Homogeneous actions with coherent simplex masses

Summary: A specified family of local homogeneous actions gives exact coherent simplex frequencies, canonical energies and full spatial linear stability, while its normalization, master-field carrier and physical selection remain assumptions.

A constructive inverse result

There is an action-level route to the coherent mass multiplier that does not require a rank-one mass endomorphism. Start with the existing quartic coherent-current integral, take its square root, and supply the explicit product-rank normalization. The resulting nonlinear Dirac action has exact nonconstant coherent solutions on the specified physical spatial geometry. Their frequency and energy per canonical norm are both $\mu D_N$.

This is an inverse construction: the action is specified to meet the mass condition, and its consequences are then derived. The original master interaction has not been shown to produce either this functional or its canonical mode-bundle field. Those are substantive remaining assumptions.

Geometry, currents and the proposed action

At physical spatial depth $d$, use

$$W=\mathbf1\oplus(T^*Q_d\otimes\mathbb C),\qquad E_N=\operatorname{Sym}^N W,\qquad D_N=\binom{N+d}{d},\qquad N=n-1\ge1. \tag{1}$$

The projective rays of $W$ label local mode data over the physical spatial point. They add no spatial dimensions. The $d$ directions of $Q_d$ remain ordinary macroscopic spatial directions. The coherent localization density is

$$v_z=z^{\odot N},\qquad \Lambda_z=D_N|v_z\rangle\langle v_z|,\qquad \int\Lambda_z\,d\nu(z)=I, \tag{2}$$

with $\|z\|=1$ and unit-volume invariant ray measure. For the full field $\Psi\in\Sigma\otimes E_N$, define

$$j_z=\Psi^\dagger(\beta\otimes\Lambda_z)\Psi, \qquad K_p(\Psi)=\int j_z^p\,d\nu(z). \tag{3}$$

Here $\beta$ is the Hermitian scalar-current matrix. Take an even integer $p\ge2$ and specify the potential density

$$\boxed{U_{N,p}(\Psi)=\mu\,[D_{pN}K_p(\Psi)]^{1/p},\qquad \mu>0.} \tag{4}$$

The simplest case is $U_{N,2}=\mu\sqrt{D_{2N}\int j_z^2d\nu}$. It is smooth wherever $K_p>0$. Its spatial action is

$$S=\int dt\left[ \frac i2\int(\Psi^\dagger\dot\Psi-\dot\Psi^\dagger\Psi)dV -\langle\Psi,\mathscr D_N\Psi\rangle-\int U_{N,p}(\Psi)dV \right]. \tag{5}$$

The canonical kinetic metric in (5) is part of the candidate. In particular, it cannot silently be assigned to a collection of constrained derivatives of one smaller spinor. The derivative-lift test below makes that distinction explicit. The coefficient $\mu$ has dimensions of energy in units $\hbar=c=1$; (4) is quadratic in field amplitude.

Full variation and the occupied eigenvalue

Vary every field component in (4), including the current inside its normalizing factor. Put

$$\begin{aligned} \mathcal M_p[\Psi] &=\mu D_{pN}^{1/p}K_p^{1/p-1} \int j_z^{p-1}\Lambda_z\,d\nu(z),\\ G_p[\Psi]&=(\beta\otimes\mathcal M_p[\Psi])\Psi. \end{aligned} \tag{6}$$

Then $\delta U=2\operatorname{Re}\langle G_p,\delta\Psi\rangle$ and

$$i\partial_t\Psi=\mathscr D_N\Psi+G_p[\Psi]. \tag{7}$$

The endomorphism in (6) acts as a scalar in Clifford indices. It is Hermitian, and the action is covariant under the specified unitary mode frames. A complete spacetime-covariant master embedding of those spatial mode components is still required; a spin-scalar contraction alone does not provide that embedding.

At a coherent field $\Psi_*=\chi\otimes v_{z_0}$, let $J=\chi^\dagger\beta\chi>0$ and $u=|z^\dagger z_0|^2$. The invariant ray measure gives

$$\int u^{pN}d\nu =d\int_0^1u^{pN}(1-u)^{d-1}du =\frac{d!\,(pN)!}{(pN+d)!}=\frac1{D_{pN}}. \tag{8}$$

Thus $K_p=D_N^pJ^p/D_{pN}$ and $U_{N,p}(\Psi_*)=\mu D_NJ$. To check the other equations, decompose the mode carrier by the number $j$ of factors orthogonal to $z_0$, with projectors $P_j$. Their ranks are $\binom{j+d-1}{d-1}$. Integrating the normalized occupation monomials gives

$$r_j^{(p)}=\frac{(N)_{\underline j}}{(pN)_{\underline j}}, \qquad r_0^{(p)}=1,\qquad \boxed{\mathcal M_p[\Psi_*]=\mu D_N\sum_{j=0}^N r_j^{(p)}P_j.} \tag{9}$$

For example, the monomial with $j$ transverse occupations contributes $\binom Nj u^{N-j}(1-u)^j$ before averaging its transverse direction. The beta integral weighted by $u^{(p-1)N}$, divided by the occupied integral, gives $(N)_{\underline j}/(pN)_{\underline j}$. Consequently every occupied equation gives

$$\boxed{G_p[\Psi_*]=\mu D_N\beta\Psi_*.} \tag{10}$$

All $r_j^{(p)}$ are positive at finite $p$. The mass endomorphism therefore has full mode rank, although its occupied eigenvalue equals the one from rank-one coherent contact. An occupied mass sequence does not determine the complete endomorphism uniquely.

The complete response retains the normalization variation

The first derivative (10) does not license freezing the current or its normalization when calculating perturbations. At the coherent field the complete derivative has the especially simple form

$$\boxed{ \begin{aligned} \delta G_p&=A_ph,\qquad B_p=0,\\ A_p&=\mu D_N\left[ \beta\otimes P_0+ \sum_{j=1}^N r_j^{(p)} \left(\beta+(p-1)\frac{\beta\chi\chi^\dagger\beta}{J}\right) \otimes P_j\right]. \end{aligned}} \tag{11}$$

Here $B_p$ denotes the coefficient of $\overline h$ in the general real-linear response. Its cancellation in (11) is a result on the coherent background, not a property of the nonlinear action at arbitrary fields.

To see the cancellation, work in the frame $z_0=e_0$. Before varying $K_p^{1/p-1}$, the radial response contains $2(p-1)\beta\chi\operatorname{Re}(\chi^\dagger\beta h_0)/J$. Since

$$\frac{\delta K_p}{K_p} =\frac{2p}{J}\operatorname{Re}(\chi^\dagger\beta h_0), \tag{12}$$

the derivative of that factor cancels the radial term exactly. Transverse phase integration leaves one ordinary term with coefficient $(p-1)r_j^{(p)}$; a conjugate term would require two nonnegative transverse occupations to sum to zero and hence occurs only in the radial entry. This proves (11), including every spinor polarization.

Each matrix in (11) is Hermitian. The identity $p r_1^{(p)}=1$ also gives the required unchanged frequency along a moving coherent ray. For a positive-chirality spinor, the response within its occupied spin line in sector $j\ge1$, divided by $\mu D_N$, is $p r_j^{(p)}$. At $N=12$, its $j=2$ value is $11/23$, $11/47$ or $11/71$ for $p=2,4,6$, respectively. Their isolated coherent masses are identical.

Full spatial linear stability and propagation

Linearize (7) around (15) in its rotating frame. Equation (11) gives

$$i\partial_t h=(\mathscr D_N-\mu D_NI+A_p(q))h. \tag{17}$$

The chosen coherent section is nowhere zero, so $A_p(q)$ is a smooth Hermitian endomorphism. For this positive-chirality field its operator norm is at most $\mu D_N$: in each transverse sector $r_j^{(p)}\le1/p$. On the compact physical candidate without boundary, $\mathscr D_N$ is self-adjoint; adding this bounded Hermitian potential preserves self-adjointness. Therefore

$$\|h(t)\|_{L^2(dV)}=\|h(0)\|_{L^2(dV)}. \tag{18}$$

This includes all spatial modes and spinor components. It does not require the varying projectors $P_j(q)$ to be parallel. It establishes full linear stability, not nonlinear orbital stability of the spatial particle.

On a neighborhood where $K_p$ stays positive, (7) is a smooth semilinear Dirac system. Its source has no physical spatial derivatives, so the principal symbol remains Clifford multiplication. For two smooth solutions, their difference obeys the usual local density balance with a source bounded by a local Lipschitz constant times the squared difference. The Clifford flux through a unit normal is bounded by that density. Integrating over a light cone and applying Gronwall's inequality therefore gives finite propagation with the original Dirac speed, for as long as both solutions remain in that smooth domain. This argument is conditional on the independent kinetic carrier in (5).

The energy-frequency identity simplifies here: quadratic homogeneity gives $\operatorname{Re}\langle\Psi,G_p\rangle =U_{N,p}$ pointwise. Thus any sufficiently regular stationary solution of this action satisfies $E=\omega Q$. Its scale copies retain the same frequency. The action consequently does not select a distinguished norm.

A reciprocal coupling diagnostic

The same action predicts more than a mass. At one physical point retain two orthogonal positive-chirality spinor polarizations, occupied on the orthogonal mode rays $e_0$ and $e_1$, with scalar intensities $A,B>0$. They are two components of one field. The exact mixed ray moment is

$$t_N=\frac1{\binom{2N}{N}},\qquad U_{N,2}=\mu D_N\sqrt{A^2+B^2+2t_NAB}. \tag{19}$$

The factor follows by integrating $|z_0|^{2N}|z_1|^{2N}$ and dividing by the self moment $1/D_{2N}$. Both sources and their reciprocal derivative are fixed:

$$\begin{aligned} \partial_AU&=\mu D_N\frac{A+t_NB}{\sqrt{A^2+B^2+2t_NAB}},\\ \partial_A\partial_BU &=-\mu D_N\frac{(1-t_N^2)AB}{(A^2+B^2+2t_NAB)^{3/2}}. \end{aligned} \tag{20}$$

Consequently a very small mixed quartic overlap does not make the root-action response small. At $A=B$, the occupied coefficient relative to its isolated value is $\sqrt{(1+t_N)/2}$, tending to $1/\sqrt2$. This is a definite consequence of normalizing by the total local moment. Different moment orders give different collective responses despite (16).

This comparison concerns coincident polarized components. Fields with disjoint physical spatial support have additive energies because (5) is local in physical space. Neither comparison supplies a completed encounter between particles on different nested sectors. Their full spatial patterns, relative phases, kinetic response and common projected interaction remain necessary for an observed coupling prediction.

What the inverse construction still assumes

The normalization in (4) is essential. For the general choice $U=C_NK_p^{1/p}$, the coherent coefficient is $C_ND_N/D_{pN}^{1/p}$. Requiring $\mu D_N$ determines $C_N=\mu D_{pN}^{1/p}$. The product rank is an exact geometric integer; choosing this function of it in the physical action remains a premise. For example, a common unnormalized square root gives the assigned-band ratio about $11.706$, rather than $3515/17$.

The action also has a nontrivial singular set. The coherent symbols are informationally complete: $j_z=0$ for every $z$ is equivalent to $\Psi^\dagger\beta\Psi=0$ as a mode matrix. This follows by expanding the homogeneous polynomial in $z$ and $\overline z$. That condition can hold for a nonzero spinor. Along a coherent curve with scalar current $J=\epsilon$, (4) becomes $\mu D_N|\epsilon|$, which has unequal one-sided derivatives at zero. The construction provides no general continuation through that set and makes no antiparticle identification.

At the field vacuum $U=O(\|\Psi\|^2)$ has first derivative zero, but the nonquadratic homogeneous functional has no single quadratic Taylor expansion. A degree-two homogeneous function that is twice differentiable at zero must equal its quadratic Taylor form everywhere. Thus one cannot assume an ordinary smooth vacuum expansion for (4).

Finally, cumulative bands are not automatically independent physical branches. Their standard inclusions do not turn the coherent contact family into compressions of one bare operator. A master derivation must supply the carrier, its kinetic norm, the interpretation and selection of $N$, and the action itself. The conditional result here is an exact spatial mass construction with a derived response, not that missing master derivation.

Verification and related calculations

The combined-band calculation now puts the distinct occupied mass branches into one root action and varies both currents. Its mixed response remains strongly band dependent; isolated mass agreement does not establish a common coupling group. The geometric current construction instead retains fixed-order tensor currents. It preserves a prescribed simplex mass term and gives a common coherent elastic vertex, with explicit spatial normalization and reciprocal evolution tests. Its master origin and the mass operator itself remain assumptions.

The current hierarchy extends the inverse construction to one common action that generates both self-mass and the interaction between grades. Its root endpoint gives an exact common lower-order probe operator on coherent targets. The shell weighting and canonical master-field carrier still need a derivation; the result does not make those premises consequences of the homogeneous action above.

coherent_root_action_checks.py compares independent projective quadrature with exact symmetric-tensor contractions, differentiates arbitrary complex eight-component fields, checks the full coherent Hessian in rotated frames, and verifies spatial normalization and both sides of (20). Exact factorial formulas cover the assigned large bands without allocating their full matrices.

The peak and smooth comparison treats the limit of increasing moment order and a smooth logarithmic replacement. The constrained-derivative calculation tests whether the field in (5) can simply be interpreted as a canonical jet of one smaller spinor; that specified identification fails finite propagation.