Reconstructing dynamics with exact simplex frequencies

Summary: An inverse polynomial classification gives nine local Hermitian dynamics with exact simplex frequencies on the specified six-dimensional Dirac pair, while canonical normalization fixes their action and the original unchanged kinetic term cannot realize that spectrum through a scalar polynomial alone.

Reconstruct the full frequency rather than only its coefficient

$$P(X)=\frac{X(X+4)(X+6)}{720},\qquad P\bigl(n(n+5)\bigr)=S(n,6). \tag{1}$$

The inverse problem here is stronger than asking for this scalar coefficient: require the complete positive frequency on that family to be $\mu S(n,6)$, with one fixed $\mu>0$.

There are explicit local Hermitian dynamics that satisfy that target. Within the two-polynomial class defined below, there are nine solutions at depth six up to a simultaneous sign of the two polynomials. The corresponding number at depth ten is eighty-one. They give the same frequencies while acting differently on the paired form components.

These are reconstructed quadratic actions. The calculation does not derive them from the algebraic master scalar current, select occupied particle modes or prove causal propagation. The positive field norm and the full complementary spinor kinetics are retained explicitly.

The constructions at the two depths are independent sector calculations. They do not require an electron state to be a restriction of a tau state; only the sectors belong to the nested spatial chain.

Preserve every form component of the spatial field

On physical $\mathbb{CP}^m$, use the specified bundle $\Lambda^{0,*}\otimes\mathcal O(m-1)$ and the spatial Dirac operator $\mathscr D=\sqrt2(\bar\partial+\bar\partial^\dagger)$. The dagger includes the physical FS metric and measure. Let $\Pi_q$ project onto exterior degree $q$, and let $\beta$ be exterior parity. Define

$$\begin{aligned} \mathscr D_0&=\sqrt2\left(\bar\partial^\dagger\Pi_1+ \bar\partial\Pi_0\right),\\ \mathscr D_c&=\mathscr D-\mathscr D_0,\qquad T_0=\frac R2\mathscr D_0,\qquad X_0=T_0^2. \end{aligned} \tag{2}$$

$$\mathscr D_0\mathscr D_c=\mathscr D_c\mathscr D_0=0, \qquad \{\beta,\mathscr D_0\}=\{\beta,\mathscr D_c\}=0. \tag{3}$$

On the complete zero/exact-one-form family, $X_0$ has values $x_n=n(n+2m-1)$. On the other form families it vanishes, while $\mathscr D_c$ retains their original kinetic equation. Harmonic components remain in the full field space.

The dimensionless target polynomial at real depth $2m$ is

$$P_m(X)=\frac{1}{(2m)!} \prod_{j=0}^{m-1}\left[X+j(2m-1-j)\right],\qquad P_m(x_n)=S(n,2m). \tag{4}$$

It has one zero at $X=0$ and distinct negative zeros at $-c_j$, where $c_j=j(2m-1-j)>0$ for $1\le j\le m-1$. No particle mode or dimensionality is reassigned.

What keeping the original kinetic term would require

First retain $\mathscr D_0$ and change only a real scalar polynomial $B(X_0)$. On its paired family,

$$H=\mathscr D_0+\beta B(X_0) \quad\Longrightarrow\quad \omega^2=\frac{4X}{R^2}+B(X)^2. \tag{5}$$

Exact target frequencies at every $x_n$ would require the polynomial identity

$$\boxed{B(X)^2=\mu^2P_m(X)^2-\frac{4X}{R^2}.} \tag{6}$$

Equality on infinitely many distinct $x_n$ forces equality as polynomials. At $X=0$, (6) gives $B(0)=0$. Its left side therefore has no linear term, while the right side has coefficient $-4/R^2$. No polynomial $B$ satisfies (6) at a finite physical radius.

A specified spectral square root can instead be defined on the positive paired spectrum:

$$B(x_n)=\sqrt{\mu^2P_m(x_n)^2-\frac{4x_n}{R^2}},\qquad B(0)=0. \tag{7}$$

For (7) to be real on every positive level, its first level imposes $\mu R\ge2\sqrt{2m}$. This condition is sufficient because $P_m(X)/\sqrt X$ increases for $X>0$. The resulting spectral operator is not a finite differential polynomial. Defining it supplies neither a local master interaction nor a causal reduction. Equation (6) identifies the constraint that a scalar-only inverse construction has to address.

A local polynomial class that meets the target exactly

Allow both coefficients on the paired family to change:

$$\boxed{ H_{a,b}=\mathscr D_c+ \mu\left[a(X_0)T_0+\beta b(X_0)\right], } \tag{8}$$

where $a,b$ are real dimensionless polynomials with $b(0)=0$. The identity $b(0)=0$ prevents an extra scalar term on the complementary form families. Using (3) gives

$$H_{a,b}^2=\mathscr D_c^2+ \mu^2\left[X_0a(X_0)^2+b(X_0)^2\right]. \tag{9}$$

Thus the exact inverse condition is

$$\boxed{Xa(X)^2+b(X)^2=P_m(X)^2.} \tag{10}$$

Every solution gives both frequencies $\pm\mu P_m(x_n)$ on the complete paired family. The other spinor families retain their original kinetic spectrum. Neither frequency sign is identified with an antiparticle or electric charge.

Both terms in (10) are nonnegative for $X\ge0$. Their leading coefficients cannot cancel. Consequently $\deg b=m$ and $\deg a\le m-1$. The constant and linear coefficients of (10) give, respectively,

$$b(0)=0,\qquad a(0)=0. \tag{11}$$

In particular none of these solutions retains the original constant coefficient $a=2/(\mu R)$ of $\mathscr D_0$ while changing only the scalar polynomial. The change to the kinetic operator is part of the reconstructed dynamics, rather than a normalization convention.

Classify all real polynomial solutions in this class

Introduce a mathematical polynomial variable $y$ and write

$$C(y)=b(y^2)+iy\,a(y^2). \tag{12}$$

For real coefficients, $C(-y)$ is the coefficient conjugate of $C(y)$. Equation (10) becomes

$$C(y)C(-y) =\frac{y^4}{[(2m)!]^2} \prod_{j=1}^{m-1}(y^2+c_j)^2. \tag{13}$$

The root at zero has multiplicity two in $C$. For each distinct pair $\pm i\sqrt{c_j}$, its two copies can be allocated in three ways: one to each root, or both to either root. Fixing the positive leading coefficient of $b$, all solutions are therefore

$$\boxed{ C(y)=\frac{y^2}{(2m)!}\prod_{j=1}^{m-1}F_{j,\epsilon_j}(y), \qquad \epsilon_j\in\{-1,0,1\}, } \tag{14}$$

with

$$F_{j,0}=y^2+c_j,\qquad F_{j,\pm1}=(y\pm i\sqrt{c_j})^2. \tag{15}$$

Every choice has the conjugation property in (12), so its even part is a real $b(y^2)$ and its odd part is $iy$ times a real $a(y^2)$. Conversely, the roots and leading coefficient in (13) leave no additional polynomial choices. A simultaneous sign change of $a,b$ gives the remaining solutions. The count up to that sign is

$$\boxed{3^{m-1}:\qquad 9\text{ at depth six},\quad 81\text{ at depth ten}.} \tag{16}$$

This classification is restricted to the class (8). It does not classify all matrix-valued, nonpolynomial or nonlinear dynamics.

At depth six, three illustrative members are:

For the second row, for example, (10) follows from $16X+(X-4)^2=(X+4)^2$. The two mixed examples have a nonzero odd-component coupling and an exact simplex frequency. The remaining members follow directly from (14), including opposite signs for each rotation.

Reconstruct the action with the correct field norm

Let $f$ be the full field coordinates in a fixed Hilbert space, and let $Z>0$ be a specified, time-independent kinetic metric. For a target self-adjoint Hamiltonian $H$, the real quadratic action

$$\mathcal S[f]=\int dt\left\{ \frac i2\left(\langle f,Z\dot f\rangle- \langle\dot f,Zf\rangle\right) -\langle f,Kf\rangle\right\}, \qquad K=Z^{1/2}HZ^{1/2}, \tag{17}$$

gives

$$iZ\dot f=Kf,\qquad u=Z^{1/2}f\quad\Longrightarrow\quad i\dot u=Hu. \tag{18}$$

All spatial adjoints and every complex variation are included. The conserved norm is $\langle f,Zf\rangle=\|u\|^2$. This construction works also for noncommuting $Z,H$; that general formula alone does not guarantee spatial locality of $K$. If $Z$ and $H$ commute, $K=ZH$. Polynomial commuting data in the present construction then give a local quadratic action.

$$\boxed{ Z(X)P(X)= \frac{X^5+22X^4+184X^3+688X^2+960X}{28\,800}. } \tag{19}$$

Using $Z(X_0)$ on the complete paired field, with value one on the complement, is a specified extension of that metric. Multiplying the complete target (8) by this $Z$ gives the corresponding local inverse action. Its scalar energy term has spatial order ten. The earlier $P/Z$ result used energy numerator $P$; (19) deliberately changes that energy to meet the inverse target. The algebraic master current has not thereby been shown to contain the additional derivatives.

A field-dependent or time-dependent $Z$ needs its complete variation and kinetic connection. Equations (17)–(18) assume fixed data; they cannot be used to omit the response of a moving geometric background. The canonical coupling calculation likewise normalizes each interaction leg using the same field metric.

Physical scope of the reconstruction

Each operator (8) is Hermitian on smooth spatial fields and has a self-adjoint realization defined by the complete harmonic decomposition. Its unitary linear evolution preserves the full canonical norm. Finite linear superpositions do not develop exponentially growing amplitudes under this fixed quadratic action.

That statement is not nonlinear particle stability. These equations permit every spectral level and do not select the assigned occupied states. They do not by themselves specify a nonlinear interaction between two particle structures. Locality as a differential expression also does not establish a finite propagation cone for a higher-order spatial equation.

The complete curved causality test now proves that every member of (8), used on all spatial modes, lacks a finite propagation speed. The test uses a local spatial commutator and retains both frequency branches. These actions therefore need a controlled effective interpretation or a different causal completion.

The inverse calculation supplies explicit actions that meet the target frequency identity and a complete classification within (8). Their physical coupling and state selection remain additional constraints. This approach is consistent with the distinction between an effective action and its deeper origin in Burgess, Section 3.3. No scale hierarchy or effective cutoff has been derived here.

Verification

inverse_simplex_dynamics_checks.py independently compares the real polynomial recurrence with all complex root allocations, exact coefficient identities and Hermitian matrix diagonalization. The six-dimensional and ten-dimensional spectra are checked at the unchanged assigned indices.

Repeated application of the actual curved zero-form and one-form differential operators checks the reconstructed action on spatial Jacobi fields. Independent curved quadrature retains their complete kinetic norms. Full complex Dolbeault complexes verify preservation of the complementary spinor equation. Noncommuting positive kinetic metrics, direct energy variations and norm conservation independently check the inverse action.