The complete shape source and surface-branch balance

The two surface-sign branches can cancel some geometric forces, but they cannot cancel the complete source of the symmetric linked carrier. At all maintained depths there is an even-order deformation driven with the same sign by both positive-frequency branches. The proposed geometric cost has zero first derivative in that direction.

This is a necessary source-balance test of a specific trial configuration. It does not exclude other shapes, other sustaining motion or the broader IDWT proposal. A source coefficient is not a signed particle inertia. The surface signs are not assigned a particle–antiparticle interpretation.

A full matrix derivative replaces separate source terms

Use the connected normal carrier with winding $d$, normal modes $\Phi_l$, and the complete projected pair

$$H_P=\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix},\qquad T_{ll}=\beta+l-\frac{d-1}{2},\qquad T_{l+1,l}=\gamma h_l. \tag{1}$$

All $h_l>0$ are spatial integrals, with $\gamma>0$. For a polynomial deformation $P(z)=z^d+\alpha_kz^{d-k}$, let $\delta D_k$ include both the texture and connection variations. Choose the orthogonal mode derivative $\delta\Phi_l=-D^{-1}(I-P)\delta D_k\Phi_l$. Then the full derivative is

$$\delta T_k=\mathcal P_k+\gamma\mathcal H_k, \qquad (\mathcal H_k)_{ij}= -\langle\delta D_k\Phi_i,D^{-1}(I-P)h\Phi_j\rangle -\langle D^{-1}(I-P)\bar h\Phi_i,\delta D_k\Phi_j\rangle. \tag{2}$$

The first term $\mathcal P_k$ originally contains the explicit polarization variation and both changes of its mode basis. The local anticommutator identity proves

$$\boxed{\mathcal P_k=0\quad\text{at the radial configuration}.} \tag{3}$$

These terms cancel only after their complete variations are included. They must not be individually omitted from an unreduced field equation. The result is not shape independence to higher order.

For a jointly unit-normalized positive-frequency state $(u,v)$, the complete first frequency derivative is therefore

$$\boxed{F_k(\beta)=2\gamma\operatorname{Re}(v^\dagger\mathcal H_k u).} \tag{4}$$

An independent calculation integrates the original full spatial texture, connection and explicit polarization sources. It agrees with (4) for every maintained control, every $k=2,\ldots,d$, both surface signs, and complex deformation orientations.

Reflection explains which sources cancel

Let $R$ reverse the normal labels and $S_0=\operatorname{diag}((-1)^l)$. The original carrier has $h_l=h_{d-2-l}$, and its real deformation matrices satisfy

$$R\mathcal H_k^T R=\mathcal H_k,\qquad S_0\mathcal H_kS_0=(-1)^{k-1}\mathcal H_k, \qquad T(-\beta)=-S_0R T(\beta)^T RS_0. \tag{5}$$

Transforming both singular vectors in (4) gives

$$\boxed{F_k(-\beta)=(-1)^kF_k(\beta).} \tag{6}$$

Equal positive-frequency populations of the two signs cancel odd-order sources and leave even-order sources unchanged. The previously known endpoint relation is the special case $k=d$. Retaining the other orders is essential at odd depth.

For mutually orthogonal surface harmonics, the surface integral of the source is their population-weighted sum: the interference terms integrate to zero. This is only an integrated necessary condition. Vanishing of an integral would not establish pointwise balance of a coherent field.

A nonzero surviving source at every depth

The endpoint order $k=d$ has the single entry

$$\mathcal H_d=\kappa_d|0\rangle\langle d-1|,\qquad \kappa_d<0. \tag{7}$$

The positive Green contraction fixes its sign. At the next order,

$$\boxed{\mathcal H_{d-1} =a_d\left(|0\rangle\langle d-2|+|1\rangle\langle d-1|\right), \qquad a_d<0.} \tag{8}$$

The sign also follows from a spatial integral. Let $(U,V)$ be the positive horizontal geometric mode for $k=d-1$, in its $\tau=c_0\alpha$ coordinate. Write $Y^+_{d-2,0}$ for the first positive-chiral component of $D^{-1}(I-P)h\Phi_{d-2}$, and $Y^-_{0,1}$ for the second radial component of $D^{-1}(I-P)\bar h\Phi_0$, in the phase conventions of (2). Then

$$a_d=-2\pi c_0\int_0^\infty r\left[ (UU_0+VV_0)Y^+_{d-2,0} -(UV_{d-2}+VU_{d-2})Y^-_{0,1}\right]dr. \tag{9}$$

The first Green source is strictly positive and the second strictly negative. Their regular decaying scalar inverses preserve those signs. Since the radial modes are positive, the bracket is positive on a set of nonzero measure. The horizontal basis-phase correction has angular order $d-1$; its only possible retained endpoint entry vanishes by the endpoint reflection identity. Thus (9) gives the same coefficient as (2).

For $\beta>(d-1)/2$, every diagonal of $T$ is positive. The matrix $S_0T^{-1}S_0$ is triangular with positive entries on and below its diagonal. Its two singular Perron vectors are strictly positive. Consequently the smallest-frequency vectors $u,v$ have matching alternating signs, with every component nonzero. Equation (7) then has $F_d>0$ when $d$ is even. Equation (8) has $F_{d-1}>0$ when $d$ is odd. Reflection gives the same sign at $-\beta$.

Equivalently, choose the largest even integer $k_*\le d$. Then

$$\boxed{F_{k_*}(\beta)>0,\qquad F_{k_*}(-\beta)>0.} \tag{10}$$

This holds for every positive hopping and each maintained positive-diagonal control, not just asymptotically. At large hopping the penultimate source has the useful leading form

$$F_{d-1}(\beta)=\frac{2(-1)^d a_d\beta}{h_0} [1+O(\gamma^{-2})]. \tag{11}$$

At $\gamma=10000$, the lowest maintained controls give:

These are derivatives in the stated polynomial coordinate. Their raw sizes are not physical coupling comparisons. A positive metric normalization preserves their nonvanishing and sign.

What the source-balance test excludes

The ordinary round tube has weight $J=(1+y_1/R)^{d-2}$. Its restricted geometric first derivative vanishes whenever $k>d-2$. In particular,

$$\partial_{\alpha_{k_*}}G_J=0,\qquad \sum_a q_aF_{k_*}(\beta_a)>0 \quad\text{if }q_a\ge0\text{ and some }q_a>0. \tag{12}$$

Thus no mixture of these positive-frequency surface branches balances this direction at the symmetric profile by adding a scalar multiple of $G_J$ and a radius-only supporting term. The geometric integral is understood in the same localized full-plane tube approximation as the curvature calculation; a finite-tube boundary supplies its separately bounded tail correction. Equation (12) is exact for that declared trial functional, rather than a claim about an unspecified boundary force.

The geometric Jacobi carrier changes the hopping integrals and removes the reflection symmetry between their two ends. It nevertheless retains negative endpoint coefficients and two negative penultimate entries in the verified controls. Its largest even source remains positive on both surface signs. The common-operator simplification therefore does not itself solve the occupied geometry.

The missing step is a complete same-field equilibrium or relative motion with its energy, momentum and canonical normalization retained. This calculation identifies a definite force that such a construction must supply; it does not assign a negative inertia or infer a neutrino rate.

Reproducible checks

linked_shape_matrices.py computes every matrix in (2) from the actual complementary spatial inverses. shape_matrix_checks.py compares them with independently integrated sources, proves their numerical reflection and angular controls, and checks (9) separately. The 382-check report includes every maintained index at depths 3, 4, 5, 6 and 10. The bottom entries remain its two neighboring component controls. Raw matrices and exact executed source versions are retained separately in the database.