Which normal shapes preserve the simplex frequency polynomial?

The complete kernel calculation identifies a useful constraint on the inverse mass operator. Each lower holomorphic coefficient of the normal texture supplies a different lower-degree contribution to its frequency polynomial. The correspondence is triangular and has nonzero diagonal coefficients. Consequently, the winding and mode count alone do not protect the specific simplex polynomial: the sustaining dynamics must also control the normal shape.

This statement concerns the prescribed spatial carrier, with winding n identified with depth d as an inverse premise. It does not derive that identification or a physical source. It continues the endpoint deformation calculation and retains the complete normal field.

Vary every lower holomorphic coefficient

Start from W=z^n and consider

$$W(z)=z^n+\sum_{p=0}^{n-1}a_p z^p. \tag{1}$$

The coefficients can be complex. They describe spatial shape changes, not amplitudes assigned to particle masses. For a sufficiently small deformation the complete index and normal kernel count remain n, while the projected matrix elements can change.

Put q=n-1 and s=n-p. The full kernel derivative in direction delta W=a z^p follows from

$$D_n\delta\Phi_l=-\delta D\,\Phi_l,\qquad P\delta\Phi_l=0. \tag{2}$$

The scalar partner inverse gives an explicit solution. Define

$$A_l=K_{l-p}^{-1}(r^pU_l),\qquad B_l=K_{l-q+p}^{-1}(r^pV_l). \tag{3}$$

For a real unit deformation, its two nonzero spinor components are

$$\begin{aligned} (\delta\Phi_l)_1={}&-r^nA_l e^{i(l+s)\theta} +(B_l'+(l-q+p)B_l/r)e^{i(l-s)\theta},\\ (\delta\Phi_l)_2={}&i(A_l'-(l-p)A_l/r)e^{i(l-p+1)\theta} -i r^n B_l e^{i(l-q+p-n)\theta}. \end{aligned} \tag{4}$$

For a complex direction a, the terms with angular shift +s multiply conjugate(a), and the terms with shift -s multiply a. Thus both real quadratures are retained. The actual first-order Dirac equation, not only its scalar square, verifies these components.

The diagonal polarization is stationary at first order

There is an exact cancellation for the projected constant matrix S=sigma_3 tensor I:

$$\boxed{\delta(P S P)=0} \tag{5}$$

in the parallel kernel frame of (2), for every lower holomorphic direction in (1). This does not follow merely from angular selection; for many p there are allowed angular entries. Use instead the exact inverse of the polarization source,

$$D_n^{-1}(I-P)S\Phi_l =\frac r{n+1} \left(e^{i(l-n)\theta}V_l,0,0, i e^{i(l+1)\theta}U_l\right)^T. \tag{6}$$

Substitute (6) into the two terms obtained by differentiating the matrix element of S. Each allowed angular entry is the sum of two opposite copies of the same radial integral, with the same complex coefficient. They cancel. This proves (5) without omitting components.

Equation (5) is a first-order result at the monomial texture. It does not assert that the polarization spectrum is invariant under arbitrary finite deformations.

The hopping has exactly two possible bands

For h=z/sqrt(1+r^2), the derivative of a projected matrix entry can be nonzero only when

$$k-l=1-s\quad\text{or}\quad k-l=1+s. \tag{7}$$

The first band is on or above the diagonal and multiplies a. The second is below the original adjacent band and multiplies conjugate(a). The constant deformation p=0 leaves only the endpoint-closing entry already calculated in the companion note.

There is a useful sign formula for every entry of the first band. Set l=k+s-1, with k=0 through p, and h_z=(1+r^2/2)/(1+r^2)^(3/2). Its coefficient is

$$\begin{aligned} A^{(p)}_{kl}=-\int_{\mathbb R^2}r^p\big[& U_k K_{l-q}^{-1}(2h_zV_l)\\ &+V_l K_k^{-1}(2h_zU_k)\big]d^2x<0. \end{aligned} \tag{8}$$

All radial functions, h_z and the scalar Green kernels are positive. Thus the inequality is strict, and no cancellation of these upper entries is possible. This formula was also evaluated independently of the differentiated kernel fields in (4).

Cofactors give a lower-degree frequency polynomial

At the undeformed texture, the canonical pair uses the lower-bidiagonal matrix T with diagonal d_j=beta+j-q/2 and adjacent entries gamma h_j. Its leading small frequency is

$$\omega_0(\beta)= \frac{\prod_{j=0}^{n-1}d_j} {\gamma^{n-1}\prod_{j=0}^{n-2}h_j}. \tag{9}$$

For n=d and the assigned relation between beta and surface degree, this numerator is the rising product giving the simplex count. The following calculation keeps those mode assignments fixed.

Differentiate the determinant of the complete matrix. The second, strictly lower band in (7) has zero first determinant derivative. Each upper entry A^(p)_(k,k+s-1) instead closes a path containing s-1 existing adjacent links. Its cofactor leaves exactly p diagonal factors outside that path. Dividing by the same leading product of large singular frequencies gives

$$\frac{\partial\omega}{\partial\operatorname{Re}a_p} =\gamma^{1-p}P_{n,p}(\beta) \,[1+O(\gamma^{-2})], \tag{10}$$

at a fixed beta where the leading derivative is nonzero. Explicitly,

$$\begin{aligned} P_{n,p}(\beta)=\frac{(-1)^{s-1}}{\prod_{j=0}^{n-2}h_j} \sum_{k=0}^{p}&A^{(p)}_{k,k+s-1} \left(\prod_{j=k}^{k+s-2}h_j\right)\\ &\times\left(\prod_{j<k}d_j\right) \left(\prod_{j\ge k+s}d_j\right). \end{aligned} \tag{11}$$

Empty products equal one. Every term has degree p. Its leading coefficient is nonzero because the A entries are strictly negative and all hopping products are positive. Therefore P_(n,p) has exactly degree p, for each p from zero through n-1.

This proves that the map from real lower texture coefficients to the lower coefficients of the leading frequency polynomial is triangular and invertible at first order. It is a geometric constraint within this carrier, not an argument that other dimensional assignments fit the mass relations. Nor does it select a physical restoring law for the a_p.

For n=6, the first three response polynomials are approximately

$$P_{6,0}=0.6149315907,\qquad P_{6,1}=-0.8243156254\beta,\qquad P_{6,2}=1.1583120795\beta^2+0.7967533699. \tag{12}$$

Their frequency derivatives carry the respective factors gamma, one, and gamma^(-1) in (10). The coefficients come from full spatial Green integrals; they have not been chosen to produce a familiar polynomial family.

For complex deformations, the leading effective off-diagonal pair amplitude receives the corresponding complex a_p contributions. Its positive frequency is the magnitude of that amplitude. Imaginary directions at a real reference can therefore change the field at first order while changing the positive frequency only at second order. Frequencies alone do not determine all complex shape data, and a zero first frequency derivative does not imply zero coupling.

A translation must move all spatial coefficients

For p=n-1, the perturbation z^n+a z^(n-1) is, to first order, the translation (z+a/n)^n. The complete kernel derivative in (4) then agrees with partial_x Phi/n or partial_y Phi/n for the two real directions. If h is translated with the same field geometry, its explicit coefficient derivative cancels its kernel-basis derivative:

$$\boxed{\delta(P h P)+P(\delta h)P=0.} \tag{13}$$

Thus translating the whole set of coefficients leaves the spectrum unchanged, as it should. Moving W while holding h fixed is a relative displacement inside the specified geometry. The p=0 splitting of a multiple zero at n>1 is not a common translation and is not removed by this identity.

The small polynomial coefficients quoted in the endpoint note must also be interpreted in spatial terms. For the individual deformation z^n+a_p z^p, its nonzero roots lie at radius |a_p|^(1/(n-p)). For the electron control at gamma=10000, the first-order coefficients that give a frequency change comparable to its original value correspond to radii about 0.00228 through 0.00286 in normal-width units. The muon comparison gives about 0.00555 through 0.00699. These are local sensitivity scales, not a solved finite-deformation spectrum.

The constant coefficient 1.42e-16 is therefore not a literal fractional length displacement of 1.42e-16. Its sixth root gives the relevant root-separation scale in that example.

What the source must preserve

A common translation is already harmless. Genuine changes of the relative normal shape alter the low-frequency polynomial while preserving the winding and normal mode count. A physical source must therefore either restore the required shape or constrain the allowed deformations through the geometry of the same field.

For example, requiring several coefficient textures to be functions of one common field geometry would change their variational source: their chain-rule contributions could not be treated as independent. That possibility is different from the simple two-current closure tested earlier. No such common-geometry action or constraint has been derived here, and a mere change of spatial coordinates would not provide one.

A common restoring energy must satisfy more than one source equation

The first two genuine shape directions already constrain a simple inverse completion. Suppose its energy separates as

$$E_L(a,Q)=E_{\rm geom}(a)+Q\,\omega_L(a), \tag{14}$$

where a denotes real shape coordinates, the same geometric energy applies to each occupied surface mode L, and Q is the canonical wave norm. This is a particular trial energy, not a derived nonlinear IDWT action. Allowing Q to vary tests fixed-point consistency; it does not identify amplitude with mass. Mass remains the selected frequency in equivalent units.

For the electron and muon to support the same reference shape a=0, stationarity at fixed respective norms would require

$$\begin{aligned} \nabla E_{\rm geom}(0)+Q_e\nabla\omega_e(0)&=0,\\ \nabla E_{\rm geom}(0)+Q_\mu\nabla\omega_\mu(0)&=0. \end{aligned} \tag{15}$$

Consequently, their frequency gradients must be parallel. This necessary condition is independent of the stiffness or higher derivatives of E_geom at that same shape.

Take a_0 and a_1 from W=z^6+a_0+a_1 z. Both change the shape; neither is the common translation considered in (13). The complete projected frequency derivatives at gamma=10000 are

Their ratios of second to first components are approximately -0.00207777 and -0.00502687. Thus these two-dimensional gradients are not parallel. Choosing Q_mu/Q_e=1.0000278872 to satisfy the first source component leaves a second-component difference of about -18.13484 times Q_e. No nonzero scalar normalization choice can satisfy both components of (15). Linear independence also survives any common invertible change of shape coordinates.

The leading formula (12) explains why this is not just a numerical coincidence. At large gamma the two components are

$$\nabla_{(a_0,a_1)}\omega_\beta =\bigl(\gamma A[1+O(\gamma^{-2})], -B\beta[1+O(\gamma^{-2})]\bigr), \quad A=0.6149315907,\quad B=0.8243156254. \tag{16}$$

For distinct fixed beta values, the determinant of the two gradient rows has leading term -gamma A B (beta_mu-beta_e), which is nonzero. The obstruction persists for all sufficiently large gamma in the controlled regime. The numerical source derivatives were also computed independently from the full normal currents.

This conclusion is limited to the separable energy (14), the shared monomial reference texture, and its stated independent deformations. The premise of a common cavity radius within a depth does not require every background shape variable to be identical. A constrained shape, or an action in which W, h and the sustaining oscillation vary together, changes the source equations. Their additional terms would have to be derived from that action.

The next inverse calculation should therefore construct the joint same-field source, retaining its oscillation and its momentum and stress. Merely adding a common restoring potential to this prescribed carrier, or adjusting its wave norm alone, does not complete it.

Reproducible evidence

holomorphic-polynomial-variation-20260914T065239_077836Z.json passes 570 checks of all complex polynomial directions, complete first normal equations, projected polarization, source derivatives, and common translations. holomorphic-cofactor-response-20260914T065811_748663Z.json passes 183 checks of the independent positive Green formula, full-matrix high-precision determinant derivatives, and the controlled large-gamma polynomial response. The general degree and nonzero coefficient statements follow from (8) and (11), beyond the tested windings five, six and ten.

holomorphic-source-compatibility-20260914T074340_237342Z.json uses those independently computed kernel and current derivatives to verify the nonparallel source gradients at gamma=1000 and 10000. Its 16 checks also compare the stored complete normal boost results with the exact momentum identity in the inertia note. No new particle normalization or observed mass has been fitted in this test.

Related research

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