Nonlinear mass reduction with spatial response

Summary: Eliminating complementary spatial Dirac modes gives a nonlinear effective action, its exact induced norm, and a positive source-leakage criterion for departures from a common linear contact scale.

The interaction coefficient and the occupied field

The occupied-graph action retains the simplex count as a nonlinear coefficient. Determining its physical consequence requires solving the spatial field and its normalization. The calculation below eliminates complementary modes of that same field while retaining their contribution to both quantities. It does not derive the graph action from the master interaction.

Let the spatial Hilbert space have a fixed physical measure and canonical time-kinetic norm. Its self-adjoint spatial Dirac operator and chirality obey

$$\mathscr D^\dagger=\mathscr D,\qquad \beta^\dagger=\beta,\qquad \beta^2=I,\qquad \{\mathscr D,\beta\}=0. \tag{1}$$

These are operators on actual spatial fields, not a replacement of spatial coordinates by coefficient rays. The calculation applies at any spatial depth for which the following spectral and regularity hypotheses hold; six dimensions alone do not establish them.

Let $P$ project onto the entire zero space in the problem being reduced, assume $\beta P=P$, and write $Q=I-P$. Require an invertible complementary operator $\mathscr D_Q=Q\mathscr DQ$. If a restricted spatial symmetry class is used, it must be invariant under the full equation, and every zero mode in that class must be retained. A negative-chirality zero mode cannot silently be inverted or omitted.

For finite Galerkin systems these are finite matrix assumptions. On a smooth closed spatial manifold, an elliptic Dirac operator with a spectral gap on $Q$ supplies the complementary inverse between the appropriate Sobolev spaces. One sufficient framework uses $H^s$, $s>d/2$, so products are controlled, and a scalar kernel bounded on that space. The local kernel is the identity. The resulting real-analytic implicit-function argument is local near zero coupling and zero frequency. It is not a claim about arbitrary coupling or continuous spectra without a gap.

Use a fixed real reciprocal scalar kernel $K$, with

$$\begin{aligned} \sigma_\psi&=\psi^\dagger\beta\psi,& V_\psi&=K\sigma_\psi,\\ U(\psi)&=\langle\sigma_\psi,K\sigma_\psi\rangle,& E_g(\psi)&=\langle\psi,\mathscr D\psi\rangle+\frac g2U(\psi). \end{aligned} \tag{2}$$

The scalar potential acts by multiplication and commutes with $\beta$. Its complete gradient is

$$\nabla_{\bar\psi}E_g=\mathscr D\psi+g\beta V_\psi\psi. \tag{3}$$

Both ordinary and conjugate variations of the scalar source are part of (3). A fixed potential evaluated on an unchanging target would give a different reduction.

Exact stationary elimination and the norm

At a real trial frequency $\omega$, form

$$\mathcal R(\psi;g,\omega)=E_g(\psi)-\omega\|\psi\|^2, \qquad \psi=u+y,\quad u=Pu,\quad y=Qy. \tag{4}$$

Solve the full complementary equation

$$Q[\mathscr D(u+y)+g\beta V_{u+y}(u+y)-\omega(u+y)]=0. \tag{5}$$

The invertibility in (1) gives a unique small local solution $y=y(u,g,\omega)$. Substitute this solution into (4), defining $\mathcal R_{\rm eff}$. Stationarity in $y$ removes its implicit derivatives from every first variation. In particular,

$$\boxed{-\partial_\omega\mathcal R_{\rm eff} =\|u+y(u,g,\omega)\|^2.} \tag{6}$$

The reduced field's norm is therefore not generally $\|u\|^2$. Equation (6) is exact on the local complementary branch. This norm is the phase-symmetry norm of the written action; no ordinary electric charge or particle-number interpretation has been assigned to it.

The stationary second variation has an equally exact expression. In real coordinates retaining both quadratures, let $H$ be the full Hessian of (4). Where $H_{QQ}$ is invertible,

$$\boxed{H_{\rm eff}=H_{PP}-H_{PQ}H_{QQ}^{-1}H_{QP}.} \tag{7}$$

This nonlinear stationary reduction uses the same elimination principle as the linear Feshbach--Schur method. The band-consistency calculation explains why eliminating modes and bare compression obey different consistency rules. Neither (7) nor a positive curvature in a restricted subspace establishes dynamical stability of the full Dirac field.

The energy-response calculation applies this Schur reduction to the canonical norm direction. It gives the exact shape correction to the branch's energy curvature and verifies opposite correction signs on the two assigned excited radial controls.

Leading effective action, including complementary normalization

For $u$ in the positive-chirality zero space define

$$\rho_u=u^\dagger u,\qquad V_u=K\rho_u,\qquad t_u=\mathscr D_Q^{-1}Q(V_uu). \tag{8}$$

The vector $t_u$ has opposite chirality. With $\epsilon=|g|+|\omega|$, expansion of (5) gives

$$y=-g t_u -g\mathscr D_Q^{-1}Q[(\omega+gV_u)t_u] +O(\epsilon^3). \tag{9}$$

The first displayed term is negative-chirality and the second is positive-chirality. Their pointwise scalar cross term and their norm cross term vanish. Substitution into the complete functional yields

$$\boxed{ \begin{aligned} \mathcal R_{\rm eff}(u;g,\omega) ={}&\frac g2U(u)-\omega\|u\|^2\\ &-g^2\omega\|t_u\|^2-g^3\langle t_u,V_ut_u\rangle +O(\epsilon^5). \end{aligned}} \tag{10}$$

One can check the cubic terms directly. Write the first two terms of $y$ as $y_1,y_2$. The kinetic cross term is $2\operatorname{Re}\langle y_1,\mathscr Dy_2\rangle$. The scalar energy contributes $2g\operatorname{Re}\langle V_uu,y_2\rangle -g\langle y_1,V_uy_1\rangle$. Because $\mathscr Dy_1=-gQ(V_uu)$, the two terms containing $y_2$ cancel. The norm term contributes $-\omega\|y_1\|^2$. These are exactly the two cubic contributions in (10).

There is no term of total degree four: the symmetry

$$y(u,-g,-\omega)=\beta y(u,g,\omega),\qquad \mathcal R_{\rm eff}(u;-g,-\omega) =-\mathcal R_{\rm eff}(u;g,\omega) \tag{11}$$

follows from uniqueness of the local complementary solution and (1). The analytic remainder in (10) can also be differentiated. Equation (6) therefore gives

$$\boxed{Q_{\rm eff}=\|u\|^2+g^2\|t_u\|^2+O(\epsilon^4).} \tag{12}$$

The reduced equation must be varied together with this norm. Keeping only the last scalar-potential term of (10) would miss a contribution of the same order to the normalized frequency.

A normalized branch and a criterion for an exact linear scale

Let $\chi$ be a unit vector in the retained zero space satisfying

$$P(V_\chi\chi)=A\chi,\qquad A=U(\chi). \tag{13}$$

This is stationarity of the restricted scalar quartic on its unit sphere. Assume a smooth parity-compatible normalized branch through $\chi$. For a one-dimensional complex zero space its local existence follows after fixing the phase; in a larger space the retained stationary problem must also be solved, with its symmetries and degeneracies accounted for. Define

$$\eta=-\mathscr D_Q^{-1}Q(V_\chi\chi),\quad a=\|\eta\|^2,\quad b=\langle\eta,V_\chi\eta\rangle, \quad C=Aa+b. \tag{14}$$

The full field is $\phi_g=\chi+g\eta+g^2\zeta+O(g^3)$, where $\operatorname{Re}\langle\chi,\zeta\rangle=-a/2$ fixes the total norm. Its complementary equation at second order gives $\mathscr D\zeta=(A+V_\chi)\eta$. Reciprocity of $K$ then gives

$$\boxed{\omega(g)=gA-4g^3C+O(g^5),\qquad E_g(\phi_g)=\frac{gA}{2}-g^3C+O(g^5).} \tag{15}$$

For example, $\operatorname{Re}\langle V_\chi\chi,\zeta\rangle =-Aa/2-Aa-b$. The scalar expansion is consequently $U(\phi_g)=A-6g^2C+O(g^4)$. Together with the exact identity $E_g=\omega-gU(\phi_g)/2$, this proves both expressions in (15). The local scalar normalization explains why nonlinear frequency and total energy must be distinguished before a physical rest mass is identified.

If $A>0$ and the multiplication operator $V_\chi\ge0$, then

$$\boxed{C\ge A\|\eta\|^2,\qquad C=0\ \Longleftrightarrow\ Q(V_\chi\chi)=0.} \tag{16}$$

Here positivity of the potential is an explicit hypothesis; a positive quadratic kernel form alone need not imply a pointwise positive potential. Under (13), vanishing source in the complement means $V_\chi\chi=A\chi$. Thus the unchanged field $\phi_g=\chi$ is an exact branch with $\omega=gA$ for every $g$ for which the stated equation is defined. Its total energy is $gA/2$.

Equation (16) gives a concrete route to exact linearity: the selected spatial field must make its scalar source preserve its zero-mode line. It also gives a necessary condition for exact linearity of the analytic branch under these positivity hypotheses. Existence of such a line does not establish its stability, occupied mode index or observed mass.

What a common simplex coefficient would have to preserve

Suppose, conditionally, that an independently derived graph interaction has supplied $g_N=\kappa D_N$, $D_N=\binom{N+d}{d}$. Applying (15) to its normalized spatial branches gives

$$\begin{aligned} \omega_N&=\kappa D_NA_N-4\kappa^3D_N^3C_N+O(\kappa^5),\\ E_N&=\frac{\kappa D_NA_N}{2}-\kappa^3D_N^3C_N+O(\kappa^5). \end{aligned} \tag{17}$$

The remainder here is at each fixed band; uniform control across growing bands needs its own estimates. Even if $A_N$ is common, the next term requires control of $D_N^2C_N$. A common bare $\kappa$ is insufficient to establish a common physical mass coefficient after spatial response.

As a strictly specified control, take the same spatial branch family at two coefficients $g_i=\kappa D_i$, so $A,C$ are identical. Then

$$\begin{aligned} \frac{\omega_2/\omega_1}{D_2/D_1} &=1-4\frac CA\kappa^2(D_2^2-D_1^2)+O(\kappa^4),\\ \frac{E_2/E_1}{D_2/D_1} &=1-2\frac CA\kappa^2(D_2^2-D_1^2)+O(\kappa^4). \end{aligned} \tag{18}$$

For the committed six-dimensional counts $D_1=S(13,6)=18564$ and $D_2=S(35,6)=3838380$, the coefficient ratio is $3515/17$. The diagnostic holds the control action fixed and verifies both corrections in (18) without adjusting either coefficient separately. This control does not identify its fields with the physical electron and muon. Their actual band-dependent profiles, normalization and master interaction remain to be obtained. It neither changes the committed mode selections nor tests alternative particle dimensionalities.

Verification

nonlinear_mass_reduction_checks.py independently solves the nonlinear complementary and normalized equations. It checks complex source derivatives, the complete real Schur Hessian, retained spaces with one to three complex modes, common-frame covariance, the exact frequency derivative (6), fourth- and fifth-order remainders, the independent norm identity $dE/dQ=\omega$, and the common-coefficient comparison (18). Physical six-dimensional checks use both chiral radial components at section degrees $0,1,3,6$, three specified scalar kernels, and two spatial cutoffs. These finite checks support the displayed conditional derivation; they do not supply the missing master kernel.