IDOS Energy, Coherence, and Supporting Dynamics — Part 15

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Using a finite sampled mean to choose $\sigma_0$ shifts the true asymptotic carrier $h_\infty/\sigma_0$ relative to that prescribed center. In two tested preparations the displacement exceeds the narrowest reported half-band by factors about 1.67 and 3.21. Degenerate coherent cross terms also change the exact mean from a diagonal-only sum. A complete comparison must hold the physical normalization fixed and state its spectral center.

The prepared trajectories need their own equations and complete energy. Assigning a positive frequency to a constant Neumann mode without restoring energy is not a solution of the unmodified wave law. Flattening frequencies while retaining the original gradient energy likewise changes the implied equations and energy per mode. Doubling equal-amplitude quadratures doubles the energy budget unless their amplitudes are rescaled. Refitting $\sigma_0$ can conceal these absolute normalization changes in fractional spectra; it cannot establish the required common action normalization.

Dispersion does not universally force temporal local-energy beats: placing each balanced mode in its own orthogonal real component plane removes them for arbitrary frequencies, at a cost of $2D$ components. The density may still vary spatially and the complete field remains multifrequency. Weighted spatial energy readouts provide another distinct branch requiring a derived response kernel. None automatically satisfies the combined uniform-density, actual coherent carrier and nested-occupation target. The exact identities, validated sample, sampling limits and alternatives are preserved in phase-spectrum-validation/report.md.

Complete Quadratic Energy And Neutral Texture Directions

The covariant support's lower-order terms can also be retained. Write $n=n_{\rm peak}F(v)$, $c_0=n_{\rm peak}F'(v)$, and let $K_s[y]$ be the complete quadratic sum of the linearized $A,B,G$ residual squares, including their undifferentiated terms. Eliminating the algebraic density perturbation gives

$$\mathcal L_2=\frac a2\theta_t^2+c_0V\theta_t+n\langle Jh,h_t\rangle +\frac12(y_t-\mathsf G b)^T\mathsf M(y_t-\mathsf G b) -\frac{b_0}{2}|b|^2-K_s[y].$$

The density-lock and horizontal symplectic terms are linear in velocities. Their Legendre contributions cancel, leaving the complete conserved energy

$$E_2=\int\left[\frac12(a\theta_t^2+y_t^T\mathsf M y_t +b^T\mathsf H b)+K_s[y]\right]dx\ge0.$$

This is $\tfrac12\delta^2(E-\omega Q)$ on the constrained phase space, with the complete off-branch Noether charge $Q$, rather than the unrestricted Hessian of $E$. The energy identity retains all spatially varying background coefficients. The phase and support density-lock terms cancel in its time derivative; the horizontal remainder vanishes by antisymmetry of $J$. Conservation requires an admissible domain with vanishing integrated flux.

The support part of the zero-energy kernel contains only translations on the smooth connected positive branch. The carrier part additionally permits $\nabla\theta=A h$, where $A_i h=2\langle J\partial_i e_0,h\rangle$. For large enough carriers this leaves infinitely many neutral texture directions. Support and texture may shift separately in this flat-jet model; their relative registration is not selected. Therefore nonnegative energy does not establish coercivity or occupied-band selection.

Within an energy-conserving linear realization, a finite-energy eigenmode $e^{st}$ with $\operatorname{Re}s\ne0$ would have to have zero $E_2$. Positivity forces $\theta_t=y_t=b=0$, and the nondegenerate horizontal equation then gives $h_t=0$. Since $s\ne0$, the eigenmode is trivial. This excludes such exponential modes, conditional on that domain; it does not exclude polynomial drift, uncontrolled continuous-spectrum behavior or nonlinear instability, and does not supply a vacuum completion.

The fixed auxiliary exponent four also admits the flatter density in this same covariant action. Take

$$F_k(v)=[1+(v^{-1/4}-1)^k]^{-2},\quad M(v)=M_0F_k(v)/v^2,\quad \gamma_A(v)=\gamma_B(v)=M(v)/4.$$

Coefficient variations multiply zero residuals and velocities on the branch. The bound $M|\nabla Y|^2/n\le129M_0/(4n_{\rm peak}L^2)$ is independent of $k$. Choosing $M_0=2n_{\rm peak}L^2\omega_{\min}/129$ and $\lambda=\omega_{\min}/(2n_{\rm peak})$ satisfies the coupled covariant cone inequalities and preserves the complete nonnegative quadratic energy for every finite integer $k\ge4$. At the transition $r=L$, the local density-lock contribution to a fixed-phase-momentum support oscillator is

$$\Delta\Omega_{\rm shape}^2=\frac{129}{64}\frac{k^2}{L^2}.$$

This is a local oscillator diagnostic, not a full spatial eigenfrequency. It replaces exponential coordinate stiffness with polynomial stiffness. A large internal disturbance frequency is distinct from a large propagation speed. Coefficients and their inverses still degenerate in the vacuum tail. The full derivation and 1,720 checks are in covariant-quadratic-energy.

A Small Carrier Can Occupy A Full Band Without Equal Weights

There is a useful distinction between occupying every simplex label and meeting the equal-participation and mixing requirements. In paired-real notation let $z_j=e^{ix_j/L}$, $U=\operatorname{diag}(1,z_1,\ldots,z_d)$, $R\in SO(d+1)$ and

$$p_N=(UR)^Ne_0.$$

Its norm is pointwise one and its Fourier support lies in $K_N=\{k\ge0:|k|\le N\}$. For a generic single fixed $R$, every coefficient is nonzero for every finite $N$. For each target label, a near-identity rotation along a chain of its occupied directions gives a nonzero leading coefficient. Thus its coefficient function is not identically zero; removing the countable union of its analytic zero sets proves the common all-degree existence statement. A bounded rational witness verifies degrees one through three at all six active depths. This is stronger than matching a few moments with only a few occupied labels.

The construction needs only $2(d+1)$ real components, but does not have equal modal weights. Its zero-corner weight is $R_{00}^{2N}$, whereas equal participation requires $1/\binom{N+d}{d}$. Strict concavity of the logarithm of that count prevents a fixed repeated rotation from satisfying this at three distinct cutoffs. This restricted obstruction does not exclude other independently prepared equal-energy constructions.

Moreover, cell-averaged family overlaps obey

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