The linked shape response after geometric normalization

The linked mass carrier gives different frequencies with nearly equal absolute shape responses within a depth. The coefficient written in its polynomial coordinate also becomes much smaller with increasing depth. Normalizing that coordinate by its actual geometric metric changes the second observation substantially: the normalized coefficient is already within a few percent of one common value across depths 3 through 10.

This comparison retains the entire finite wave branch and its spatial normal modes. It uses the stated positive geometric metric, whose physical coefficient remains to be derived from the one field. It does not establish measured interaction strengths or neutrino suppression.

One spatial integral replaces the longer source variation

Let $(p_d,q_d)$ be the positive radial components of the normalized endpoint normal mode $\Phi_0$:

$$p_d'=-\frac{a}{2r}p_d-wq_d,\qquad q_d'=\frac{d-1-a/2}{r}q_d-wp_d,\qquad 2\pi\int r(p_d^2+q_d^2)dr=1. \tag{1}$$

Use $(U_d,V_d)$ for the distinct, horizontal geometric zero mode of $P=z^d+\alpha$, with $M_d=2\pi\int r(U_d^2+V_d^2)dr$. The metric calculation gives

$$\delta W_h=U_d,\qquad \delta A_h=2V_d\bigl(\sin((d-1)\theta),\cos((d-1)\theta)\bigr). \tag{2}$$

Define

$$h_z(r)=\frac{1+r^2/2}{(1+r^2)^{3/2}},\qquad \left[-\partial_r^2-r^{-1}\partial_r+\frac{a^2}{4r^2}+1\right]Y_d =2h_zp_d. \tag{3}$$

This is the positive-chiral Green inverse associated with the actual bounded multiplier $h=z/\sqrt{1+r^2}$. It retains both linked coefficients through $a,w,p_d,q_d$. The operator is at least one; its regular, decaying inverse is positive on this positive source.

The constant-polynomial variation of the projected multiplier has one entry, $\delta T=\gamma\kappa_d\alpha|0\rangle\langle d-1|$. Its coefficient simplifies to

$$\boxed{\kappa_d=-4\pi\int_0^\infty r\,(U_dp_d+V_dq_d)Y_d\,dr.} \tag{4}$$

To obtain (4), use (2) in the complete normal Dirac variation. The positive-chiral component that can contract with $D^{-1}h\Phi_{d-1}$ has radial coefficient $U_dp_d+V_dq_d$. Every other angular component is orthogonal. Reflection between the two endpoint modes makes the two wave-variation terms equal. The geometric basis phase used in (2) has angular order $d$, exceeding every difference of retained labels; it has no within-kernel matrix element. Thus the change of basis does not alter the endpoint entry.

This also explains its sign in the stated coordinate convention: $U_d,V_d,p_d,q_d,Y_d$ are positive, so $\kappa_d<0$. Changing the deformation phase changes the corresponding real source; the sign is not an assignment of negative inertia.

The metric removes a coordinate dependence

If the physical geometric kinetic term were $\mathcal T_d M_d|\dot\alpha|^2$, with a positive coefficient $\mathcal T_d$, the canonical real coordinate would be $x=\sqrt{2\mathcal T_dM_d}\operatorname{Re}\alpha$. The geometric part of its normalized vertex is therefore

$$\boxed{\widehat\kappa_d=\frac{\kappa_d}{\sqrt{M_d}},\qquad g_{N,d}=\frac{\gamma\widehat\kappa_d}{\sqrt{2\mathcal T_d}} \,2\operatorname{Re}(v_0^*u_{d-1}).} \tag{5}$$

Here $(u,v)$ is the complete jointly normalized finite wave branch. The factor $\mathcal T_d$, spatial units and the surface normalization must be supplied before treating $g_{N,d}$ as a physical vertex. Equation (5) displays those missing units rather than setting them to one without notice.

Near the origin, write $w=c_0r^d+\cdots$. Switching to $\tau=c_0\alpha$ changes $\kappa_d$ to $\kappa_d/c_0$ and $M_d$ to $M_d/c_0^2$, leaving $\widehat\kappa_d$ unchanged. An independent adaptive Green solve gives:

The raw depth-three to depth-ten coefficient ratio is about 622. The metric-normalized ratio is about 1.035. Thus a small raw polynomial coefficient does not establish a weak physical coupling. In this comparison the depth-five entry is also close to the others.

The mass-branch grouping within one depth remains: the exact source is $\gamma\widehat\kappa_d\,2\operatorname{Re}(v_0^*u_{d-1})$, and the last factor approaches a common sign at large hopping. Its finite correction is retained. No division by the particle's mass is used.

The Gaussian limit follows from the linked equations

The near-constant values have a mathematical explanation. In the following limit, $d$ is the winding of the stated coefficient rule; this is not a proposal for alternative particle assignments.

First, $0\le a(r)\le r^2$. Integrating the first equation of the linked profile from $r$ to $\sqrt d$, using $w\le1$, gives

$$w_d(r)\le\left(\frac{r}{\sqrt d}\right)^d e^{(d-r^2)/2},\qquad r\le\sqrt d, \qquad c_0\le(e/d)^{d/2}. \tag{6}$$

Hence $w_d\to0$ on every fixed bounded interval and $a_d(r)\to r^2$. With $b_d(r)=w_d/(c_0r^d)$, integration of $b_d'/b_d=-a_d/r$ gives

$$b_d(r)\longrightarrow e^{-r^2/2}. \tag{7}$$

The stable zero-mode variable $Q_d=1-r^df_d$ is positive, decreases from one to zero, and satisfies

$$-Q_d'(r)=4r^{2d-1}\int_r^\infty s^{1-2d}w_d(s)^2Q_d(s)ds.$$

Bounding $w_d^2$ by $\min(c_0^2s^{2d},1)$ and splitting the integral at $\sqrt d$ gives, for fixed $R<\sqrt d$,

$$0\le1-Q_d(R)\le R^{2d}\left[c_0^2+\frac{d^{-d}}{d-1}\right]\longrightarrow0. \tag{8}$$

The geometric mode in the $\tau$ coordinate therefore has $U_d/c_0\to e^{-r^2/2}$. Its exact metric identity gives

$$\int(V_d/c_0)^2d^2y \le\frac1d\int(U_d/c_0)^2d^2y.$$

For control of the unbounded domain, monotonicity of $a$ gives $b_d(r)\le b_d(R)(R/r)^{a_d(R)}$ for $r\ge R$. Taking a fixed sufficiently large $R$ supplies an integrable uniform tail bound. Thus

$$\boxed{M_d/c_0^2\longrightarrow\pi.} \tag{9}$$

For the normal mode, its weighted polarization identity implies

$$\int a_dp_d^2\,d^2y+(d+2)\int q_d^2\,d^2y\le2. \tag{10}$$

The lower component tends to zero in norm. The first term bounds the upper-component tail outside any fixed $R$ by $2/a_d(R)$. Combining that bound, (1), and unit normalization gives

$$p_d\longrightarrow p_\infty= \frac{e^{-r^2/4}}{\sqrt{2\pi}},\qquad q_d\longrightarrow0. \tag{11}$$

The positive Green operators in (3) have lower bound one and converge locally to $-\Delta_r+r^2/4+1$. Their quadratic forms and the convergent sources select its regular decaying inverse. The preceding tail bounds control the source integral. Equivalently, the limiting Green function solves

$$\left(-\Delta_r+\frac{r^2}{4}+1\right)Y_\infty =2h_zp_\infty. \tag{12}$$

Evaluation of the limiting response

Put $t=r^2/2$. The normalized radial functions $p_j=(2\pi)^{-1/2}e^{-t/2}L_j(t)$ diagonalize the operator in (12) with eigenvalues $2(j+1)$. The Laguerre generating identity is

$$\sum_{j=0}^\infty L_j(t)z^j =\frac1{1-z}\exp\left(-\frac{tz}{1-z}\right),\qquad |z|<1.$$

See NIST DLMF, equation 18.12.13 for this mathematical identity. Expanding (12), using $\int_0^\infty e^{-2t}L_j(t)dt=2^{-j-1}$, reduces (4) to

$$\lim_{d\to\infty}\frac{\kappa_d}{c_0} =-2\int_0^\infty\frac{e^{-t}-e^{-2t}}{\sqrt{1+2t}}dt. \tag{13}$$

One way to perform the sum is to use $(j+1)^{-1}=\int_0^1s^jds$ in the generating function and then note $h_z=(1+t)/(1+2t)^{3/2}=\partial_t[t/\sqrt{1+2t}]$. All sums occur inside their radius of convergence.

Finally,

$$\int_0^\infty\frac{e^{-ut}}{\sqrt{1+2t}}dt =\sqrt{\frac{\pi}{2u}}e^{u/2}\operatorname{erfc}(\sqrt{u/2}),$$

so (9) and (13) give the closed result

$$\boxed{\lim_{d\to\infty}\widehat\kappa_d =e\operatorname{erfc}(1)-\sqrt{2e}\operatorname{erfc}(1/\sqrt2) =-0.312271559800\ldots.} \tag{14}$$

The limit belongs to this particular bounded multiplier and linked profile. It is not a universal measured coupling constant. Changing the spatial interaction rule would require a new calculation.

Verification

linked_endpoint_response.py solves the normal mode and positive Green boundary problem independently of the earlier finite-difference source implementation. endpoint_metric_checks.py compares them with the complete linked variation, verifies coordinate invariance, and retains all maintained controls at depths 3, 4, 5, 6 and 10 on both surface signs. The bottom entries remain the two component controls of the maintained geometric-mean assignment.

The 97-check report also compares the integral and closed form in (14) with an independent spatial solution of (12). Additional winding controls 12, 16, 20 and 24 approach that limit. Outer-domain and profile-tolerance refinements check the largest maintained depth and winding 24 separately. These are mathematical controls, not alternative particle assignments. The source reports and raw radial arrays preserve the executed equations. Neither the metric comparison nor its Gaussian limit supplies the missing same-field physical kinetic coefficient $\mathcal T_d$, the occupied curved equilibrium, or a complete particle-pair interaction.