Complete normal currents and the two-wave coupling

The gapped normal carrier admits a concrete polarized two-wave current calculation. Its large-gap limit has two distinct angular contributions: the overlap of separate self currents and a mixed current between the waves. Keeping both removes a false inference of zero interaction from a zero of the separate-current overlap.

Under common parameters, the normal response is nearly independent of the occupied surface degree within a winding. Actual surface overlaps retain additional degree and orientation dependence. This is a conditional geometric interaction on the inverse spatial carrier, not a derived physical IDWT force or a rate of particle encounters.

The operator, source and complete normal corrections are defined in the carrier, the inertia calculation, and the endpoint source response. The local quartic used here is a declared probe of those fields. Its failure to generate the prescribed normal texture remains unresolved.

Keep the complete same-field phase average

For two distinct carrier frequencies, write the field at a given position as A exp(-i omega_a t)+B exp(-i omega_b t). With the same two Hermitian normal matrices Gamma_i, define

$$J_i^A=A^\dagger\Gamma_iA,\qquad J_i^B=B^\dagger\Gamma_iB,\qquad K_i=A^\dagger\Gamma_iB. \tag{1}$$

The quartic U=g sum_i J_i^2/2 is an interaction of this one field. Expansion and averaging over the relative oscillation phase gives the cross term

$$\boxed{\overline U_{AB} =g\left(\boldsymbol J^A\cdot\boldsymbol J^B +\sum_i|K_i|^2\right).} \tag{2}$$

The second term is interference of the two waves in the same local current. It is not an exchanged particle. Equation (2) is an algebraic phase average for distinct frequencies. At equal or resonant frequencies the relevant phases must instead remain in the dynamics. The unaveraged quartic retains them in either case.

For the normal kernel both currents in (1) vanish at leading chiral order. Inserting the complete correction eta/Lambda gives their first nonzero terms. The calculation retains all normal Fourier powers and all four normal spinor components in both members of the canonical pair.

The normal angular factors

Write vartheta for a normal polarization orientation and theta for the ordinary polar coordinate in the normal plane. Let q=n-1, A_0=U_0, B_0=V_0, and define the positive scalar responses

$$\begin{aligned} Y&=K_0^{-1}\left[ \frac{2(1+r^2/2)}{(1+r^2)^{3/2}}U_0\right],\\ Z&=K_{n+1}^{-1}\left[ \frac{r^2}{(1+r^2)^{3/2}}V_0\right]. \end{aligned} \tag{3}$$

The inverse K_m is the complete half-line inverse, not a normal-mode cutoff. In the large-gamma endpoint limit, the complex self current scaled by Lambda is

$$j_1+i j_2= -2(-1)^q\gamma A_0Y e^{in\vartheta} +2(-1)^q\gamma B_0Z e^{-in\vartheta}e^{i2n\theta}. \tag{4}$$

Both terms survive at leading order. Define the positive common spatial coefficient

$$C_n=4\int_{\mathbb R^2} \left(A_0^2Y^2+B_0^2Z^2\right)d^2x. \tag{5}$$

Angular integration of (4), and a separate contraction of the mixed spinors, then give

$$\begin{aligned} D_{ab}&:=\int\boldsymbol j_a\cdot\boldsymbol j_b\,d^2x \longrightarrow C_n\gamma^2\cos(n\Delta\vartheta),\\ X_{ab}&:=\int\sum_i|k_{i,ab}|^2\,d^2x \longrightarrow C_n\gamma^2\cos^2(\Delta\vartheta/2). \end{aligned} \tag{6}$$

Here Delta vartheta=vartheta_b-vartheta_a. The coefficients k_i are the first mixed-current terms scaled by Lambda, matching the scaling of j_i. The endpoint result follows from the actual wavefunctions, with their canonical norm. It contains no division by particle frequency.

Numerically C_5=0.0935501816, C_6=0.0920762937 and C_10=0.0868345076. Thus the winding-five normal channel has no large suppression under the same dimensionless normal action parameters. This is not a derivation of neutrino interactions. The full depth-five geometry, physical units and cross-sector interaction still have to be supplied.

At gamma=10000, the aligned electron and muon normal source profiles differ by about 0.00328 in relative L2 norm. Their direct and mixed normal overlap coefficients are respectively about 9.20738332e6 and 9.20740815e6, close to the common endpoint value 9.20762937e6. At Delta vartheta=pi/12 the depth-six separate-current overlap is nearly zero, but the mixed contribution is about 0.98294 of its aligned endpoint scale. That preparation is not a zero of (2).

The cross term can also be negative at other orientations; the complete quartic, including the separate self terms, remains nonnegative for positive g. Neither the sign of this cross energy nor an energy gradient alone establishes the dynamics of an encounter.

Restore the actual depth-six surface patterns

For the monogenic surface family with one common spinor vacuum z_0,

$$\chi_{L,u}=Y_{L,u}z_0,\qquad Y_{L,u}=\frac{(u_1z_1+u_2z_2)^L}{\sqrt{V B(L)}}, \quad B(L)=\frac{\Gamma(L+1)\Gamma(5/2)}{\Gamma(L+5/2)}, \quad V=\frac{8\pi^2}{3}. \tag{7}$$

These are actual spinor fields on the conditional four-sphere surface feature in six ordinary dimensions. The two ordinary normal coordinates remain spatial coordinates. This surface/normal description does not impose a special three-plus-three split of electron space.

The two terms in (2) generally multiply different surface overlaps: integral rho_a rho_b and integral |chi_a^dagger chi_b|^2. For the common-vacuum family (7) those overlaps coincide pointwise. Their exact integral is

$$I_{ab}(x)=\frac{B(a+b)}{V B(a)B(b)} \frac{\sum_{j=0}^{\min(a,b)}\binom aj\binom bj x^j} {\binom{a+b}{a}},\qquad x=|u^\dagger v|^2. \tag{8}$$

The earlier surface spinor overlap contains an additional tangent-current factor. It is absent from (8) because the present matrices act in the normal spinor factor. Reusing that tangent-current integral unchanged would give the wrong coefficient.

The complete reference cross coefficient for (7) is I_ab(D_ab+X_ab). For a common degree-two mathematical probe and the electron and muon surface assignments, the ratio of responses is:

The normal orientations are aligned and all action parameters and norms are common. The particle-frequency ratio is 206.75894 at the tested finite gamma. The degree-two probe is a depth-six mathematical mode, not a photon. These results illustrate comparable responses for some preparations and substantial geometric variation for others; they do not establish an orientation-independent universal coupling.

The physical normalization removes an apparent width suppression

Let the surface radius be R and the normal profile width be a. Both are ordinary spatial lengths. The normal width is a field-profile scale on an unbounded plane, not a compactified dimension. Equal physical wave speeds give Lambda=R/a. Canonical normalization of a d-dimensional product profile requires

$$\Psi_{\rm phys} =a^{-1}R^{-(d-2)/2}\psi_{\rm dimensionless}. \tag{9}$$

Its first nonzero normal current therefore scales as

$$J_{\rm phys} =a^{-2}R^{-(d-2)}\Lambda^{-1}j =a^{-1}R^{-(d-1)}j. \tag{10}$$

The complete volume element contributes a^2 R^(d-2). Hence

$$\boxed{\int J_{\rm phys}^2\,d^dx =R^{-d}\int j^2\,d^2r\,dA} \tag{11}$$

at this leading order. The apparent Lambda^(-2) suppression is exactly offset by the increasingly concentrated physical profile. If g_d is the physical quartic coefficient in frequency-times-volume units, the pair coefficient is

$$\overline U_{ab} =\frac{g_d}{R^d}I_{ab}(D_{ab}+X_{ab}). \tag{12}$$

Thus narrowing this normal profile alone does not weaken the physical interaction at fixed g_d and R. Taking Lambda large at fixed dimensionless quartic g instead changes the corresponding physical coupling. That distinction is necessary when comparing the normal gap with the source and response. Across different d, the physical normalization and sector measure still require a common master-field derivation; the numerical C_n values alone do not settle that relation.

Separate the two ordinary spatial profiles

The complete normal fields can also be placed at different centers. Let s be their separation along the axial normal direction, in units of the common normal width a. This axial displacement is an ordinary Cartesian translation. It is distinct from displacing the radial coordinate of the round surface, which would change the profile of the feature rather than translate the whole feature.

The separate and mixed terms are evaluated on the actual complex fields at x+s/2 and x-s/2, with all their spinor components and normal angles. For aligned normal polarization, the depth-six electron/muon comparison gives the following normal cross coefficient, divided by the common endpoint value 2 C_6 gamma^2:

The dimensionless normal width cancels from the leading coincident strength in (11), but the physical separation is a s. It therefore still sets the range of this overlap. This distinction does not supply a physical value for a or an interaction rate.

Polarized coefficients are more varied than this aligned example. At relative angle pi/6 the complete cross term is negative at coincident centers, positive at separations one and two, and negative again at separation three. The separate and mixed contributions have different spatial envelopes. A sign-changing cross energy alone does not prove a bound particle pair or a stable dynamical trajectory.

Direct differentiation of the full complex spatial fields gives the gradient of this cross energy. Independent displaced-profile differences agree with it. Interchanging the two profiles leaves their cross energy unchanged and reverses its separation derivative. A common rotation of both polarized fields and the separation vector also leaves it unchanged. These checks establish reciprocity of this interaction contribution; the inertial and supporting-field terms needed for motion remain absent.

Scope and verification

The normal report holomorphic-normal-pair-20260914T062613_882235Z.json passes 256 checks. It includes all mixed currents, spatial refinement, the full first normal equation, reciprocity, common physical rotations, and an independent phase average of the complete complex local spinors. It also checks the winding-five control with no inserted suppression.

holomorphic-surface-pair-20260914T062853_406360Z.json passes 75 checks of actual ordinary surface integration, arbitrary complex coherent lines, canonical norms, and the physical volume factors.

holomorphic-normal-separation-20260914T064513_632049Z.json passes 138 checks of complete Cartesian overlap, polarization, physical rotations, reciprocity and the spatial energy derivative. The very small separation-four tail uses a denser radial profile and independent Cartesian quadratures. The preceding report and its source remain preserved: its last radial tail comparison missed the required three-parts-per-million tolerance.

The calculations concern prescribed overlapping or separated reference profiles and an initial weak-interaction coefficient. They do not yet supply self-supported particles, a physical background, a dynamical encounter trajectory, or reciprocal evolution of the normal texture. Those require the same-field source and its kinetic and stress terms. The negative wave-only inertia found in the companion calculation is not repaired by the present pair overlap.

Related research

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