IDOS Energy, Coherence, and Supporting Dynamics — Part 27

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with fixed commuting label operators and constant frame/scale, $R_i=0$ forces constant pair occupations and then $F=0$ by commuting derivatives. A positive scalar density weight cannot turn its zero spatial connection into the circulating current of a scalar product-vortex trial profile.

Curved preparations remain available in the same carrier. In six physical coordinates $(x_1,y_1,x_2,y_2,x_3,y_3)$, retain the full prefix with $D$ labels, change three unit-label occupations to $p_j=D^{-1}+\alpha x_j$ and the zero label to $p_0=D^{-1}-\alpha\sum_jx_j$, and add phases $\beta y_j$ to those three pairs. On a finite positive patch,

$$F=\alpha\beta\sum_{j=1}^3dx_j\wedge dy_j,\qquad \sum_i\|R_i\|^2= \frac{\alpha^2}{4}\left(\sum_j\frac1{p_j}+\frac3{p_0}\right) +\beta^2\sum_jp_j(1-p_j).$$

At the patch center, this family has $\sum_i\|R_i\|^2\ge3\sqrt{2(D-1)/D}|\alpha\beta|$. Thus finite curvature has a nonzero residual cost in this family. Occupation and, for higher prefixes, phase penalties also enter. This is an explicit configuration, not a stationary or persistent particle solution. The same construction gives ranks $2,2,4,4,6,10$ at the maintained depths, a generic even-rank property rather than a special depth-five effect.

For rigid translation $e=e^{J\theta}e_*(x-X)$, $a_0=\dot\theta-\dot X\cdot a_*$ and $K=-e^{J\theta}\dot X_iD_i e_*$. Horizontal kinetic energy contains no common-phase rate. The primitive clock term supplies $-\int n a_i\,dx\,\dot X_i$; density elimination can additionally produce the mixed velocity Hessian $-na_i/b$. It vanishes on the centered branch. If density and texture translate together, the integrated connection is constant in $X$ and its collective curvature vanishes even when local $F$ is nonzero. Source-dependent weighting/deformation and reciprocal dynamics are still needed to derive a center gyro force. No scalar-vortex precession frequency or fitted lock follows from these identities.

The source report actual-phase-curvature/report.md has 56 exact checks; phase-curvature-review/report.md independently verifies 151 conditions, including higher-prefix residuals and the fixed-total-charge qualification.

Actual-Count Homogeneous Initial Forces

The exact partition reduction has been implemented with sparse analytic gradients and tested on all eleven maintained same-family pairs, at envelope ratios $0.001,1$ and relative phases $0,\pi/2,\pi$. Each preparation starts from its two isolated count frequencies. Fixing charge adjusts the common phase rate while retaining the relative horizontal velocity and its energy. The controls set $Q=n_0=I=\ell=\gamma=\eta=1$ and each within-family frequency unit to one. They are not calibrated cross-sector forces.

All 66 preparations develop acceleration outside their initial two-group coefficient subspace. The homogeneous variance term alone suffices; this is also present for an isolated nonzero uniform prefix and is not an interaction-only force. Degree-preserving redistribution and the gyroscopic term preserve both initial groups exactly. Tiny recorded violations from those terms are numerical subtraction error. At equal-envelope quadrature, the mean cutoff is $(N_a+N_b)/2$, so the flattened mass gradient vanishes for integer midpoints; this arithmetic is not specific to depth five.

The largest carrier uses 59,301 real pairs. The full checked batch took 59.43 seconds and 122.47 MiB peak resident memory; one top potential and all-gradient evaluation took 0.150–0.177 seconds. Twelve small exact-oracle controls and all actual-preparation checks passed. Independent review verified full-label multiplicities, separate scalar potentials/gradient differences, constraints and all 66 initial kinetic identities. These are initial-force calculations, not long-time integrations or spatial collisions. The sources and complete decompositions are in sparse-carrier-force-batch; the independent audit is audit-team-20261015/sparse-review/report.md.

Fixed Total Charge With Spatially Varying Clock Compressibility

In the frozen-shape collective family, let $u$ be the common phase rate, $v$ the center velocity, $a(x;X)$ the spatial connection and $H=h+\gamma V$. For positive $b(x;X)$, density elimination gives

$$L=\int b n_0\left(e^{(u-v\cdot a-H)/b}-1\right)dx, \qquad n=n_0e^{(u-v\cdot a-H)/b},\qquad Q=\int n\,dx.$$

For integrable positive $n_0$, bounded $H,a$, and $b$ bounded above and away from zero, every finite $v$ and $Q>0$ determines a unique finite $u$. These are sufficient conditions, not a characterization of all physical tails. Set $w=n/b$, $W=\int w$ and $\bar a=\int wa/W$. At fixed total charge, the Routh function $R=L-Qu$ obeys

$$u_{v_i}=\bar a_i,\qquad R_{v_i}=-\int na_i\,dx,\qquad R_{v_iv_j}=\int w(a_i-\bar a_i)(a_j-\bar a_j)\,dx.$$

Thus this clock contribution to the velocity Hessian is positive semidefinite, including spatially varying $b$. A direction is null exactly

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