Self-source from complementary Dirac modes

Summary: An explicit canonical normal-mode reduction produces a spectral square-root source and its full response; the projected count, total norm and complete scalar square determine what remains missing from a physical mass law.

Derive a source before interpreting it as mass

The current hierarchy reconstructs an action with a simplex self-source. A complementary calculation starts from an explicit canonical Dirac pair and derives the interaction obtained by following its occupied modes. The reduction explains a square-root law, but also fixes its matrix meaning, its occupation dependence and its frequency response. Those details cannot be chosen independently afterward.

All retained and complementary variables below are conditional modes of the same field. Their normal kinetic matrix must ultimately follow from the physical spatial Dirac operator and its geometry. No extra interaction particle is introduced. The matrix rank counts coefficient modes; it adds no physical spatial dimensions and imposes no compactification.

Let a Hermitian retained current be $R$ on an $r$-dimensional coefficient space. Consider the complementary Hamiltonian

$$H(R)=\epsilon\tau_x\otimes I+g\tau_z\otimes R, \qquad \epsilon>0, \qquad E_i=\sqrt{\epsilon^2+g^2r_i^2}. \tag{1}$$

Here $r_i$ are the eigenvalues of $R$, $\tau_x,\tau_z$ are Pauli matrices, and $\epsilon$ is a specified normal kinetic frequency. It is not an inserted physical particle mass. The two signs of the Dirac spectrum are retained without assigning them a particle–antiparticle interpretation.

Canonical occupation and the spectral root

Represent the occupied complementary field by a matrix $Y$ whose columns are explicitly orthogonal spectator modes of the same field. With one common canonical norm $q$ per positive eigenmode, its density is

$$Y^\dagger Y=qI_r,\qquad YY^\dagger=qP_+(R),\qquad Q_f=\operatorname{Tr}Y^\dagger Y=qr. \tag{2}$$

A single pure coefficient vector does not automatically give the complete projector in (2). The physical spectator modes and their occupation need an origin in the master field. A quantum filled sea is not assumed.

The positive spectral space maximizes $\operatorname{Tr}(Y^\dagger H Y)$ at this fixed Gram matrix: diagonalizing $H$ bounds the occupied trace by the sum of its $r$ largest eigenvalues. This is an energy maximum, rather than a passive ground state. Its dynamical viability must be checked in the coupled system and in its actual spatial domain.

Subtracting the specified $R=0$ reference energy gives the adiabatic law

$$\begin{aligned} U_{\rm ad}(R)&=q\operatorname{Tr} \left[\sqrt{\epsilon^2I+g^2R^2}-\epsilon I\right],\\ K(R)&=\frac{\partial U_{\rm ad}}{\partial R} =qg^2R(\epsilon^2I+g^2R^2)^{-1/2}. \end{aligned} \tag{3}$$

For a scalar current this is exactly the smooth saturating law, with $\mu=q|g|$ and $\sigma=\epsilon/|g|$. For a matrix current it is a trace of spectral roots. It is not $\mu\sqrt{\operatorname{Tr}R^2+\sigma^2}$. Those two functions agree on an isolated rank-one current but give different finite interactions.

For example, let $R=AP_z+BP_w$, $A,B>0$, and $\tau=|z^\dagger w|^2$. Its two nonzero eigenvalues are

$$\lambda_\pm=\frac{A+B\pm \sqrt{(A-B)^2+4AB\tau}}2. \tag{4}$$

At positive $\epsilon$, (3) gives a nonnegative cross energy which increases with $\tau$ and vanishes for orthogonal rays. At zero gap, $\operatorname{Tr}|R|=\operatorname{Tr}R=A+B$: the positive-current cross energy vanishes at every orientation. The Hilbert–Schmidt root instead gives $\sqrt{A^2+B^2+2AB\tau}-A-B$, which is generally negative.

The zero-gap and zero-probe limits also fail to commute at a current with a null eigenspace. At every positive gap an infinitesimal probe orthogonal to a rank-one target has zero first derivative, whereas a finite positive probe has a nonzero slope after taking the root limit. An isolated source coefficient therefore does not determine the pair law.

The complete frequency response

Use the convention $e^{-i\Omega t}$, with a positive imaginary part of $\Omega$ for the retarded continuation. In an eigenbasis of $R$, the linear response acts entry by entry on $\delta R$:

$$\begin{aligned} \delta K_{ij}&=\chi_{ij}(\Omega)\delta R_{ij},\\ \chi_{ij}(\Omega)&= \frac{2qg^2T_{ij}(E_i+E_j)}{(E_i+E_j)^2-\Omega^2},\\ T_{ij}&=\frac{E_iE_j+\epsilon^2-g^2r_ir_j}{2E_iE_j}. \end{aligned} \tag{5}$$

To derive (5), linearize the complete density equation $i\dot D=[H,D]$ about $D=qP_+$. In the full Dirac eigenbasis,

$$(\delta D)_{ab}= \frac{(D_b-D_a)(\delta H)_{ab}} {\Omega-(H_a-H_b)}. \tag{6}$$

Only transitions between the equally occupied positive space and the unoccupied negative space contribute. Their squared scalar transition matrix element is $T_{ij}$; combining both directions gives (5). Taking the scalar current of $\delta D$ supplies the second factor of $g$.

At zero frequency this is the divided difference of $qg^2r/\sqrt{\epsilon^2+g^2r^2}$. The diagonal is

$$\chi_{ii}(\Omega)= \frac{4qg^2\epsilon^2}{E_i(4E_i^2-\Omega^2)}. \tag{7}$$

The full response is even in frequency. In a smooth positive frame, the trace Berry connection vanishes, although the matrix connection can be nonzero. The leading susceptibility correction is proportional to $\Omega^2$ and is positive along a Hermitian probe below the poles. Its eliminated real-time derivative term has the sign associated with expansion about the constrained energy maximum. This is a low-frequency expansion of the explicit canonical system, not a fundamental higher-time- derivative theory or a proof of spatial causality.

Retain the actual master projection

The smooth coherent scalar projection defines $D_N=\binom{N+d}{d}$ and the normalized current symbol $j_N(z)=D_N\langle z^N|C_N|z^N\rangle$. A complementary degree-$M$ space sees the actual insertion

$$V_{N\to M}(C_N)=D_M\int j_N(z)P_{z,M}\,d\nu(z). \tag{8}$$

The ray measure describes coefficient directions over physical space. It is not a new spatial sector. The exact contractions of the linked projection note evaluate (8) without replacing its weights by shell weights.

For an isolated coherent current $C_N=JP_{u,N}$, the eigenvalue at transverse occupation $j$ and its multiplicity are

$$v_j=J\frac{D_ND_M}{D_{N+M}} \frac{(M)_{\underline j}}{(N+M)_{\underline j}}, \qquad h_{j,d}=\binom{j+d-1}{d-1},\quad 0\le j\le M. \tag{9}$$

One derivation sets $t=|z^\dagger u|^2$. The sum of coherent probabilities in the $j$th transverse shell is $\binom Mj t^{M-j}(1-t)^j$, and its measure is $d(1-t)^{d-1}dt$. The beta integral, divided by $h_{j,d}$, gives (9). The trace identity is particularly restrictive:

$$\boxed{\sum_{j=0}^M h_{j,d}v_j=D_MJ.} \tag{10}$$

Thus the zero-gap, completely occupied positive complementary space gives cross source $q|g|D_M$. It counts the populated complementary band, independently of the retained degree $N$. Obtaining $D_N$ requires the additional rule $M=N$. At fixed total complementary norm $Q_f$, $q=Q_f/D_M$ removes that enhancement. At common norm per mode, $Q_f=qD_M$ grows with the count. The scalar gap normalization explains the corresponding distinction in a global self-consistent model.

A single coherent normal source has a different preparation from the complete population in (2). Its full target insertion approaches $qD_NP_{z,N}$ at fixed canonical norm as the normal degree tends to infinity. That construction derives an explicit contact approximation and its divergent scalar self-energy cost; it does not require matching the populated band to the target.

For a positive kinetic gap, insert (9) into (3). At $J=q=g=1$, matched depth-six bands $N=12,34$ give the following cross-only ratios. The width is $\sigma=\epsilon/|g|$ in the same declared current units.

These are controlled model values, with no fitted physical gap. The smallest insertion at $N=34$ is approximately $2.80\times10^{-15}$ in these units. Adiabatic following must compare the actual frequencies of current variation with the relevant transition gaps. A stationary field's overall phase cancels from its current; its phase frequency alone is not a time scale for that test.

Restore the complete scalar square

The cross interaction alone is not the entire scalar master kernel. At zero normal kinetic gap, take a coherent retained current $JP_{u,N}$ and a uniformly populated complementary current $qI_M$. The complete projected square gives

$$\begin{aligned} \lambda_N&=\frac{D_N^2}{D_{2N}},\\ E_{\rm total}&=\frac g2 \left[J^2\lambda_N+2JqD_M+q^2D_M^2\right],\\ \boxed{\omega_N}&=\boxed{g\left[J\lambda_N+qD_M\right]}. \end{aligned} \tag{11}$$

Here $g>0$ is the common scalar-square coefficient in the same normalized mode units, and $\omega_N$ is the retained positive-spinor phase frequency. The complementary self-insertion is $qD_MI_M$. It commutes with the complete complementary density, so that density stays invariant while its complex occupied modes acquire their actual phases.

The total canonical norm is $J+qD_M$. Above the specified complementary- only reference, the excess energy is

$$E_{\rm excess}=gJqD_M+\frac g2J^2\lambda_N, \qquad \frac{\partial E_{\rm excess}}{\partial J}=\omega_N. \tag{12}$$

For $M=N$ and $J=q=1$, the depth-six source-frequency ratio is $204.6505814$, and the excess-energy ratio is $205.6913738$. The simplex ratio is $D_{34}/D_{12}=3515/17=206.7647059$. The retained self-term in (11) therefore matters. Dropping it would change the action under investigation.

The full energy also contains the complementary self-energy, proportional to $q^2D_M^2$. Identifying only the retained phase or a subtracted energy with a particle's physical rest mass requires an explicit physical preparation and energy prescription. A coefficient that contains a count does not settle that identification.

The complete normal band at finite kinetic gap

The full self-consistent normal band can be calculated without constructing a $D_M$-dimensional dense matrix. In the coherent target frame, let $c_j$ be the scalar eigenvalue of the normal current on transverse shell $j$, with multiplicity $h_{j,d}$. Its coherent symbol is $D_M\sum_jc_j\binom Mj t^{M-j}(1-t)^j$. Integrating the product of two such symbols gives the exact self-insertion matrix

$$\boxed{\mathcal K_{kj}= \frac{D_M^2d}{h_{k,d}}\binom Mk\binom Mj \mathrm B(2M-k-j+1,d+k+j),} \tag{12a}$$

where $\mathrm B$ is the Euler beta function. Every entry is positive, $\sum_j\mathcal K_{kj}=D_M$, and $h_{k,d}\mathcal K_{kj}=h_{j,d}\mathcal K_{jk}$. The latter is reciprocity with the actual shell multiplicities. The weighted matrix is a positive Gram matrix of the coherent shell symbols.

Write $w_k=v_k/J$ from (9) and $\sigma=\epsilon/g$. Equal positive occupation $q$ in each normal eigenspinor now requires

$$r_k=Jw_k+\sum_j\mathcal K_{kj}c_j, \qquad c_k=\frac{qr_k}{\sqrt{\sigma^2+r_k^2}}. \tag{12b}$$

For positive $J,q,\sigma$, monotone iteration from $r_k=Jw_k$ is bounded above by $Jw_k+qD_M$. Positivity and strict sublinearity of the scalar response give a unique positive solution by the component-ratio argument also used in the linked balanced-response calculation. The target's complete source is

$$\boxed{\frac{\omega_N}{g} =J\lambda_N+\sum_kh_{k,d}w_kc_k.} \tag{12c}$$

This calculation includes the normal kinetic energy as well as both self-terms and the cross-term:

$$\begin{aligned} \frac{E}{g}={}&\sum_k \frac{h_{k,d}q\sigma^2}{\sqrt{\sigma^2+r_k^2}} +\frac12J^2\lambda_N +J\sum_kh_{k,d}w_kc_k\\ &+\frac12\sum_{k,j}h_{k,d}c_k\mathcal K_{kj}c_j. \end{aligned} \tag{12d}$$

At the stationary normal solution, differentiation at fixed canonical normal occupations gives $\partial E/\partial J=\omega_N$. Independent fifty-digit energy differences verify that statement, even when the normal energy is much larger than the retained derivative. The total canonical norm remains $J+qD_M$; finite gap does not erase the occupation.

For matched depth-six bands and $J=q=1$, the complete source ratios at common widths $\sigma=1,10^3,10^6$ are respectively $204.6505816$, $204.9364800$ and $6505.0628511$. These common-width controls are not physical fits. They show that the kinetic scale changes the ratio continuously and cannot be ignored or chosen independently for each target.

The normal self-field also changes the smallest insertion. At width one it is approximately $18564.0002$ for $N=12$ and $3838380.0000$ for $N=34$, rather than the very small values of the cross-only operator (9). Normal response must use the complete self-consistent Hamiltonian when that is the declared action.

At vanishing retained current, a uniform positive normal branch obeys $r=D_Mc$ and exists for $qD_M>\sigma$, with $c=\sqrt{q^2-(\sigma/D_M)^2}$. This is the uniform instance of the canonical scalar gap mechanism. It still requires the stated occupation and a physical normal spectrum; it does not select a particle norm.

Reciprocal time evolution checks the elimination

A finite control retains an eight-component spatial Dirac spinor with two coefficient directions, one specified momentum, and both complementary normal signs with two occupied spectator columns. Its exact energy is

$$E=p\langle X,\alpha_1X\rangle+ \operatorname{Tr}\left[Y^\dagger \left(\epsilon\tau_x\otimes I+g\tau_z\otimes C_X\right)Y\right], \qquad (C_X)_{ab}=\sum_{s,t}X_{sa}\beta_{ts}\overline{X_{tb}}. \tag{13}$$

Both fields evolve by the complete complex gradient of this declared cross-action. It is a check of (3), not a claim that the retained and normal self-terms of (11) have been included in this control.

Scaling $(\epsilon,g,q)$ to $(\Lambda\epsilon,\Lambda g,q/\Lambda)$ keeps (3) fixed while increasing the normal transition frequencies. For $p=0.35$, baseline $(\epsilon,g)=(0.8,0.6)$ and time interval $0\le t\le20$, the maximum complete retained-field error relative to the adiabatic evolution is

The observed convergence is approximately second order in $1/\Lambda$. Independent time refinement changes the full fields by less than $7.1\times10^{-9}$ for the last two cases. The full norms, occupied Gram matrix, energy and total coefficient-rotation moment are conserved to the reported numerical accuracy. This is a one-momentum mode control, not a localized particle or a spatial scattering calculation.

The joint mass and coupling question

Equation (3) is a derived effective source for the specified canonical normal-mode preparation. Equations (10)–(12) identify the missing band matching, occupation and physical mass interpretation. The spatial normal spectrum itself must also come from the same physical manifold. The localization conditions remain relevant when searching for actual finite-norm spatial particles.

A further balanced-population calculation retains paired positive and negative normal occupations. Their stationary scalar contributions cancel while their moving polarization gives a nonzero geometric response. That result supplies a way for a shared response to coexist with count-dependent stationary sources under the same scalar interaction. It does not select those populations or derive the physical mass operator.

Verification

dirac_current_reduction_checks.py checks the full spectral variation, independent Dirac Liouvillian, canonical occupations, actual coherent master insertion and all terms of (11). Its finite equations are implemented in dirac_current_reduction.py. dirac_current_reciprocal_flow.py stores the complete complex trajectories, and dirac_response_trajectory_checks.py independently checks their refinement and conserved total rotation moment.

projected_normal_self_consistency_checks.py checks the finite-gap complete square with independent beta quadrature, signed full tensor insertions, canonical shell reciprocity and fifty-digit energy derivatives. The exact shell reduction and positive normal solver are in projected_normal_self_consistency.py.