IDOS Energy, Coherence, and Supporting Dynamics — Part 5

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$$F_s(\rho)=s\sigma_0\log(\rho/\rho_{\rm ref}),\qquad L=\int\left[F_s\eta(\dot\psi)-\kappa\rho- \frac\gamma2|A\psi|^2\right]dx.$$

Here $A$ denotes the fixed spatial annihilators and commutes with $J$. All components are varied. With radial-phase projector $P=[\psi\psi^T+(J\psi)(J\psi)^T]/(2\rho)$ and $Q=1-P$, the full symplectic matrix is $W=-J[F_s'P+(F_s/\rho)Q]$. Thus

$$\dot\psi=J\left[\frac P{F_s'}+\frac{\rho Q}{F_s}\right] \left[\kappa\psi+\gamma A^*A\psi\right].$$

A constant-density coherent field in $\ker A$ rotates at its actual rate $\Omega=s\kappa\rho/\sigma_0$. Equal physical modal energies $\varepsilon$ give

$$E=\kappa\rho V=D\varepsilon,\qquad |\Omega|=\frac{D\varepsilon}{\sigma_0V},\qquad \frac E{|\Omega|}=\sigma_0V.$$

The global phase charge is instead $C=\int F_s\,dx$. Equating it with $E/|\Omega|$ would be an additional assumption. This changed kinetic law closes the count-to-carrier bridge without a distinct clock phase. For comparison, a separate canonical clock with $H=e^{sJ_0/A}\omega_0\sum I_j$ rotates at $sH/A$, while its band phases rotate at $e^{sJ_0/A}\omega_0$; those rates generally differ.

For the logarithmic carrier the complete constrained energy satisfies

$$(H-\Omega C)[\psi]-(H-\Omega C)[\psi_0] =\kappa\int[\rho-\rho_0-\rho_0\log(\rho/\rho_0)]dx +\frac\gamma2\|A\psi\|^2\ge0.$$

This controls density and annihilator residuals. It does not select equal occupations, control every neutral polarization, or prove continuum well-posedness. Both rotation senses retain the inequality. Vacuum is singular; for multiple paired components $F_s=0$ is another symplectic degeneracy. In the general local radial class, exact $\Omega=h'/F'=s h/\sigma_0$ forces $h=C_0e^{sF/\sigma_0}$. A finite $F$ at a zero-energy vacuum then permits only trivial $h$. Regularization therefore changes the clock relation or introduces background energy. Complete supporting and exterior energies cannot be omitted.

A finite common-operator existence example assigns one orthogonal internal plane to each label of a maximum band. Fixed commuting internal matrices $K_j$ carry those labels, and $\sum_j(D_j-K_j)^2$ selects exactly one paired mode per label. Every lower occupation has uniform density; the same logarithmic action preserves every modal occupation and gives the count clock for equal energies. Its cost is $R=D_M$ internal planes. Lower $D_N$ are occupied counts or IDOS values of the separate grading $\sum_jD_j$, not distinct-frequency thresholds of the flat operator. Neither the selected cutoff nor common modal energy follows from this construction. It supplies no special depth-five effect.

Transition Closure Also Tests Uniform Energy

For orthonormal family embeddings $U,D$, a local contact $C$, compression $M=U^*CD$, and omitted-space map $L=(1-UU^*)CD$, the exact identity is

$$1-M^*M=D^*(1-C^*C)D+L^*L.$$

A unitary compression of a contraction therefore requires zero leakage and norm saturation. An inverse construction uses pointwise orthonormal frames $F_U,F_D$ and one texture $T=F_UVF_D^*$, where $V$ is a single product of the three fitted count-angle rotations. With common normalized envelope $w$ and energy-density multiplier $\rho_B$, the normalized family vertex is $V$, but its omitted norm is

$$L^*L=\frac{\operatorname{Var}_w(\rho_B)}{\langle\rho_B\rangle_w^2}1.$$

Uniform participating energy removes this leakage in the constructed contact. Deriving the frames, background and texture remains necessary; their variation supplies reciprocal response. The canonical family kinetics used for this contact example are distinct from the logarithmic kinetics above, so combining them requires a fresh variation. Different particles retain their distinct isolated frequencies and detuned transfer dynamics. The bottom double-count and tau adjacency problems remain unresolved.

The reviewed source records are preserved in MariaDB under this run's inverted-clock-closure, common-nested-ladder, nested-transition-action and homogeneous-coherent-band directories. Their checks cover 139, 235, 22 and 11,376 exact conditions respectively; principal adds 306 prefix controls and 6,302 simplex-operator controls. General conclusions rest on the displayed proofs, not the number of finite assertions.

A Local Spatial-Current Completion With Positive Inertia

A second inverse construction places the count in a spatial-gradient constitutive function and retains canonical field kinetics. It gives an actual coherent clock, complete positive translation inertia and a regular bounded vacuum functional together. It changes the local clock premise: a localized profile has energy density divided by frequency equal to its varying action density, rather than a constant $\sigma_0$. The integrated normalization survives. This branch must therefore be compared with, rather than identified with, the uniform-density logarithmic construction above.

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