Nonlinear spinor frequencies under one common surface action

The curvature parent now has nonlinear stationary spinor branches near the assigned depth-six modes. The radius, intermediate gap, interaction coefficient and canonical norm are common to both branches. Their frequencies are read directly from the stationary equation. The interaction deforms the spatial pattern and changes the frequency ratio; it does not preserve the simplex ratio automatically.

This is a conditional inverse action on a specified round field feature. The existence argument below concerns its complete surface equation. The numerical calculation resolves its low parent modes and explicitly measures forcing outside that space. A physical construction of the feature, its two ordinary normal profiles, gap and complete inertia is still required.

The common action and its complete field

Use the actual spinor parent of A spatial spinor parent for the curvature frequency, on a four-sphere of radius R in ordinary six-dimensional space. Let H_0 be its full twelve-component chain Hamiltonian, each component a complete rank-four spatial spinor. The chain components are hypothesized modes of the same field. Their physical carrier has not been derived.

At the two ends, indexed e=0,1, define

$$J_a=\sum_e\Psi_e^\dagger\gamma_a\Psi_e, \qquad U[\Psi]=\frac12\int J_aJ_a\,dA, \qquad \mathcal E=\langle\Psi,H_0\Psi\rangle+gU[\Psi]. \tag{1}$$

Here gamma_a is the actual tangent Clifford symbol, not a scalar density. The kinetic term is canonical and fixes Q=integral sum_j |Psi_j|^2 dA. The local quartic varies into N_e=J_a gamma_a Psi_e at both ends; its other chain components vanish. Its exact stationary equation is

$$\boxed{H_0\Psi+gN[\Psi]=\omega\Psi,\qquad Q=1.} \tag{2}$$

Mass is omega in equivalent units. Q is a stated common normalization, not an amplitude fitted separately to an electron or a muon. The endpoint current in (1) remains an interaction choice. It is distinct from the conserved spatial flux fixed by the chain kinetic term.

The rotation family in A complete rotation family for the depth-six fixed point retains all four spinor components, complex coefficients and every harmonic compatible with the initial rotation labels and parity. Its base degree is L=14 for the electron branch and L=36 for the muon branch. These correspond to the already assigned simplex indices N=12 and N=34.

For each scalar degree K, apply both exact B_K spectral projections. Their full normalized chain eigenvectors embed the positive and negative low parent coefficients c_+ and c_- canonically. If E_0 and E_1 denote the two endpoint matrices on these orbital coefficients,

$$\Psi_0=E_0(c_++c_-),\qquad \Psi_1=E_1(c_+-c_-). \tag{3}$$

The complete chain norm is ||c_+||^2+||c_-||^2. All intermediate components enter that identity, even when their pointwise size is small. Projecting (1) gives the adjoint endpoint maps and retains the negative parent response; it is not a projection onto a scalar ray.

A local existence statement for the complete surface equation

This statement uses the entire first-order chain on the compact surface, not a finite harmonic truncation. Compactness here describes the specified round field feature. It is not compactification of ordinary spatial dimensions.

Within the fixed rotation family, the base monogenic mode has a one-dimensional positive eigenspace. At all larger compatible degrees the smallest positive chain frequency is strictly larger. To see the latter, the absolute entries of the triangular inverse are products of positive factors G/(beta+shift). They decrease strictly as beta increases above the largest negative shift. Its largest singular value therefore decreases, by positivity of the corresponding Gram matrix. The smallest chain frequency, its reciprocal, increases. Every other positive singular frequency at a larger degree is at least as large. At the base degree, the other chain frequencies exceed its simple smallest one. Negative frequencies cannot equal this positive value.

The full parent is self-adjoint and elliptic: its principal symbol squares to c^2|p|^2 on every chain component. The base frequency is therefore an isolated simple eigenvalue in the stated rotation family. Fix its overall phase and Q=1. The derivative of (2), including the norm equation and frequency unknown, is invertible on the orthogonal complement of this mode. For Sobolev index s>2 on the four-sphere, the pointwise cubic is smooth from H^s to H^s and hence to H^(s-1). The complementary elliptic inverse maps H^(s-1) back to H^s. The implicit function theorem then gives a unique nearby normalized branch, up to overall phase, for sufficiently small real g.

This establishes conditional local existence. It supplies neither an explicit allowed interval for g nor stability against other rotation families. It also does not turn the externally specified surface or gap into a self-supported six-dimensional particle.

The first two frequency coefficients

Let u be the full normalized base mode and omega_L its linear frequency. With the overall phase fixed, write

$$\Psi=u+g\eta+O(g^2),\qquad \omega=\omega_L+ga_L+g^2b_L+O(g^3),\qquad \langle u,\eta\rangle=0. \tag{4}$$

Full differentiation of (2) gives

$$a_L=\langle u,N(u)\rangle=2U(u),\qquad \eta=-(H_0-\omega_L)^{-1}_{\perp}N(u), \tag{5}$$

where the inverse includes the orthogonal projection on both sides. The real Hessian of U is symmetric and DN(u)u=3N(u), so

$$\boxed{b_L=-3\sum_{\nu\ne\omega_L} \frac{|\langle u_\nu,N(u)\rangle|^2}{\nu-\omega_L}.} \tag{6}$$

The sum includes both signs, all spatial polarizations and all chain branches. A calculation retaining only the initial harmonic would miss this second coefficient and the profile change in (5).

For the monogenic base spinor, let Y_L be its unit scalar harmonic and let K_L be the sum of the squared endpoint coefficients. Direct Clifford multiplication gives the current square of Y_L z as |Y_L|^4(1-x_5^2) on the unit four-sphere. Integration over its actual angular coordinates yields

$$\boxed{a_L=K_L^2\frac{B(2L)}{V B(L)^2} \left(1-\frac1{4L+5}\right),\qquad B(q)=\frac{\Gamma(q+1)\Gamma(5/2)}{\Gamma(q+5/2)}, \quad V=\frac{8\pi^2}{3}.} \tag{7}$$

At radius R, multiply (7) by R^(-4). This is a geometric matrix element of the declared action, not an inserted mass count. At R=c=1 and G=10000, its values are approximately 0.6299168140 for L=14 and 2.3403634515 for L=36.

For comparisons use the common linear electron frequency omega_e^0 as the unit and define epsilon=g/(R^4 omega_e^0). Then both branches have the same epsilon. Assigning epsilon separately relative to each particle frequency would change the physical action between particles.

Frequency, energy and the omitted response

Because the interaction is quartic, every complete stationary solution obeys

$$\mathcal E=\omega Q-gU. \tag{8}$$

Equation (8) is checked in the numerical solutions. It is not the complete particle inertia: the sustaining geometry and its energy and stress are not present in this surface action. Their contribution cannot be discarded when testing the proposed inertial mechanism.

The calculation also reconstructs the complete spatial source. Its component outside the retained scalar-spinor harmonics is measured as the norm of the actual difference of fields. The high chain source is measured by subtracting the full canonical low-mode projection from the injected endpoint source. These direct differences avoid subtracting nearly equal squared norms.

Within a retained degree band, the first high-chain correction is bounded by |g| times this source norm divided by its spectral distance from omega. The simple bound G-max_j|L_j|-|omega| is positive for the tested bands. This is a first-response estimate; the local existence argument concerns the complete nonlinear equation separately.

Independently refined results

The common parameters are R=c=1, G=10000 and Q=1. Frequencies below are in the same linear electron unit, omega_e^0=1.3366047052613313e-13. The linear muon value in this unit is approximately 206.7622950.

These are illustrative values of one inverse action. They are not independently selected physical couplings. A positive coefficient in this range gives a larger fractional shift to the electron branch, so it lowers the muon-to-electron frequency ratio further below the limiting simplex ratio. Choosing a sufficiently small coefficient makes the discrepancy small; it does not explain that choice.

In the same units, the second coefficients in (6) are approximately -0.46708250654 and -0.09489144102. Both signs of the parent enter their calculation. Two-sided nonlinear continuation agrees with these coefficients and the analytic first derivative (7).

The report spinor-fixed-point-20260914T014635_511200Z.json passes 148 checks. It retains source hashes and the complete complex coefficients at J=8, 12 and 16, with separately increasing radial and polar quadratures. The largest J=12 to J=16 profile difference is 2.59e-10; the frequencies agree to the quoted digits. Direct complex action variations, the real Hessian including conjugates, its phase identity, the full chain norm and (8) are independently checked.

At J=16 the largest relative spatial source remainder among these controls is below 6e-8. The largest first high-chain response bound is below 1.5e-20 in canonical field norm at the assumed gap. The earlier report spinor-fixed-point-20260914T014107_593148Z.json passes 138 checks and remains preserved; its subtraction of nearly equal source norms overestimated the smallest spatial remainders. The current report uses their directly reconstructed spatial fields.

No experimental coupling or physical mass-operator derivation follows from the conditional existence statement alone.

Related research

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