. For dispersive $K$, harmonic mixing can return an initially dark vector to the contact directions. For example $K=\operatorname{diag}(1,4)$ and $A=(1,1)$ have no persistent dark subspace despite a single contact. Conversely, full rank is not sufficient for sharing: diagonal quartic contacts give independent Duffing oscillators. Finite-time averages must also retain interaction energy; harmonic modal energy alone is not the complete participating energy.

Flatness of the quadratic part does not ensure nonlinear single-frequency motion. For $q=u\cos\Omega t+v\sin\Omega t$, put $x_m=a_m\cdot u$, $y_m=a_m\cdot v$. The third harmonics vanish only if

$$\sum_m a_m(x_m^3-3x_my_m^2)=0,\qquad \sum_m a_m(3x_m^2y_m-y_m^3)=0.$$

The remaining fundamental vector equations must also agree at one $\Omega$. The radial quartic $\lambda(q^Tq)^2/4$ supplies an exact prepared rotating family for $u\perp v$, $|u|=|v|=R$, with $\Omega^2=\omega^2+\lambda R^2$. Equal-weight quadrature frames give equal time-averaged complete modal allocations $E/D$, but do not attract generic preparations or prove uniform local spatial energy. Its phase charge is $S=\Omega R^2$, while

$$E=\omega^2R^2+\frac34\lambda R^4,\qquad E-\Omega S=-\frac14\lambda R^4.$$

Thus energy sharing, a pure carrier clock and the required action normalization remain joint tests of one action. A trajectory calculation of sharing and a separate prescribed balanced-density calculation cannot be combined as though they established that joint solution.

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