Derivative constraints and coherent mass propagation

Summary: Restricting the homogeneous coherent action to normalized derivatives of one spinor preserves its massive plane waves but gives an exactly nonlocal linear response, so this specified lift does not preserve finite propagation.

The carrier must have a physical realization

The homogeneous coherent action has exact simplex mass branches and finite propagation when its components are independent canonical fields in the specified spatial mode bundle. A different proposal identifies those components with derivatives of one smaller spinor. That proposal must retain both their compatibility conditions and their induced kinetic norm.

Here this identification is tested in a flat constant-coefficient control on ordinary physical spatial coordinates. The calculation does not choose a new geometry for the electron or compactify any directions. It excludes this particular derivative realization, not all reductions of the master field on the curved physical sector.

A complete, normalized derivative lift

For $|\alpha|\le N$, define the coherent-normalized derivatives

$$(T_N\psi)_\alpha =\sqrt{\frac{N!}{(N-|\alpha|)!\,\alpha!}} \ell^{|\alpha|}\partial^\alpha\psi, \tag{1}$$

where $\ell$ is a fixed physical normalization length. The coefficients are the same normalized cumulative occupations used in the local-current derivative calculation. They are not independent functions: differentiating a lower component fixes the appropriate higher one.

On a Fourier component of wavevector $k$, the complete coefficient vector is the unnormalized coherent tensor generated by $(1,i\ell k)$. Summing every occupation gives

$$\boxed{Z_N(k)=T_N(k)^\dagger T_N(k) =(1+\ell^2|k|^2)^N.} \tag{2}$$

Indeed, the degree-$j$ contribution is $\binom Nj(\ell^2|k|^2)^j$, by the multinomial theorem. When the lifted field has the canonical time derivative, the parent field's time metric is $Z_N$, not the identity.

The free constant-coefficient Dirac operator commutes with this lift. Restricting the complete action, rather than just substituting the source in one component, gives

$$iZ_N\partial_t\psi =Z_N\mathscr D\psi+T_N^\dagger G_p[T_N\psi]. \tag{3}$$

This is a specified new action for the parent field. It is distinct from prolonging the original master equation by its product rule.

Exact massive plane waves survive

For a plane wave $\psi=\chi e^{i(k\cdot x-\omega t)}$, the lifted field is pointwise coherent. If $\chi^\dagger\beta\chi>0$, the complete source is $G_p[T_N\psi]=m_N\beta T_N\psi$, with $m_N=\mu D_N$. Therefore (3) reduces exactly to

$$(\alpha\cdot k+m_N\beta)\chi=\omega\chi, \qquad \omega=\sqrt{|k|^2+m_N^2} \tag{4}$$

on its positive-energy branch. The induced norm cancels consistently in both the equation and $E/Q=\omega$. These are plane-wave controls, not localized particle solutions.

The complete perturbation response, however, contains information that the dispersion relation (4) does not test.

Compress the full Hessian with the same kinetic norm

Take the constant coherent background in the slack occupation, with unit positive-chirality spinor $\chi_0$. Put $P_\chi=\chi_0\chi_0^\dagger$, $s=\ell^2|k|^2$ and

$$\begin{aligned} r_j&=\frac{(N)_{\underline j}}{(pN)_{\underline j}},\\ a_{N,p}(s)&=\frac{\displaystyle\sum_{j=0}^N\binom Nj r_js^j} {(1+s)^N},\\ b_{N,p}(s)&=(p-1)\left[a_{N,p}(s)-(1+s)^{-N}\right]. \end{aligned} \tag{5}$$

The full coherent Hessian of the independent action is complex-linear. Its degree-zero spin matrix is $m_N\beta$; its degree-$j>0$ matrix is $m_Nr_j[\beta+(p-1)P_\chi]$. Multiplying each by its contribution to (2) proves the complete canonical compression

$$\boxed{ Z_N^{-1}T_N^\dagger A_pT_N =m_N[a_{N,p}(s)\beta+b_{N,p}(s)P_\chi]. } \tag{6}$$

The rotating-frame generator adds $\alpha\cdot k-m_NI$ to (6). It is Hermitian at each real $k$. Thus the calculation does not uncover an exponentially unstable Fourier mode. Its obstruction concerns spatial locality and finite propagation.

For $N=1$, the response along the occupied spin line happens to have $a+b=1$. This cancellation does not hold in all the other spinor polarizations. Retaining them is essential to the test.

The denominator cannot cancel

Let $P_{N,p}(s)$ be the numerator of $a_{N,p}$. The finite Vandermonde identity gives

$$\boxed{ \begin{aligned} P_{N,p}(-1) &=\sum_{j=0}^N(-1)^j\binom Nj \frac{(N)_{\underline j}}{(pN)_{\underline j}}\\ &=\frac{[(p-1)N]!^2}{[(p-2)N]!\,(pN)!}>0. \end{aligned} } \tag{7}$$

One elementary proof draws $N$ objects without replacement from $pN$ objects, of which $N$ are marked. The sum is inclusion-exclusion for drawing no marked objects. The direct probability is $\binom{(p-1)N}{N}/\binom{pN}{N}$, equal to the last expression. This supplies an independent combinatorial derivation of the nonzero remainder.

Along a single physical Fourier direction, $a_{N,p}(\ell^2k^2)$ therefore has poles of order $N$ at $k=\pm i/\ell$. It is a nonpolynomial rational multiplier. A translation-invariant local generator has a convolution kernel supported at the origin, hence a polynomial Fourier symbol. Equation (7) proves that this compressed response is not local.

For a concrete spin channel in six dimensions, use the exterior-form Clifford representation and choose a positive two-form polarization orthogonal to the zero-form $\chi_0$. Its partner under the first Clifford matrix is a negative one-form, also orthogonal to $\chi_0$. This two-state channel is preserved by that one-direction Dirac symbol; $P_\chi$ vanishes on it. The nonlocal coefficient $a_{N,p}$ cannot be removed by the occupied-spin term in (6).

An explicit instantaneous tail

Already at $N=1$,

$$a_{1,p}(\ell^2k^2) =\frac1p+\frac{1-1/p}{1+\ell^2k^2}. \tag{8}$$

In one physical direction the second term has convolution kernel

$$K(x)=\frac{1-1/p}{2\ell}e^{-|x|/\ell}. \tag{9}$$

For any nonnegative nonzero smooth perturbation supported in $[-1,1]$, the convolution $K*f$ is strictly positive for $x>1$. The local Dirac derivative and local mass terms vanish there initially, while (9) gives a nonzero initial time derivative. The same rational symbol in $d$ physical coordinates is a nonlocal Helmholtz resolvent and has the same support obstruction.

Finite-speed differentiable evolution would require its generator to preserve initial support: outside that support, the solution is zero for a nonzero interval of time. Its derivative at time zero must then also vanish there. Equations (7)–(9) violate this necessary condition. The pole proof applies to every finite even $p\ge2$ and every $N\ge1$, including the unchanged degrees $12$ and $34$.

The nonlocality arises because the independent nonlinear source does not preserve the derivative constraints. Restricting its action projects the response back to the allowed derivatives through $Z_N^{-1}T_N^\dagger$. That spatial projection is not an instantaneous local operation.

Why canonical independent jets impose a strong restriction

There is a general local explanation. Consider a first-derivative field $(f,g)$ with $g=\ell\partial_xf$, in a fixed flat canonical metric. Suppose an algebraic source $(F_0,F_1)$ both comes from a real local potential and preserves this constraint for every smooth $f$. Then

$$F_1=\ell\partial_xF_0 =(D_fF_0)g+\ell(D_gF_0)\partial_xg. \tag{10}$$

The second derivative $\partial_xg$ is arbitrary at a point. Since the source is algebraic in $(f,g)$, (10) forces $D_gF_0=0$ and $F_1=DF_0(f)g$. Work in real and imaginary components. A real potential has a symmetric real Hessian. Its upper off-diagonal block is zero, so the lower one must also vanish:

$$D_fF_1=(D^2F_0)(f)[\,\cdot\,,g]=0 \quad\hbox{for every }g. \tag{11}$$

Thus $F_0$ is affine. With phase covariance its constant term and conjugate-linear part vanish; it is one fixed Hermitian complex-linear map, repeated on the derivative components. A nonlinear local gradient on independent canonical jets generally fails this compatibility test.

This argument has stated limits: it assumes an algebraic source, a fixed flat canonical fibre metric and exact first-derivative compatibility. A dynamical complementary field, a different induced kinetic structure, or a frequency-dependent elimination is not covered by it. Such ingredients must be derived from modes of the same master field; they are not mediator particles or unspecified extra physical dimensions.

Consequence for the mass investigation

The homogeneous action remains a valid conditional construction on its independent kinetic bundle. Its interpretation as the particular normalized derivative lift (1) is excluded as an exact finite-speed law by the full response, despite the plane-wave success (4).

A viable master reduction must therefore specify how the carrier is realized, retain its actual norm and complementary response, and pass a propagation test on perturbations. Equality of carrier dimensions or an exact rest-frequency sequence does not supply those missing steps.

Verification

coherent_jet_causality_checks.py constructs every small-band derivative component, compresses the complete spinor Hessian with its induced norm, and measures the omitted response. It also verifies massive plane waves, exact rational pole coefficients through degree $34$, and the nonzero convolution outside compact initial support. The analytic support argument, rather than a finite numerical cutoff, establishes the failure of finite propagation for this lift.