Exact metric identities for the linked spatial deformations

The bounded normal carrier's geometric deformations have a simple metric once changes of component basis are removed. Its radial equations imply an exact identity between that metric and the second change of the geometric profile. This supplies the normalization needed to compare shape responses without confusing a small coordinate coefficient with a small physical vertex.

The metric is a stated positive kinetic comparison for the spatial coefficients. Its physical coefficient and origin in the one master field remain to be derived. It is not a calculation of the complete particle's inertia. The normal coordinates are ordinary unbounded spatial coordinates.

Linked geometric zero modes

Use the bounded linked profile with winding equal to the tested depth $d$:

$$W=w(r)e^{id\theta},\quad A=\frac{a(r)}r e_\theta,\qquad w'=\frac{d-a}{r}w,\quad a'=2r(1-w^2). \tag{1}$$

More generally, the common rule is $W=e^{-u}P(z)$, $A=\nabla^\perp u$, $\Delta u=2(1-|W|^2)$. Consider the real polynomial deformation

$$P(z)=z^d+\alpha_kz^{d-k},\qquad \partial_{\alpha_k}u=f_k(r)\cos k\theta,\quad 1\le k\le d.$$

Its linear equation is

$$\left(-\partial_r^2-r^{-1}\partial_r+k^2/r^2+4w^2\right)f_k =4w^2r^{-k}. \tag{2}$$

The regular solution behaves as $r^k$ at zero and as $r^{-k}$ at infinity. Its unprojected field variations are those obtained by varying both $W$ and $A$ in the common rule. Retaining both is essential for the metric and the wave source.

Removing a change of component basis

Define the real positive metric by minimizing over an infinitesimal basis phase $a_0(x,y)$:

$$M_k=\inf_{a_0}\int_{\mathbb R^2} \left(|\delta W-i a_0W|^2+ \frac14|\delta A-\nabla a_0|^2\right)d^2y. \tag{3}$$

This specifies what is meant by the geometric kinetic comparison; it has not been inferred from the wave-induced response coefficient. Variation with respect to $a_0$ gives

$$(-\Delta+4w^2)a_0 =4\operatorname{Im}(W^*\delta W)-\nabla\cdot\delta A.$$

Equation (2) then gives $a_0=-f_k\sin k\theta$. The horizontal variations simplify to

$$\begin{aligned} \delta W_h&=U_ke^{i(d-k)\theta},\\ (\delta A_{h,r},\delta A_{h,\theta}) &=2V_k(\sin k\theta,\cos k\theta),\\ U_k&=w(r^{-k}-f_k),\qquad V_k=\frac12(f_k'+kf_k/r),\\ M_k&=\boxed{2\pi\int_0^\infty r(U_k^2+V_k^2)dr.} \end{aligned} \tag{4}$$

The imaginary polynomial variation has the same metric. Their real cross term vanishes at the radial configuration by rotational symmetry.

Differentiating (4), using (1) and (2), gives another first-order pair:

$$U_k'=\frac{d-k-a}{r}U_k-2wV_k,\qquad V_k'=\frac{k-1}{r}V_k-2wU_k. \tag{5}$$

Consequently the integral of $[r^2(U_k^2-V_k^2)]'$ vanishes, giving

$$\int (d-k+1-a)U_k^2\,d^2y =k\int V_k^2\,d^2y. \tag{6}$$

For $k=1$, $f_1=a/(dr)$ is exact. The polynomial coordinate moves the center by $-\alpha_1/d$, and (4) gives $M_1=\pi/d$. The actual translation coordinate therefore has metric $\pi d$, half the flat geometric cost $2\pi d$.

The second profile and the same bounded weight

Expand the exact planar solution through second order:

$$u=u_0+\alpha_k f_k\cos k\theta +\alpha_k^2(g_{0,k}+g_{2,k}\cos2k\theta)+O(\alpha_k^3).$$

The angular mean obeys

$$(-\Delta+4w^2)g_{0,k}=2U_k^2. \tag{7}$$

The bounded polarization weight used in the mass carrier also obeys an exact identity:

$$(-\Delta+4w^2)(d+1-a)=4. \tag{8}$$

To verify it, differentiate $a'=2r(1-w^2)$, giving $\Delta a=4-4(d+1-a)w^2$. Pairing (7) and (8), with their regular and decaying boundary conditions, yields

$$\int g_{0,k}\,d^2y =\frac12\int(d+1-a)U_k^2\,d^2y =\boxed{\frac{k}{2}M_k}. \tag{9}$$

The last equality uses (6). Thus the same linked weight that gives the carrier's equally spaced projected polarization also fixes this geometric metric identity. No separately fitted stiffness-to-metric ratio enters it.

Holomorphic moments fix the leading forces

The asymptotic expansion of $u$ agrees with $\log|P|$ up to decaying corrections. Green's identity against a holomorphic monomial therefore gives

$$\int z^k B\,d^2y=2\pi\sum_{i=1}^d z_i^k, \qquad B=\Delta u, \tag{10}$$

where $z_i$ are the zeros of $P$, counted with multiplicity. For the first variation in $\alpha_k$, Newton's polynomial identity gives $\partial_{\alpha_k}\sum_i z_i^k=-k$. Equivalently,

$$\boxed{\int_0^\infty r^{k+1}wU_k\,dr=\frac{k}{2}.} \tag{11}$$

The boundary calculation in (10) matters: $f_k$ itself has a harmonic $r^{-k}$ tail. Integrating it against a growing tube polynomial and discarding its boundary term would give a wrong force. The decaying combination $U_k$ avoids that problem.

Numerical normalization and verification

Near the center, $w=c_0r^d+\cdots$. The computation uses $\tau_k=c_0\alpha_k$ to avoid poorly scaled polynomial coefficients. Metrics and second profile integrals acquire the same factor $c_0^{-2}$; physical conclusions cannot depend on this coordinate conversion.

At $k=d$, the dimensionless metrics in this coordinate are:

linked_moduli_metric.py solves a stable equation for $Q_k=1-r^kf_k$, keeping its exponentially decaying tail without subtracting nearly equal numbers. The second boundary problem is rescaled before solving; this removes the large coordinate condition number at depth ten without changing its equation.

curved_normal_checks.py checks every $k=1,\ldots,d$ at depths 3, 4, 5, 6 and 10, including the translation profile, (6), (9) and (11). Its 357-check report also contains the complete spatial first-variation and nonlinear curvature checks described in the curved geometry calculation. The underlying spatial arrays are preserved separately from the report.