Self-mass from a hierarchy of currents

Summary: One canonically normalized nonlinear action generates a simplex self-source and a common lower-order probe response; its shell weights and physical carrier remain inverse-model premises.

A self-interaction that also determines coupling

The target is a single field law that gives different masses and similar coupling responses within a spatial depth. The construction below generates its self-source from the same nonlinear interaction used for a second field pattern. It contains no separately prescribed quadratic mass term. Its weighting is reconstructed from the desired simplex rule, however; it has not been derived from the existing master scalar interaction.

The fixed-order current construction supplies the exact contractions used here. Retaining the whole hierarchy allows those currents to generate the isolated self-source as well.

Let $Q_d$ be an actual $d$-dimensional spatial sector and use the bundle

$$W=\mathbf1\oplus(T^*Q_d\otimes\mathbb C),\qquad E_N=\operatorname{Sym}^N W,\qquad D_N=\binom{N+d}{d},\qquad N=n-1. \tag{1}$$

The first summand homogenizes the cumulative oscillator occupations. The other $d$ directions describe ordinary spatial tangent components. Tensor coefficients and their projective rays add no physical dimensions. Only the spatial sectors are nested; the particles are not nested objects.

For a finite set of occupied grades, declare canonical spinor fields $\Psi_N\in\mathcal S_d\otimes E_N$ with action

$$S=\int dt\,dV\left[ \sum_N\frac i2\bigl(\Psi_N^\dagger\dot\Psi_N -\dot\Psi_N^\dagger\Psi_N\bigr) -\sum_N\Psi_N^\dagger\mathscr D_N\Psi_N-U\right]. \tag{2}$$

Here $\mathscr D_N$ is the full covariant spatial Dirac operator, including the coefficient-bundle connection. This independent canonical grade decomposition needs an origin in $\Psi_\infty$. It is not implied by an equality of representation dimensions or by sector inclusion. An infinite grade sum would additionally require a domain and convergence estimates.

Exact reduced currents

Splitting $k$ symmetric indices from the remaining $N-k$ gives the natural isometry $T_{Nk}:E_N\longrightarrow E_k\otimes E_{N-k}$. In normalized occupation bases its only nonzero coefficient is

$$(T_{Nk})_{\alpha,\gamma;\alpha+\gamma} =\sqrt{\frac{\binom k\alpha\binom{N-k}\gamma} {\binom N{\alpha+\gamma}}}. \tag{3}$$

Let $\beta$ be the Hermitian scalar insertion of the spatial Dirac equation. Trace the spectator indices and contract the complete spinor indices with $\beta$, defining a Hermitian matrix $\rho_{N,k}$ on $E_k$. For example, with $Y=T_{Nk}\Psi_N$,

$$(\rho_{N,k})_{ab} =\sum_{q,s,t}\overline{Y_{t,bq}}\,\beta_{ts}Y_{s,aq}, \qquad \operatorname{Tr}\rho_{N,k} =J_N=\Psi_N^\dagger\beta\Psi_N. \tag{4}$$

At $k=0$, the current is the scalar $J_N$. These currents can have either sign; no spinor component has been removed to make them positive. Their adjoint insertion is

$$B_{Nk}(H)=T_{Nk}^\dagger(H\otimes I)T_{Nk},\qquad B_{Nk}(I)=I. \tag{5}$$

For a coherent coefficient field $\Psi_N=\chi\otimes z^{\otimes N}$, $\|z\|=1$, every contraction has the particularly simple form

$$\rho_{N,k}=J\,P_{z,k},\qquad J=\chi^\dagger\beta\chi,\qquad P_{z,k}=|z^{\otimes k}\rangle\langle z^{\otimes k}|. \tag{6}$$

This identity is about a full spinor field at a physical point. It does not assert that an arbitrary spatially varying coherent field solves the Dirac equation.

The common action and the Hockey-Stick sum

Combine the grades before varying the action:

$$R_k=\sum_{N\ge k}\rho_{N,k},\qquad h_{k,d}=\binom{k+d-1}{d-1},\qquad \boxed{U=\sum_{k\ge0}h_{k,d}\,\Phi(\|R_k\|_{\rm HS}).} \tag{7}$$

The same weights and scalar function $\Phi$ apply to every grade at this depth. For an isolated coherent field, $\|R_k\|_{\rm HS}=|J|$, and the complete source is

$$\boxed{\frac{\delta U}{\delta\overline\Psi_N} =D_N F(J)\,\beta\Psi_N, \qquad F(J)=\operatorname{sgn}(J)\Phi'(|J|).} \tag{8}$$

Indeed, the matrix derivative in each channel is proportional to $P_{z,k}$, and $B_{Nk}(P_{z,k})z^{\otimes N}=z^{\otimes N}$. The coefficient is therefore

$$\sum_{k=0}^N h_{k,d}=\binom{N+d}{d}=D_N. \tag{9}$$

Conversely, within the class of common weighted current sums, demanding (9) at every $N$ forces $h_{N,d}=D_N-D_{N-1}$, with $D_{-1}=0$. This uniqueness is an inverse statement. The counting identity does not show that the master dynamics assigns these weights to its interaction channels. The projection calculation derives the different weights actually supplied by a smooth coherent scalar projection and keeps its kinetic normalization explicit.

Three self-interaction laws

The polynomial choice $\Phi(r)=\kappa r^2/2$ gives

$$M_N(J)=\kappa D_NJ,\qquad U_N=\frac\kappa2D_NJ^2. \tag{10}$$

Its mass-like source is the simplex count times a self-current scale. That scale remains continuous unless the spatial dynamics selects $J$. At fixed canonical density, changing the spinor's scalar balance changes the self-source without necessarily making its other response directions small.

The homogeneous root $\Phi(r)=\mu r$, with $\mu>0$, instead gives

$$M_N=\mu D_N\quad(J>0),\qquad U_N=\mu D_NJ. \tag{11}$$

The complete law is nonsmooth at a zero matrix current. In particular, nonzero scalar-null spinors require attention; the positive-current formula is not a globally smooth full-Dirac action.

Other homogeneous coherent actions retain the same occupied mass while giving different complete responses. The chosen self-mass sequence does not uniquely specify the interaction.

A smooth comparison uses one positive density scale $\sigma$:

$$\begin{aligned} \Phi_\sigma(r)&=\mu\bigl(\sqrt{r^2+\sigma^2}-\sigma\bigr),\\ \boxed{M_N(J)}&=\boxed{\mu D_N\frac{J}{\sqrt{J^2+\sigma^2}}},\\ U_N&=\mu D_N\bigl(\sqrt{J^2+\sigma^2}-\sigma\bigr). \end{aligned} \tag{12}$$

The subtraction fixes the vacuum energy in the same action. It is not adjusted separately for different branches. With canonical density units, $\sigma$ has units of $J$, $\mu$ has frequency units, and $\kappa$ in (10) has frequency divided by density units.

For a zero-kinetic positive-$\beta$ spatial field of uniform density $\rho=Q/\mathcal V$, the total energy and phase frequency are

$$E_N=\mathcal V D_N\Phi(\rho),\qquad \omega_N=D_N\Phi'(\rho)=\frac{dE_N}{dQ}. \tag{13}$$

Both ratios are $D_M/D_N$ at a common physical norm and volume, but energy and frequency are different quantities except at the homogeneous root. The norm $Q$ has not been identified with electric charge or particle number. Physical radius, norm selection and the rest-energy interpretation remain part of the mass problem.

The fixed depth-six assignments are $N=12,34$, with $D_{12}=18564$, $D_{34}=3838380$ and ratio $3515/17$. The same algebra applies at the assigned depth-five and depth-ten grades; it does not select a common scale between different physical depths.

Full coherent response

Choose a local mode frame with $z=e_0$. Let $j$ count occupations outside that ray, and put $D_j=\binom{j+d}{d}$. Two exact contraction sums are

$$\begin{aligned} \sum_{k=0}^N h_{k,d} \frac{\binom{N-j}{k}}{\binom Nk}&=\frac{D_N}{D_j},\\ \sum_{k=0}^N h_{k,d} \frac{\binom kj}{\binom Nj}&=D_N\frac d{d+j}. \end{aligned} \tag{14}$$

Unavailable binomials are zero. The first is the falling-factorial moment of one occupation in a uniform composition of $N$ into $d+1$ slots. For the second, combine $h_{k,d}\binom kj$ into a constant times $\binom{k+d-1}{d+j-1}$ and apply Hockey-Stick again.

For (10), a complete perturbation $\eta_j$ has source derivative

$$\begin{aligned} \delta\mathcal N_0 &=\kappa D_N\left[J\beta\eta_0 +2\operatorname{Re}(\chi^\dagger\beta\eta_0)\beta\chi\right],\\ \delta\mathcal N_j &=\kappa D_N\left[\frac J{D_j}\beta +\frac d{d+j}\beta\chi\chi^\dagger\beta\right]\eta_j, \qquad j\ge1. \end{aligned} \tag{15}$$

Thus the occupied block retains a conjugate variation. The transverse blocks are complex-linear and Hermitian. For (12), replace $\kappa$ by $\mu/\sqrt{J^2+\sigma^2}$, and multiply the second term of the occupied block by $\sigma^2/(J^2+\sigma^2)$.

At the positive root endpoint $\sigma=0$, that occupied term cancels exactly. Every coherent Hessian block is then complex-linear and Hermitian. On a smooth nonvanishing coherent spatial solution, the complete linearized Dirac evolution therefore conserves its perturbation norm under the stated self-adjoint boundary conditions. This includes spatial and spinor perturbations outside the occupied ray; it is not a nonlinear stability theorem or a statement at scalar-null fields.

Coupling and the remaining operator question

Because the grades enter the combined currents $R_k$ in (7), the action also fixes their reciprocal interaction. A fixed probe of degree $M$ sees an identical coherent target current through every order it can access whenever $N\ge M$. The complete probe calculation states the required density and ray conditions, the full target response, and the root law's delicate vacuum limit.

This supplies a joint inverse model for mass and coupling. It still needs the master field to produce the canonical carrier, the shell weighting, the scalar law and the selected spatial states. Merely counting independent copies of a field does not do so: those copies contribute to the kinetic norm as well as to the potential.

The full-field localization condition now excludes flat finite-norm stationary fields in the convex hierarchy's positive-current cone under its stated virial hypotheses. A fixed split into positive and negative insertion channels has the same obstruction. The calculation retains spatially changing sign projectors explicitly, giving a necessary geometric term for a possible textured solution.

Verification

The later localized current construction retains the hierarchy while adding matrix partners and a local binding interaction. It supplies actual localized spatial solutions, their complete boost stress and an exact frequency-proportional canonical inertia at a common norm. This is a different reconstructed action: it does not remove the positivity obstruction for the action studied above, nor derive the shell weights or physical mode carrier from the master field.

An explicit complementary Dirac reduction now derives a spectral square-root source from specified canonical normal occupations and checks its complete response. Its actual scalar projection counts the populated normal band; restoring both self-terms identifies the remaining band matching, energy and occupation questions.

graded_current_tower_checks.py checks the exact sums, complete complex gradients, both reciprocal responses, spinor and spatial-frame covariance, the fixed assignments, and scalar-null controls. graded_current_actions_checks.py checks the root and smooth variations, every coherent response block, and arbitrary full-spinor weak probes. The executable laws are in graded_current_tower.py and graded_current_actions.py.