Causal constraint on reconstructed simplex dynamics

Summary: An exact commutator on the actual curved spatial geometry proves that the reconstructed all-band polynomial dynamics have no finite propagation speed, while leaving open causal nonlinear dynamics with selected simplex particle frequencies.

What the inverse frequency match does not settle

The inverse construction gives local Hermitian equations with exact frequencies $\pm\mu S(n,2m)$ on their complete first Dirac pair. Their full canonical norm is conserved. However, every member of that polynomial class fails a finite-propagation test when used as a fixed equation for all spatial modes.

This conclusion follows directly on the specified physical $\mathbb{CP}^m$ geometry with line connection $\mathcal O(m-1)$. It does not require a flat approximation, a positive-frequency projection or a group-velocity interpretation. The physical radius $R$ is arbitrary and finite. These remain ordinary macroscopic spatial directions; no direction is compactified.

The result constrains an inverse dynamical ansatz. A nonlinear particle mass is not required to equal the frequency of every small disturbance of one fixed quadratic background. Causal dynamics that select several nonlinear particle states remain a different problem. No particle depth or assigned mass index is changed by this test.

Finite propagation requires a bounded local commutator

Let $H$ be self-adjoint on the complete spatial field and $U(\tau)=e^{-i\tau H}$ its unitary evolution. A finite propagation speed $c$ means that every initial field supported in $K$ satisfies

$$\operatorname{supp}U(\tau)\psi \subseteq\{x:\operatorname{dist}(x,K)\le c|\tau|\}. \tag{1}$$

For an ordinary first-order Dirac operator, its bounded commutator with a Lipschitz scalar function expresses its local spatial speed. See McIntosh and Morris, Section 1 for the first-order propagation theorem. The necessary estimate used here can be proved without assuming that $H$ is first order.

Let $f$ be a bounded real function with Lipschitz constant $L_f$. For $\tau\ne0$, put $r=c|\tau|$ and $\delta=L_fr$, and partition the field space using the multiplication projections

$$E_j=\mathbf1_{\{j\delta\le f<(j+1)\delta\}},\qquad f_\delta=\sum_j j\delta E_j. \tag{2}$$

Zero values of $c$ or $L_f$ follow by using an arbitrarily larger positive bound or by observing that a constant function commutes. Two bins whose indices differ by at least three are separated by more than $r$. Equation (1) therefore gives $E_jU(\tau)E_k=0$ when $|j-k|\ge3$.

For each remaining index difference $\ell$, the operator $\sum_jE_{j+\ell}U(\tau)E_j$ has norm at most one: its domain blocks are orthogonal, its range blocks are orthogonal and $\|U(\tau)\|=1$. It follows that

$$\begin{aligned} \|[f_\delta,U(\tau)]\| &\le\delta\sum_{\ell=-2}^{2}|\ell|=6\delta,\\ \|[f-f_\delta,U(\tau)]\|&\le2\delta. \end{aligned} \tag{3}$$

The deliberately nonoptimal constant is sufficient:

$$\boxed{\|[f,U(\tau)]\|\le8c|\tau|L_f.} \tag{4}$$

For a smooth field $\psi$ such that $\psi$ and $f\psi$ lie in the domain of $H$, differentiate at $\tau=0$. Equation (4) implies

$$\boxed{\|[H,f]\psi\|\le8cL_f\|\psi\|.} \tag{5}$$

Thus an unbounded commutator with one bounded smooth spatial function excludes a finite propagation speed for the full unitary equation. The argument applies on a compact spatial manifold as well: a finite diameter does not remove the small-time requirement.

Evaluate that commutator on the actual spatial field

Retain the complete inverse operator

$$H_{a,b}=\mathscr D_c+ \mu\left[a(X_0)T_0+\beta b(X_0)\right],\qquad X_0=T_0^2,\quad T_0=\frac R2\mathscr D_0. \tag{6}$$

The other spinor families still have their operator $\mathscr D_c$. All inverse candidates have

$$Xa(X)^2+b(X)^2=P_m(X)^2,\qquad b(X)=\frac{X^m}{(2m)!}+O(X^{m-1}). \tag{7}$$

The overall negative sign allowed in the classification changes the sign of the following commutator but not its norm.

Use the global spatial function $t=|Z_0|^2/|Z|^2$. For the specified physical FS metric,

$$|\nabla t|^2=\frac{4t(1-t)}{R^2},\qquad L_t=\frac1R. \tag{8}$$

It is a legitimate smooth state of the complete field. It need not be a positive-frequency eigenstate of (6). Multiplication by $t$ has the exact recurrence

$$\begin{aligned} t\phi_n&=\gamma_n\phi_{n+1}+\frac12\phi_n+ \gamma_{n-1}\phi_{n-1},\\ \gamma_n&=\frac12 \sqrt{\frac{(n+1)(n+2m-1)} {(2n+2m-1)(2n+2m+1)}}. \end{aligned} \tag{9}$$

The last term is absent at $n=0$. The coordinate is an actual function on physical space; the index $n$ is a mathematical spatial mode label.

On zero forms, $\beta=1$ and $X_0\phi_n=x_n\phi_n$, with $x_n=n(n+2m-1)$. The term $a(X_0)T_0$ changes form degree and $\mathscr D_c$ annihilates both $\phi_n$ and $t\phi_n$. Consequently the zero-form component of the full commutator is

$$\begin{aligned} \Pi_0[H_{a,b},t]\phi_n =\mu\bigl\{& \gamma_n[b(x_{n+1})-b(x_n)]\phi_{n+1}\\ &+\gamma_{n-1}[b(x_{n-1})-b(x_n)]\phi_{n-1}\bigr\}. \end{aligned} \tag{10}$$

This projection only supplies a lower bound on the complete response. It is the local form-degree projection, not a projection onto one frequency sign. The odd component and all complementary equations remain present and cannot cancel an orthogonal zero-form component.

Define

$$\begin{aligned} C_n^2={}&\gamma_n^2[b(x_{n+1})-b(x_n)]^2\\ &+\gamma_{n-1}^2[b(x_{n-1})-b(x_n)]^2. \end{aligned} \tag{11}$$

Equations (9)–(10) imply

$$\|[H_{a,b},t]\phi_n\|\ge\mu C_n. \tag{12}$$

For every root allocation in the inverse class,

$$\gamma_n\longrightarrow\frac14,\qquad b(x_{n+1})-b(x_n)= \frac{n^{2m-1}}{(2m-1)!}+O(n^{2m-2}). \tag{13}$$

The other difference has the opposite leading sign. Hence

$$\boxed{C_n\sim \frac{n^{2m-1}}{\sqrt8\,(2m-1)!}\longrightarrow\infty.} \tag{14}$$

But a finite propagation speed would require, by (5) and (8),

$$\mu C_n\le\frac{8c}{R} \quad\text{for every }n. \tag{15}$$

This is impossible for any fixed nonzero $\mu$ and finite positive $R,c$. It proves the failure of finite propagation on the full curved equation, including all nine coefficient pairs at depth six and all eighty-one at depth ten.

The asymptotic indices in this proof are test fields allowed by the proposed equation. They are not alternative particle assignments.

What can and cannot change this conclusion

The original first-order operator has $[\mathscr D,t]$ equal to local Clifford multiplication by $dt$, up to the convention for $i$. Its norm is bounded by $1/R$. This provides a direct curved-space control for the test above.

The same obstruction also applies to the direct insertion $H=\mathscr D+\mu\beta P_m(X_0)$, even though that insertion keeps the original first-order kinetic term. On the test field $\phi_n$, its zero-form commutator is still (10) with $b=P_m$; the ordinary Dirac commutator contributes only to the odd form. Thus promoting the polynomial mass coefficient to an operator on every spatial disturbance needs a causal completion as well. A coefficient obtained only on selected nonlinear stationary particle states is not covered by that promotion.

A fixed smooth multiplication potential cannot cancel (10); it commutes with $t$. In particular, adding the specified local scalar term $g(\psi^\dagger\beta\psi)\beta\psi$ has vanishing linearization at the zero field. Its small-amplitude equation still has the same obstruction. For a smooth local nonlinearity $F(\psi^\dagger\beta\psi)\beta\psi$, a nonzero $F(0)$ adds only a multiplication term to that linearization.

Likewise, a fixed positive commuting kinetic metric in the inverse action gives $iZ\dot f=ZH_{a,b}f$ and the same evolution $i\dot f=H_{a,b}f$. That normalization does not repair (15). A different same-field dynamical completion would need to change the propagation equation itself.

A controlled effective reduction may retain the polynomial over a limited regime, provided its causal parent, retained modes, frequency range and approximation error are derived. A cutoff alone is not a proof of causal support. The local Dirac retarded response already illustrates how causal propagation can survive a same-field reduction when its full frequency dependence is kept. The instantaneous interaction test addresses a different source of acausality: an explicitly nonlocal instantaneous spatial potential.

Most importantly, (14) does not exclude a universal causal nonlinear equation whose distinct self-consistent particle solutions have selected simplex masses. That equation would have to generate the particle spectrum through its nonlinear spatial states and normalization, without imposing this unrestricted high-order dispersion on every disturbance.

The inverse scalar-law construction develops that alternative with the original Dirac kinetics. It tests the full spatial residual and the requirement that all particle candidates obey one function of the scalar current.

Verification

inverse_simplex_causality_checks.py checks exact finite differences for every root allocation at depths six and ten. Independent Jacobi polynomials and repeated application of the actual curved differential operator verify the full formula (10).

Physical curved quadrature and its refinement independently check the normalized recurrence. The paired matrix equation retains both frequency branches and verifies the complete norm lower bound. The original Dirac operator supplies the bounded local commutator control. Finite block contractions also check the range-partition estimate; the continuum proof is (2)–(5), not a conclusion drawn from those finite numerical examples.