Controlling the full source of the reciprocal spinor pair

The reciprocal polynomial evolution has a controlled weak-coupling interpretation in the complete surface equation. Its cubic source contains four temporal frequencies. Each source has finite angular support, including the non-vacuum spinor components and the opposite parent frequencies. Explicit nonzero detunings permit a first spatial correction and a finite-slow-time averaging argument.

This concerns the conditional depth-six surface action in Nonlinear spinor frequencies under one common surface action and Reciprocal shape response in the spinor family. It retains all 48 complex field components on the specified four-sphere. It does not construct the supporting field in the two ordinary normal directions, establish particle stability, or determine the physical coupling coefficient. The surface is a proposed field feature in ordinary space, not compactification.

The complete cubic source

Set the common radius and wave speed to one. Let H be the complete Hermitian chain Hamiltonian, with common gap G. At its two endpoints, the interaction is

$$U=\frac12\int J_aJ_a\,dA,\qquad J_a=\sum_e\Psi_e^\dagger\gamma_a\Psi_e,\qquad N_e=J_a\gamma_a\Psi_e. \tag{1}$$

All other components of N vanish, but the free Hamiltonian propagates this forcing through every chain component. The endpoint current in (1) remains an inverse interaction premise, distinct from the canonical kinetic flux. This is a local same-field action without mediators.

For two complete holomorphic polynomial shapes A(x) and B(x), of degrees a and b, respectively, write the leading field as

$$\Psi^{(0)}=u_a A(x)z_0e^{-i\omega_a t} +u_b B(x)z_0e^{-i\omega_b t}. \tag{2}$$

The vectors u_a and u_b are the complete unit canonical chain states. Both polynomial coefficient vectors have norm one. Define their two endpoint vectors e_a,e_b and

$$k_a=e_a^Te_a,\quad k_b=e_b^Te_b,\quad k_{ab}=e_a^Te_b. \tag{3}$$

In actual Cartesian coordinates on the unit four-sphere, use z_1=x_1+ix_2, z_2=x_3+ix_4 and the vacuum spinor z_0. Its tangent current is j_0=(-x_2,x_1,-x_4,x_3,0). Full Clifford multiplication gives

$$v(x)=\gamma(j_0)z_0=z_0-x_5\Gamma_xz_0 =(F,-x_5z_2,-x_5z_1,0)^T,\qquad F=1-x_5^2,\quad\|v\|^2=F. \tag{4}$$

In particular, the non-vacuum components have squared norm x_5^2 F. They cannot be replaced by the vacuum projection F z_0.

The complete source at each endpoint is a sum of S_e(x)v(x) with the following temporal factors:

$$\begin{array}{c|l} \text{frequency}&S_e\\\hline \omega_a&[e_{a,e}k_a|A|^2+(e_{a,e}k_b+e_{b,e}k_{ab})|B|^2]A\\ \omega_b&[e_{b,e}k_b|B|^2+(e_{b,e}k_a+e_{a,e}k_{ab})|A|^2]B\\ 2\omega_a-\omega_b&e_{a,e}k_{ab}A^2\overline B\\ 2\omega_b-\omega_a&e_{b,e}k_{ab}B^2\overline A. \end{array} \tag{5}$$

Direct Fourier decomposition of the actual Clifford current verifies all four terms for arbitrary complex polynomials. No restriction to real amplitudes or one fixed relative phase is used.

The complete spectral support is finite

Under a common rotation of the two complex planes, the four source classes have vacuum orbital charges

$$q=(a,b,2a-b,2b-a). \tag{6}$$

The other spinor components carry the compensating orbital charges q+1,q+1,q+2. Their total spinor rotation character is the same within each class. Since a and b differ, the four classes are mutually orthogonal. The free Hamiltonian preserves them.

The polynomial degrees in (5), including the degree-two vector (4), give the following upper bounds on scalar harmonic degree K:

$$K_{\max}=(a+2\max(a,b)+2,\ b+2\max(a,b)+2, 2a+b+2,\ 2b+a+2). \tag{7}$$

In each class K has the parity of q. For the two positive central charges, K is at least q. For either combination class a conservative lower bound is max(0,|q|-2), adjusted to that parity. This bound can include harmonics with zero source; including them only weakens a resolvent estimate.

For each such K, retain both B eigenvalues and both signs of every parent frequency. The smallest positive parent frequency is obtained from the full six-link singular problem. The two signs of B give the same singular frequencies by reflection and diagonal sign changes. The other singular frequencies are at least G-(K_max+4), by the singular-value perturbation bound relative to the gap shift. Consequently, their detuning from a source frequency Omega is at least G-(K_max+4)-|Omega|. This is positive for the cases below.

At each central frequency, remove only its complete resonant polynomial family. Its projection is precisely the force derived from the Hamiltonian in Reciprocal shape response in the spinor family. The opposite parent frequency at that same degree remains forced. For the combination frequencies, the enumerated signed low branches have nonzero detuning in all reported cases. Thus no additional resonant source has been silently omitted.

The positive combination frequency for degrees 14 and 36 illustrates the geometric restriction. Its charge is 58, so K is conservatively at least 56. A frequency comparison that ignores this charge would include lower-degree modes to which the actual source cannot couple.

Bound the first complementary response

Let F_j be the complete nonresonant source at frequency Omega_j, with its central resonant projection removed when necessary. Let R_j=(Omega_j-H)^{-1} on that source's finite invariant complementary space. The inverse is needed only there, not on unrelated angular sectors. If delta_j bounds every retained detuning, then

$$\|R_jF_j\|_2\le\frac{\|F_j\|_2}{\delta_j}. \tag{8}$$

The first response with zero initial correction is

$$\xi^{(1)}(t)=-ig\int_0^t e^{-iH(t-r)} \sum_j e^{-i\Omega_jr}F_j\,dr. \tag{9}$$

For fixed shapes, orthogonality of the four rotation classes gives

$$\boxed{\|\xi^{(1)}(t)\|_2 \le2|g|\left[\sum_j\frac{\|F_j\|_2^2}{\delta_j^2}\right]^{1/2}.} \tag{10}$$

The bound is uniform in t. At a central frequency one can also resolve the same-degree opposite-parent projection separately. Its detuning is 2 omega_a or 2 omega_b; its orthogonal remainder uses the larger applicable gap. Their squared bounds add. This retains the small opposite-parent component without letting its small detuning inflate every other component.

All source differences are formed directly in ordinary coordinates. For example, subtracting a resonant polynomial R at a central frequency gives endpoint squared residual

$$\sum_{e=0,1}|S_eF-u_{e}R|^2 +\sum_e|S_e|^2x_5^2F, \tag{11}$$

plus the complete intermediate-chain contribution sum |u_i R|^2. This avoids subtracting two nearly equal squared norms. The opposite-parent subtraction changes the upper and lower chain polynomials to R+R_minus and R-R_minus, respectively.

Verified magnitudes under one common action

Use G=10000 and the common linear electron frequency omega_e^0=1.3366047052613313e-13. Define epsilon=g/omega_e^0 for every pair; it is not reset using each pair's smaller frequency. The following are coefficients C in the frozen bound ||xi^(1)|| <= C |epsilon|:

These bounds include self-source deformation as well as cross-source response. A small cross-overlap alone therefore does not make the entire complementary source small. Degree two is a mathematical control within this depth-six construction, not a photon. Its much smaller spectral gap makes the same averaging approximation require a weaker common interaction.

For the degree-14/36 angle-0.7 pair, the source norms at the four frequencies are approximately 0.79405147, 2.89797426, 0.02166322 and 0.08574321. The resonant norms at the first two frequencies are 0.64642895 and 2.35324052. The remaining central norms are 0.46113702 and 1.69130538. These are actual full-field L2 norms.

spinor-pair-source-20260914T045034_198193Z.json passes 459 checks. Independent coherent moment formulas, complete Clifford currents, arbitrary complex shapes, conjugate derivatives, two ordinary-coordinate quadratures and 80-digit signed detunings verify the calculation. Check counts are numerical evidence for the stated implementation, not a replacement for the support argument above.

Follow the changing shapes

The reciprocal shapes vary with slow time s=gt. Integrating (9) by parts with the complete varying F_j gives the bound

$$\|\xi^{(1)}(t)\|_2\le |g| \left[\mathcal R(s)+\mathcal R(0) +\int_0^s\mathcal R_1(\sigma)\,d\sigma\right], \tag{12}$$

where R(s) is an upper bound for the Hilbert norm of the orthogonal sum of R_jF_j(s), and R_1 uses dF_j/ds. The latter derivative follows from both complete complex shape equations, including their conjugate variations. This remains a first forced response, not the full nonlinear error at a specified finite epsilon.

For the independently refined degree-14/36, angle-0.7 trajectory, the frozen coefficient grows from 0.84919762 to 1.25663633 through slow time 20. The integral of the corresponding derivative coefficient is 16.00941368. Equation (12) therefore gives an endpoint first-response estimate of 9.05762382 |epsilon|. Two slow-time quadratures differ by 2.60e-6 in that integral. The estimate is numerically refined, not interval certified. Report spinor-source-trajectory-20260914T045545_833184Z.json passes 31 checks, including full spatial and conjugate-derivative refinements at deformed points along the trajectory.

There is also a direct inverse calculation for aligned coherent pairs. It expands all four sources in their complete signed-rotation spinor harmonics through (7), with four further degrees used as a zero-source check. Both signed B projections and every canonical parent eigenvector are retained. The full chain inverse is computed at 80-digit precision. For the aligned degree-14/36 pair, the four norms of omega_e^0 R_j F_j are approximately 2.19168906, 0.02932807, 0.00542109 and 0.00059714. They give the stronger frozen bound 4.38378412 |epsilon|, compared with the conservative gap bound 6.04185324 |epsilon| for that alignment. This aligned example is different from the angle-0.7 trajectory above.

spinor-source-resolvent-20260914T051854_908897Z.json passes 569 checks. Source reconstruction retains every generated spinor component and has L2 residual below 1.7e-12 in the assigned-degree examples. Independent quadrature, additional angular degrees, complete Dirac-square identities and a directly applied ordinary angular equation at a smaller, well-conditioned gap verify the inverse calculation.

A finite-slow-time theorem for the complete surface equation

Normalize H and all rapid frequencies by omega_e^0, and time by its inverse. The full equation is

$$i\partial_t\Psi=\widehat H\Psi+\epsilon N(\Psi). \tag{13}$$

Fix any finite slow interval 0 <= s <= S and an initial pair in the specified resonant polynomial spaces. Suppose its four complete source classes have the nonzero complementary detunings just verified. Then the exact solution of (13), with that initial pair, converges to the reciprocal resonant evolution as epsilon approaches zero, uniformly for 0 <= t <= S/epsilon in H^3 on the four-sphere.

Here is the argument, including the full field space. The chain Hamiltonian is self-adjoint and commutes with the componentwise spherical Laplacian, because B^2=-Delta+9/4. Its group is therefore unitary in every spectral Sobolev norm. H^3 on the four-dimensional surface is an algebra, and the smooth tangent Clifford coefficients make N locally Lipschitz in H^3. The usual product bound follows from Leibniz' rule, H^3 embedding into L-infinity and the applicable H^1-to-L^4 derivative estimates. It applies to all 48 complex components.

The finite polynomial averaged system preserves the two norms, so its solution and every derivative remain bounded on a fixed S. Its source has the finite support (7). In normalized units define

$$\begin{aligned} W(t,s)&=\sum_j e^{-i\widehat\Omega_jt} (\widehat\Omega_j-\widehat H)^{-1}F_j(s),\\ \widetilde W(t,s)&=W(t,s)-e^{-it\widehat H}W(0,0),\\ \Psi_{\rm app}(t)&=\Psi^{(0)}(t,\epsilon t) +\epsilon\widetilde W(t,\epsilon t). \end{aligned} \tag{14}$$

The second line enforces the original initial field exactly. The leading residual is minus epsilon times the complementary source; the fast derivative of the correction cancels it. The remaining residual is

$$\epsilon^2 i\partial_s W -\epsilon\{N(\Psi^{(0)}+\epsilon\widetilde W) -N(\Psi^{(0)})\}, \tag{15}$$

and hence is bounded by A epsilon^2 in H^3, uniformly on the interval. A is finite because the source support, its inverse detunings and the shape derivatives are all bounded there.

Choose a fixed H^3 ball containing the approximate trajectory, and let L be the local Lipschitz constant of N on a slightly larger ball. Duhamel's formula and Gronwall give

$$\|\Psi(t)-\Psi_{\rm app}(t)\|_{H^3} \le\frac{A\epsilon}{L}(e^{LS}-1). \tag{16}$$

Taking epsilon sufficiently small closes the ball bootstrap and continues the exact local solution through S/epsilon. The correction in (14) is also O(epsilon), proving the claimed convergence. No harmonic cutoff is imposed on the exact solution. Newly generated modes and subsequent nonlinear responses are covered by the full Sobolev estimate.

This theorem supplies a conditional asymptotic justification for the shape calculation. A coarse Sobolev bound can make its explicit smallness threshold extremely restrictive. It does not certify that an example such as epsilon=0.01 accurately follows the average for slow time 20. The informative L2 first-response coefficients in the table do not replace that nonlinear estimate. Neither result derives the surface, the master-field interaction, a depth-five neutral branch, or the particle's full inertia. Mass remains the selected frequency; the slower shape evolution is its interaction response.

Related research

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