The spatial source and the endpoint response of the simplex carrier

The full normal wave supplies a concrete source test for the prescribed texture W=z^n. Its simplest local matrix-current closure does not reproduce that texture: the dominant current has winding zero. That same component has a precise spectral effect. A constant deformation W=z^n+a opens a coupling between the two ends of the projected chain.

The electron and muon assignments have almost the same absolute response to this deformation under one common normal operator, while their frequencies follow different simplex counts. This is a conditional example of the combination of different mass-frequencies and similar response within a depth. The sustaining field, its stiffness, and the physical coupling law are not derived by this example.

The starting spatial operator and all its normal modes are defined in the gapped carrier. Its complete first normal response and wave stress are calculated in the normal inertia note. No independently interacting scalar field or particle-exchange mechanism is introduced.

The actual local source of the normal coefficient

Let Gamma_1 and Gamma_2 denote the two Hermitian normal potential matrices called beta_1 and beta_2 in the carrier note. At a fixed surface mode, define the local currents

$$J_a(x)=\psi(x)^\dagger\Gamma_a\psi(x),\qquad a=1,2. \tag{1}$$

The pair components are summed in this contraction. Both Gamma matrices reverse normal chirality. Therefore the complete kernel state psi_0 has J_a=0 pointwise. With the full complementary correction eta,

$$\psi=\psi_0+\Lambda^{-1}\eta+O(\Lambda^{-2}),\qquad J_a=\Lambda^{-1}j_a^{(1)}+O(\Lambda^{-2}), \tag{2}$$

$$j_a^{(1)}=2\operatorname{Re} (\psi_0^\dagger\Gamma_a\eta). \tag{3}$$

Thus the finite-kernel state alone would miss the source entirely. For the normal component ordering used in that note, the complex current is particularly simple:

$$j_1^{(1)}+i j_2^{(1)} =2\sum_\chi \left(\eta_{\chi,0}^*\psi_{0,\chi,1} -\psi_{0,\chi,2}^*\eta_{\chi,3}\right). \tag{4}$$

Every forced angular Fourier coefficient of (4) is retained. The complete first-order Dirac residual, radial refinement and independent contractions of the four-component spinors verify the result.

A specific local quartic closure

A minimal same-field source comparison is the declared quartic

$$U=\frac g2(J_1^2+J_2^2). \tag{5}$$

Its variation gives the local term g(J_1 Gamma_1+J_2 Gamma_2)psi. Consequently, reproducing the assumed normal potential Lambda(Gamma_1 Re W+Gamma_2 Im W) would require

$$\Lambda W=g(J_1+iJ_2), \quad\text{or at leading order}\quad \Lambda^2 W=g(j_1^{(1)}+i j_2^{(1)}). \tag{6}$$

This is a test of (5), not a claim that (5) is the established master interaction. It keeps a real common coefficient g and does not fit individual particle amplitudes.

Write the complete complex source as sum_m j_m(r) exp(i m theta). For W=r^n exp(i n theta), equation (6) requires only harmonic m=n. The calculated source instead has a dominant m=0 term. At gamma=10000:

The fraction uses the actual full-plane L2 norm of the source, summing all its Fourier powers. Orthogonality makes the unwanted angular components impossible to remove by changing a single real coefficient. An independent fit weighted by the actual wave density also has a relative residual essentially equal to one. This avoids giving the unbounded target coefficient an artificial finite global norm.

There is also a separate asymptotic issue: a polynomial current of a single localized wave decays at infinity, whereas W=z^n grows. A fuller same-field background or a different spatial completion would have to address both the local angular source and the outer spatial behavior. These observations constrain this source comparison; they do not rule out an ambient IDWT field or every possible self-interaction.

The dominant source has an analytic endpoint formula

Put q=n-1 and use the normalized radial functions U_l,V_l of the complete normal kernel. The reflection symmetry gives V_q=U_0 and U_q=V_0. Define the positive scalar response

$$Y=K_0^{-1}\left[ \frac{2(1+r^2/2)}{(1+r^2)^{3/2}}U_0\right], \tag{7}$$

where K_0 is the full radial operator in the normal inertia note. For large gamma at fixed beta, the two canonical vectors approach the opposite endpoints, with their normalization fixed jointly. Then

$$j_0(r)=-2(-1)^q\gamma U_0(r)Y(r) \,[1+O(\gamma^{-2})] \tag{8}$$

in the integrated comparison. The leading expression is independent of beta. Positivity of the scalar Green function makes its coefficient nonzero. A smaller harmonic at m=2n also survives the endpoint limit; it is retained in the source norm and cannot be dropped in an exact polarization calculation.

A constant shape deformation closes the chain

Now vary the prescribed normal coefficient within the same ordinary spatial operator:

$$W_a(z)=z^n+a. \tag{9}$$

Here a is a coefficient of a spatial deformation, not an amplitude chosen to represent mass. It can be complex. The winding at infinity and the finite normal kernel count remain n for sufficiently small a. The scalar partner remains -Delta+|W_a|^2. The variation below is performed in the complete normal space.

Choose the normal kernel basis with zero projected first variation. For real a its derivative at zero satisfies

$$D_n\dot\Phi_l=-\Gamma_1\Phi_l,\qquad P\dot\Phi_l=0, \qquad \dot\Phi_l=-D_n^{-1}\Gamma_1\Phi_l. \tag{10}$$

The scalar partner inverse gives its explicit radial components. Set A_l=K_l^{-1}U_l and B_l=K_{l-q}^{-1}V_l. The two nonzero spinor components of the derivative are

$$\begin{aligned} (\dot\Phi_l)_1={}&-r^n A_l e^{i(l+n)\theta} +(B_l'+(l-q)B_l/r)e^{i(l-n)\theta},\\ (\dot\Phi_l)_2={}&i(A_l'-lA_l/r)e^{i(l+1)\theta} -i r^n B_l e^{i(l-2n+1)\theta}. \end{aligned} \tag{11}$$

Each component changes the original mode's angular label by plus or minus n. Since the complete kernel has labels 0 through n-1, this immediately gives P dot_P P=0 in the chosen frame, and zero first variation of the projected constant polarization. For the bounded multiplier h=z/sqrt(1+r^2), angular selection leaves exactly one entry:

$$\boxed{\dot h_{0,q}=\kappa_n,\qquad \dot h_{kl}=0\ \text{for all other entries}.} \tag{12}$$

Integration by parts gives an independent expression for it:

$$\boxed{\kappa_n=-2\int_{\mathbb R^2}U_0(r)Y(r)\,d^2x<0.} \tag{13}$$

For complex a, the new entry is a kappa_n to first order. The ordinary adjacent links run from l to l+1; (12) runs from the last mode back to the first. It therefore closes the otherwise open chain. Numerically, kappa_6=-0.6149315907 and kappa_10=-0.5584778429.

Independent frequency derivatives and the common depth response

Let (u,v) be the two complete, jointly normalized projected chain vectors at a=0. The full kernel variation gives

$$\frac{d\omega}{d\operatorname{Re}a} =2\gamma\kappa_n\operatorname{Re}(v_0^*u_q). \tag{14}$$

The complete field's Hellmann-Feynman derivative instead gives

$$\frac{d\omega}{d\operatorname{Re}a} \longrightarrow\int_{\mathbb R^2}j_0(r)\,d^2x \qquad (\Lambda\longrightarrow\infty). \tag{15}$$

The independently assembled expressions agree. In the simplex regime, both approach (-1)^q gamma kappa_n, independent of beta. At the common gamma=10000 they give:

The two depth-six derivatives differ by about 2.8e-5 of their common value. Their frequencies differ by about 206.76. No division by the particle frequency was used to obtain the common response. Both assignments use the same spatial normal profile, gamma, canonical norm and deformation coefficient. The winding-to-depth identification and the cross-depth physical scales remain inverse premises.

The sensitivity is also a demanding constraint. The deformation size that produces a first-order frequency change comparable to the unperturbed frequency is about 1.42e-16 for the first depth-six branch, 2.93e-14 for the second, and 1.06e-24 for the depth-ten comparison. These are dimensionless coefficients in (9), not measured physical lengths or errors. They identify a direction that the sustaining dynamics must control; they do not by themselves prove instability.

For example, if a retained same-field shape coordinate had a regular quadratic restoring energy K |a|^2/2, its static displacement under the wave source would depend on K. Neither K nor that reduced energy has been derived here. Supplying an arbitrarily large stiffness would be an additional physical premise, not a completion of the source test.

Verification and the remaining step

holomorphic-normal-source-20260914T060906_643958Z.json passes 104 checks. They include every angular source, independent complete spinor contractions, spatial refinement, rotation covariance, the exact quartic variation and the dominant endpoint formula.

holomorphic-kernel-variation-20260914T061555_811538Z.json passes 42 checks. It verifies the complete first normal deformation equation, the absence of projected polarization changes, the endpoint coefficient, and agreement of the two frequency derivatives. At winding one, an independent exact translation of the full Gaussian kernel verifies the coefficient by direct quadrature.

The next coupling calculation must keep mixed currents between two oscillating waves as well as their separate self currents. A shared deformation derivative is useful evidence about this trial operator, but is not yet a complete encounter law, a suppression rule for neutrinos, or a derived physical master interaction.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.