Reciprocal wave source on the linked normal geometry

The bounded connected carrier gives a concrete spatial frequency operator. Its occupied wave also supplies a reciprocal source for the supporting texture and connection. Computing both sources reveals a useful common response and a remaining dynamical constraint: the wave drives a genuine deformation along which the proposed geometric functional is constant. Thus solving the geometry without its occupied wave does not yet give a stationary particle.

The calculation concerns the linked normal operator on an ordinary unbounded plane. All spinor and canonical-pair components are retained. The coefficients have not been assigned an independent physical interaction field or a physical energy separate from the one IDWT field. The forces below are model derivatives in the stated units, not measured couplings or encounter rates.

Complete normal response

Write the full field as psi=Phi+eta/Lambda+O(Lambda^(-2)), where Phi contains the complete n-mode kernel and both canonical components. The first complementary equation is

$$D\eta=(\omega-V)\Phi,\qquad \eta\perp\ker D. \tag{1}$$

For the connected profile, the positive-chiral radial inverses are

$$\begin{aligned} K_{j,+}&=-\partial_r^2-r^{-1}\partial_r +(j-n+a/2)^2/r^2+1,\\ K_{j,-}&=-\partial_r^2-r^{-1}\partial_r +(j+1-a/2)^2/r^2+1. \end{aligned} \tag{2}$$

These are the complete radial operators for each forced angular label. They replace both the earlier monomial scalar inverse and the bounded unconnected two-component inverse. Regular origin conditions and decaying outer conditions select their inverses. The weighted polarization source also has the exact inverse given in the carrier note.

At leading nonzero order, the sources of the normal coefficients are

$$\begin{aligned} \mathcal J_W&=\sum_\chi\left[ \Phi_\chi^\dagger(\Gamma_1+i\Gamma_2)\eta_\chi+ \eta_\chi^\dagger(\Gamma_1+i\Gamma_2)\Phi_\chi\right],\\ \mathcal J_{A_i}&=\operatorname{Re}\sum_\chi \Phi_\chi^\dagger\alpha_i\tau_3\eta_\chi. \end{aligned} \tag{3}$$

The factor Lambda multiplying the normal operator cancels the factor 1/Lambda in the field correction. One must additionally vary the explicit bounded polarization term whenever a changes. Its leading contribution is one half of Phi^dagger S chi_1 Phi times delta a.

An independent radial check replaces w(r),a(r) by w(s r),a(s r), leaving h(r) fixed. Recomputing every normal mode and projected frequency gives the same derivative as integrating (3), including the explicit polarization variation. At depth six the relative frequency derivative is about 1.1915 for both assigned branches. Keeping only W would miss most of this response; the connection supplies a substantial contribution.

Deform the linked geometry itself

The full planar geometric rule can be written with a holomorphic polynomial P and a real function u:

$$W=e^{-u}P(z),\qquad \boldsymbol A=\nabla^\perp u, \qquad \Delta u=2(1-e^{-2u}|P|^2). \tag{4}$$

The radial reference has P=z^n and a=r u'. Move along this same rule with P=z^n+epsilon z^(n-k), where 1<=k<=n. For real epsilon, write delta u=f_k(r) cos(k theta). Linearizing (4) gives

$$\boxed{ \left[-\partial_r^2-r^{-1}\partial_r+k^2/r^2+4w^2\right]f_k =4w^2 r^{-k}.} \tag{5}$$

The solution is regular as r^k at zero and approaches r^(-k) at infinity. Its associated linked variations are

$$\begin{aligned} \delta W&=w\left[(r^{-k}-f_k/2)e^{i(n-k)\theta} -(f_k/2)e^{i(n+k)\theta}\right],\\ \delta A_r&=k f_k\sin(k\theta)/r,\qquad \delta A_\theta=f_k'\cos(k\theta). \end{aligned} \tag{6}$$

For a purely imaginary epsilon replace the cosine by a sine and retain the corresponding complex coefficients. This preserves the full phase information. For k>=2 the center is unchanged, and the explicit polarization weight varies through delta a=r delta A_theta.

The k=1 case is an independent control. Equation (5) has the exact solution f_1=a/(n r). If the center and h move too, (6) is a common translation. The complete frequency derivative then vanishes. Numerical refinement verifies that cancellation. Fourier contraction is used directly for the smallest responses; subtracting large angular samples would otherwise obscure the translation check through roundoff.

For arbitrary zeros of P the same first-order geometric rule has cost G=2 pi n. Moving those zeros therefore supplies genuine constant-cost directions. It preserves the complete normal gap and kernel count, but need not preserve the projected particle frequency.

One spatial endpoint coefficient gives the response grouping

For k=n the polynomial changes by a constant. Angular selection now makes the projected variation especially simple. The polarization variation has angular difference n, larger than the largest difference n-1 between retained labels, and gives no projected entry. The variation of the degree-one multiplier h can reach only the opposite endpoints:

$$\boxed{\delta T=\gamma\,\kappa_n\epsilon\, |0\rangle\langle n-1|.} \tag{7}$$

Here kappa_n is a real spatial integral, independent of beta or any particle frequency. It is not the binding coefficient in the separate polarization-core action.

There is an explicit independent formula for this integral. Let f=f_n, U=U_0, V=V_0 and define

$$\begin{aligned} b_0&=w(r^{-n}-f/2)U+\tfrac14(nf/r+f')V,\\ b_1&=\tfrac14(nf/r-f')U+\tfrac12wfV. \end{aligned} \tag{8}$$

Let (Y_0,Y_1) be the two positive-chiral radial components of D^(-1)(I-P) h Phi_(n-1), with their known angular factors removed. They are computed from (2). Reflection between the endpoint modes gives

$$\boxed{\kappa_n=-4\pi\int_0^\infty r(b_0Y_0+b_1Y_1)\,dr.} \tag{9}$$

For a jointly normalized pair (u,v), the complete projected response is

$$\boxed{\frac{\partial\omega}{\partial\epsilon} =2\gamma\kappa_n\operatorname{Re}(v_0^*u_{n-1}).} \tag{10}$$

Both vectors have norm approaching 1/sqrt(2) on the small-frequency branch. Their endpoint product tends to (-1)^(n-1)/2, independently of beta. Thus different masses within one depth approach a common absolute shape response by the same spatial rule. No division by mass or separately chosen particle coefficient produces this grouping.

The computed geometric derivatives are approximately -0.0802981 at n=5, -0.0339063 at n=6 and -0.000529886 at n=10. These are defined in the same chosen dimensionless coefficient convention; they do not derive physical relative sector scales or neutrino suppression.

At n=6 and gamma=10000, the independently refined shape derivatives are about 339.061284 for beta=15.5 and 339.053384 for beta=37.5. The frequencies differ by approximately the assigned simplex ratio, while this response differs by about 2.33e-5 in relative terms. The sign follows the retained phase and endpoint convention. It is not a sign assigned to inertia.

The force in (10) is nonzero along the constant-cost geometric direction. Therefore an additive geometric functional alone does not hold the occupied radial configuration stationary. The wave and sustaining motion must be solved together, or a different explicit common geometric rule must be tested. The near-equal response is useful evidence about this candidate operator, not a completed physical particle-pair coupling.

A precise finite-hopping limitation on the mass law

Put d_l=beta+l-(n-1)/2>0 and let

$$\omega_{\rm lead}=\frac{\prod_l d_l}{\gamma^{n-1}\prod_l h_l}, \qquad M(\beta)=\omega_{\rm lead}T^{-1}. \tag{11}$$

The exact smallest positive frequency is omega=omega_lead/||M||_2. After fixed alternating signs are removed, every entry of M is a positive product of the diagonal factors outside one interval, divided by fixed positive hopping factors. Its lower-left entry is one. All other nonzero entries increase with beta. The positive matrix M^T M has a strictly increasing largest eigenvalue. Hence, for n>=2 and finite positive gamma,

$$0<\frac{\omega}{\omega_{\rm lead}}<1, \qquad \frac{\omega}{\omega_{\rm lead}} \text{ strictly decreases as }\beta\text{ increases}. \tag{12}$$

The exact ratio of two projected frequencies is therefore strictly smaller than their simplex-product ratio when the second beta is larger. The finite canonical matrix does not give an exactly proportional simplex spectrum at finite hopping. The first correction is explicit:

$$\frac{\omega}{\omega_{\rm lead}} =1-\frac{1}{2\gamma^2} \left(\frac{d_0^2}{h_0^2} +\frac{d_{n-1}^2}{h_{n-2}^2}\right)+O(\gamma^{-4}). \tag{13}$$

The full normal complementary correction remains separately controlled by the spatial gap. Equation (12) describes this projected operator; it is not an exclusion of other IDWT dynamics or mode assignments.

Reproducible checks

connected-source-20260914T130824_340126Z.json passes 50 checks of the full source equation, independent radial frequency derivatives, linked nonradial variations, common translations, complex rotation and refinement. The finest normal field residual is about 8e-8 relative to its complete source.

endpoint-mass-response-20260914T132230_696804Z.json passes 57 checks comparing (9)-(10) with the complete spatial source in depths five, six and ten, along with the full finite-matrix correction. Both reports preserve raw arrays and the exact source hashes used.