Complete positive-band source balance and its energy cost

The fast bands already present in the spatial carrier can supply a source of the opposite sign. There is an exact constructive way to balance all centered sources after time averaging. It also exposes a normalization problem: the resulting energy per canonical wave charge cannot remain equal to the selected small frequency in the tested model.

This calculation retains the full finite carrier, both surface signs, the density-gradient multiplier and all maintained controls at depths 3, 4, 5, 6 and 10. It is an integrated, time-averaged condition. It does not establish a pointwise coherent equilibrium or include the remaining normal continuum. All modes considered here have positive frequency and positive energy in their stated canonical normalization.

An exact constructive balance from the full trace

Let $s_j>0$ be the singular values of $T$, and let $f_{jk}=\partial_{x_k}s_j$ for $k=2,\ldots,d$. The complete density-gradient derivative has angular shifts $1-k$ and $1+k$. Neither overlaps the diagonal or the adjacent shift already present in $T$. Therefore

$$\operatorname{Re}\operatorname{Tr}(T^\dagger\partial_{x_k}T)=0, \qquad \boxed{\sum_j s_j f_{jk}=0.} \tag{1}$$

The second identity is the derivative of $\operatorname{Tr}(T^\dagger T)=\sum_js_j^2$. The first polarization derivative vanishes independently by the local anticommutator. Equation (1) includes every centered direction, rather than only its strongest source.

For positive canonical populations $q_j$, the averaged source and energy are

$$F_k=\sum_j q_j f_{jk},\qquad E=\sum_jq_js_j, \qquad\mathcal Q=\sum_jq_j. \tag{2}$$

Here $\mathcal Q$ means the action's positive phase normalization, not electric charge. In the first-order positive-frequency model it is the wave norm; in the positive oscillator action it is the sum of its positive circular-mode charges.

The explicit choice

$$\boxed{q_j=\frac{s_j}{\sum_rs_r}} \tag{3}$$

gives $\mathcal Q=1$, every averaged $F_k=0$, and

$$\boxed{\frac E{\mathcal Q}=\frac{\sum_js_j^2}{\sum_js_j}.} \tag{4}$$

It works within either surface sign separately. At large hopping, (4) is set by the fast frequencies, not the small simplex branch. For example, the geometric density-gradient carrier's lowest controls give values between roughly 3300 and 4400 in the stated dimensionless units at $\gamma=10000$, whereas their small frequencies are many orders smaller.

Distinct frequencies remove their cross terms under time averaging. The opposite surface harmonics are orthogonal in the integrated source. A coherent field can still exert oscillating local forces; (3) has not removed that further dynamical problem.

A lower bound for any positive averaged balance

The cost is not peculiar to the choice (3). Include every positive band on both surface signs. Separate the two small branches from the fast bands. Write their complete real source vectors as $f_j$.

Choose a real linear source functional $a$ for which both small branches obey

$$a\cdot f_j\ge s>0\quad\text{on the small branches}. \tag{5}$$

Such a functional exists because their source vectors cannot cancel with nonnegative weights in the complete source test. The computation constructs it explicitly: first divide each row by $|f_k^+|+|f_k^-|$, obtaining vectors $p_+,p_-$, then take $a=p_+/\|p_+\|+p_-/\|p_-\|$ in those coordinates. Its two contractions are positive unless the vectors are exactly opposite. All of these statements are checked with high-precision signed singular vectors, without changing the physical operator.

Define

$$c=\max_{j\ { m fast}}\max(0,-a\cdot f_j),\qquad \omega_F=\min_{j\ { m fast}}s_j,\qquad \omega_L=\min_{j\ { m small}}s_j.$$

Any positive source-balanced population must then satisfy

$$s\mathcal Q_L\le c\mathcal Q_F, \qquad \boxed{\frac{\mathcal Q_F}{\mathcal Q} \ge\frac{s}{s+c}.} \tag{6}$$

It follows that

$$\boxed{\frac E{\mathcal Q}\ge \omega_L+(\omega_F-\omega_L)\frac{s}{s+c} \equiv E_{\rm bound}.} \tag{7}$$

This is a finite-band analytic inequality. It can be checked directly, mode by mode, as a dual certificate. With $t=(\omega_F-\omega_L)/(s+c)$, every retained mode obeys $s_j+t\,a\cdot f_j\ge E_{\rm bound}$. Summing against positive populations and imposing the balanced source proves (7).

The row normalization makes this construction unchanged by positive rescaling of any individual shape coordinate. It is an analysis of a fixed operator, not a new fitting rule for its coefficients.

Cross-depth cost

At $\gamma=10000$, examples for the geometric density-gradient carrier are:

A small fast population can still carry a large energy because its frequency is so high. The inequality exceeds the small frequency for every maintained control and both normal-carrier choices in this test. The explicit balance (3) obeys the bound, as it must.

Thus the retained positive bands do provide an averaged supporting source, but cannot preserve the identification $E/\mathcal Q=\omega_L$ under this bare canonical wave model. A complete same-field construction would need a different joint energy and source structure, with any binding contribution and normalization actually derived. The calculation does not assign negative particle inertia or establish a constraint on unspecified modes outside this retained system.

Verification

positive_band_support.py computes the full band sources and the dual certificate. positive_band_support_checks.py verifies angular orthogonality, the complete weighted trace, positivity, the explicit balanced populations, every dual inequality and independent precision refinements. Its 302-check report uses the independently verified density-gradient matrices and their recorded input hashes. All raw matrices and exact source versions are preserved separately.