Coherent phase and polarization tests of the supporting multiplet

The curvature-moment carrier has a positive complete averaged shape Hessian. A coherent field occupying those bands is a stricter test than its averaged density. Its different frequencies produce oscillating sources, and changing their phases cannot generally remove them.

Within one fixed surface harmonic, a determinant identity tests every polarization of the positive squared-carrier doublet. For resolved distinct beat frequencies and nonzero occupation of all bands, no such polarization choice makes even the two endpoint sources vanish at every time when at least three bands are occupied. The result does not exclude responsive geometry or additional surface-harmonic content. Continuous supporting motion remains a possible part of the model.

A normalized coherent supporting field

Take $T u_j=s_jv_j$, $T^\dagger v_j=s_ju_j$, with real, unit-normalized singular vectors and $s_j>0$. In the positive retained oscillator action $L_2=\|\partial_t[B(\alpha)c]\|^2-c^\dagger H_P^2c$, the squared coefficient operator has the degenerate eigenvectors

$$\phi_j=(\eta_j u_j,\zeta_j v_j),\qquad |\eta_j|^2+|\zeta_j|^2=1. \tag{1}$$

Retain arbitrary complex $\eta_j,\zeta_j$, including their phases. The coherent coefficients and their spatial field are

$$c(t)=A\sum_{j=1}^d e^{-is_jt}\phi_j,\qquad\Psi=Bc, \qquad A^2=\frac1{2\sum_js_j}. \tag{2}$$

Its total positive canonical phase charge is one. Each band has charge $q_j=2s_jA^2$, exactly the frequency-weighted population that cancels the averaged source. The amplitudes in (2) fix that common normalization; they do not assign the masses. Every frequency is still $s_j$.

For a shape coordinate $a$, its instantaneous source is

$$\mathcal F_a(t)=c^\dagger K_a c, \qquad K_a=\partial_a(H_P^2)=\{H_P,H_a\}. \tag{3}$$

The diagonal terms sum to zero by the complete trace identity. The remaining terms oscillate at the positive frequency differences $|s_j-s_r|$. Averaging (3) removes them; it does not prove they are absent from the actual field equation.

An endpoint determinant keeps both polarizations

The endpoint derivatives have $T_x=g_*|0\rangle\langle d-1|$, $T_y=i g_*|0\rangle\langle d-1|$, where $g_*=\gamma\kappa_d\ne0$. For a distinct pair of bands define

$$a=g_*(v_j)_0(u_r)_{d-1},\qquad b=g_*(v_r)_0(u_j)_{d-1}.$$

In the two pure chiral blocks, the real/imaginary endpoint source vectors are respectively

$$\mathbf U_{jr}=\begin{pmatrix}s_j a+s_r b\\i(s_j a-s_r b)\end{pmatrix}, \qquad \mathbf V_{jr}=\begin{pmatrix}s_r a+s_j b\\i(s_r a-s_j b)\end{pmatrix}. \tag{4}$$

Their determinant is

$$\boxed{\det[\mathbf U_{jr},\mathbf V_{jr}] =2i(s_r^2-s_j^2)ab\ne0.} \tag{5}$$

The nonvanishing endpoint components do not require an assumption about an arbitrary eigenvector. The symmetric tridiagonal matrices $T^\dagger T$ and $TT^\dagger$ have nonzero adjacent entries. If an eigenvector's endpoint vanished, their row recurrence would force every component to vanish. The same recurrence makes each eigenspace one-dimensional; their eigenvalues are simple. Thus every factor in (5) is nonzero for the stated finite carrier and nonzero endpoint coefficient.

For the general polarization (1), the pair's source coefficient is

$$\eta_j^*\eta_r\,\mathbf U_{jr} +\zeta_j^*\zeta_r\,\mathbf V_{jr}. \tag{6}$$

If the beat gaps are distinct, each Fourier coefficient must vanish separately for a time-independent source. Equation (5) would then require both $\eta_j^*\eta_r=0$ and $\zeta_j^*\zeta_r=0$. Every pair would have to occupy opposite pure chiral blocks. At most two nonzero bands can satisfy that condition pairwise. For $d\ge3$, this cancellation is impossible. At depth two, the two bands can occupy opposite blocks, and the centered source vanishes exactly, including its diagonal sum. See the depth-two calculation.

The distinct-gap condition is explicit. At $\gamma=10000$, the computed gaps are distinct for all maintained controls and both surface signs, with separations well resolved under the stated spatial and normal integration refinements. This is not a claim of nondegeneracy at every possible parameter value. The two blocks in (1) are not assigned a particle–antiparticle interpretation.

A phase-independent combinatorial lower bound

Let $\ell_{jr}>0$ be the smallest squared singular value of the two columns in (5), and put $p_j=|\eta_j|^2$. Equation (6) implies

$$\|\text{pair source}\|^2\ge \ell_{jr}\left[p_jp_r+(1-p_j)(1-p_r)\right]. \tag{7}$$

The sum on the right is linear in each $p_j$ when all other entries are fixed, so its minimum over $[0,1]^d$ occurs at a binary assignment. Consequently the endpoint force obeys

$$\boxed{ \overline{\mathcal F_x^2+\mathcal F_y^2} \ge\frac1{2(\sum_js_j)^2} \min_{\epsilon_j\in\{0,1\}} \sum_{j<r:\,\epsilon_j=\epsilon_r}\ell_{jr}\ge0.} \tag{8}$$

Any two-color assignment of $d$ bands leaves at least $\lfloor(d-1)^2/4\rfloor$ same-color pairs. This gives a simpler positive bound by replacing every $\ell_{jr}$ with their minimum. The combinatorics concerns compatibility of the source polarizations; it is not an additional factor inserted into the mass formula.

The bound is strictly positive for three or more occupied bands. It is zero at depth two, where an opposite-block assignment attains zero source. For higher depths the bound can be very small. It excludes exact static cancellation in this restricted coherent state, but does not by itself exclude weak forcing or small sustained geometric motion. A displacement or stability claim needs the actual kinetic response to (3).

Constructive polarization choices can suppress the source

The same calculation can find useful coherent comparison states. Assign each occupied band to either pure chiral block, keeping its frequency and canonical population unchanged. Cross-block pair sources vanish exactly. Within-block pair sources remain. Exhaustively testing the $2^d$ assignments gives the smallest force variance in this discrete family.

Here the variance includes every centered real and imaginary deformation, with each covector component normalized by its actual positive geometric metric. Relative to equal chiral mixing, representative source RMS ratios are approximately:

These are constructive polarization controls of the same operator, not mass fits. The search covers all pure assignments, but not the full continuous polarization space; its minimum is an upper bound on that larger optimization problem. The field's dynamics has not been shown to select minimum variance. Nor are these internal supporting-source ratios particle encounter rates or a derivation of neutrino interaction suppression.

The comparison does show why retaining polarization matters: a positive averaged shape Hessian and the same frequency spectrum can coexist with very different oscillating sources. The next dynamical calculation must retain that source and its reciprocal geometry, rather than replacing it by the zero time average.

Verification

coherent_support_checks.py verifies (5), the complete finite binary bound in (8), arbitrary complex polarization controls and resolved beat gaps across all maintained sectors. Its 704-check report records the exact source hashes and refinement settings. The bound is computed with stable positive Gram eigenvalues, retaining small endpoint components at high precision.

polarized_support_checks.py uses every centered source matrix, compares them with direct differentiation of the full squared operator, and checks the selected assignments by an independent full polarization projection. Low- and high-depth spatial refinements test the force suppression. These are conditional field-source calculations; no complete particle inertia is inferred. The 82-check report preserves the full discrete searches and their independent controls.

The generalization to more polarization channels shows why the channel count must be established physically before this threshold can be interpreted as a law about spatial depth or inertia.