Geometric response of balanced normal populations

Summary: Paired normal Dirac populations leave a stationary scalar source unchanged while supplying a common geometric response across target grades, including a finite kinetic gap and the complete probe self-interaction.

A stationary source need not measure a moving polarization

The complementary Dirac reduction derives a stationary source from occupied normal modes of the same field. Its full positive-space occupation has an even frequency response. There is a different preparation in the same scalar interaction: add equal populations in the positive and negative normal branches, with a nonuniform distribution over coefficient directions.

The added populations cancel in the stationary scalar current and energy. Their polarization nevertheless changes when the retained geometry moves. The resulting leading geometric response is independent of the retained mode degree for a specified common probe. The calculation below includes both Dirac signs, their positive canonical norms, a finite normal kinetic gap and the complete probe scalar self-interaction.

The normal signs are not assigned an antiparticle interpretation. Their occupied columns must represent actual orthogonal modes of the same field; their geometry and populations have not been dynamically selected. Matrix directions describe coefficients over ordinary spatial sectors, rather than extra physical dimensions or compactification.

Balanced populations at zero kinetic gap

First take a positive Hermitian normal insertion $R$ and Hamiltonian $H=g\tau_z\otimes R$, with $g>0$. Let $U\ge0$ commute with $R$. Choose the two normal densities

$$\mathcal D_+=U+qI_r,\qquad \mathcal D_-=U, \qquad q>0. \tag{1}$$

Each density can be realized by an explicit square-root matrix of occupied spectator columns. Their scalar current, energy and canonical norm are

$$\begin{aligned} C_f&=\mathcal D_+-\mathcal D_-=qI_r,\\ E_f&=g\operatorname{Tr}(RC_f)=qg\operatorname{Tr}R,\\ Q_f&=\operatorname{Tr}(\mathcal D_++\mathcal D_-) =qr+2\operatorname{Tr}U. \end{aligned} \tag{2}$$

Thus $U$ changes the positive canonical norm even though its stationary scalar energy cancels between the two normal signs. That norm is retained; it is not identified with electric charge or particle number. The mixed normal preparation is not a passive energy minimum.

The complete density equations are

$$i\dot{\mathcal D}_\pm=\pm g[R,\mathcal D_\pm]. \tag{3}$$

For a perturbation $\delta R\,e^{-i\Omega t}$, work in a common eigenbasis of $R$ and $U$, with eigenvalues $r_i,u_i$. Applying the density linearization to both signs gives

$$\boxed{\delta(gC_f)_{ij} =\frac{2g^2\Omega(u_j-u_i)} {\Omega^2-g^2(r_i-r_j)^2}\,\delta R_{ij}.} \tag{4}$$

The uniform excess $q$ cancels from (4). The balanced response is odd in frequency and vanishes statically when the relevant spectral gaps are nonzero. For small frequency,

$$\delta(gC_f)_{ij} =-\frac{2\Omega(u_j-u_i)}{(r_i-r_j)^2}\delta R_{ij} +O(\Omega^3). \tag{5}$$

This is a geometric, phase-shifted response to motion. A single complex frequency amplitude need not itself be Hermitian; the physical real drive combines its positive and negative frequency components. Degenerate eigenspaces with unequal occupations need their own transport analysis.

The actual common probe of the scalar master kernel

Let the retained coherent current have degree $N$, norm $J$ and ray $z$, and let the normal probe have fixed degree one. Set $r=d+1$. The exact scalar projection gives

$$\begin{aligned} V_{N\to1}(C_N)&=a_N\left[J I+N\rho_{N,1}\right], &a_N&=\frac{d+1}{N+d+1},\\ C_N=JP_{z,N}&\quad\Longrightarrow\quad R_N=a_NJI+B_NP_z, &B_N&=a_NJN. \end{aligned} \tag{6}$$

Here $\rho_{N,1}$ is the one-index contraction with trace $J$. The parallel ray and its orthogonal complement are independent of $N$. Their eigenvalue gap $B_N$ depends on $N$. At depth six and $J=1$, $B_{12}=84/19$ and $B_{34}=238/41$.

Choose the same probe preparation at the two target grades:

$$U=u_\perp I+\Delta u P_z, \qquad \Delta u=u_\parallel-u_\perp, \qquad u_\parallel,u_\perp\ge0. \tag{7}$$

Near $z=e_0$, use the normalized local ray $z=(1,x+iy,0,\ldots)/\sqrt{1+x^2+y^2}$. On its active two-direction plane, the actual external current vertices are

$$\partial_xR_N=B_N\sigma_x,\qquad \partial_yR_N=B_N\sigma_y. \tag{8}$$

Insert both vertices in (4). The response of the $x$ energy gradient to a moving $y$ is

$$\boxed{ \lim_{\Omega\to0} \frac{\operatorname{Tr} [(\partial_xR_N)\,\delta(gC_f)[\partial_yR_N]]} {-i\Omega}=4\Delta u.} \tag{9}$$

The two vertex factors $B_N^2$ cancel the spectral denominator. No division by the target mass and no inverse reconstruction of the raw current has been introduced. This is a common leading geometric response for the specified common population. Its finite-frequency corrections still depend on the normal gaps.

Equivalently, the balanced population contributes the Berry one-form $2\Delta u\,i z^\dagger dz$. Its curvature on this local plane is $-4\Delta u\,dx\wedge dy$. The induced energy gradient is minus the curvature acting on the ray velocity, giving the sign in (9). The uniform population contributes the trace connection of a complete coefficient frame, which has zero curvature in this local calculation.

The same algebra works on a chosen two-direction plane at other physical depths. Consequently it does not by itself explain weak depth-five coupling. A physical rule for which normal patterns occur and how their sectors engage remains necessary.

Ordinary spatial rotation is a more specific test

An arbitrary unitary change of the $d+1$ coefficient entries is broader than a rotation of physical space. In $W=\mathbf1\oplus(T^*Q_d\otimes\mathbb C)$, an ordinary spatial rotation acts on the tangent entries and leaves the homogenizing scalar entry fixed. Let $A,B$ be two real antisymmetric tangent rotation generators, extended by zero on that scalar entry. Then

$$\partial_AR_N=B_N[A,P_z],\qquad \boxed{\mathcal G(A,B)=4\Delta u\, \operatorname{Im}\langle Az,Bz\rangle.} \tag{9a}$$

Equation (9a) is the actual spatial-rotation instance of (9). It also survives the finite-gap self-consistency below. The result concerns the coefficient-current component of the full response; spinor and spatial kinetic contributions must accompany it in a physical moving field.

For a concrete check, take $z=(1,\sqrt2,i,0,\ldots)/2$, where the first entry is the scalar slot. Let $A$ rotate tangent directions one and three, and $B$ rotate tangent directions two and three. Then $\mathcal G=\sqrt2\Delta u$. Replacing $i$ by $1$ leaves the norm and occupied degrees unchanged but makes this geometric component zero. Independent complete normal-response calculations verify both controls at the assigned degrees and each stated finite kinetic gap.

This zero follows from the actual complex polarization, rather than an inserted neutral projector. It does not make an entire particle noninteracting: other response directions and the scalar density channel remain present. Nor does the calculation assign the real preparation to one physical depth.

A specified uniform complex-ray distribution supplies a useful depth comparison. With $r=d+1$, its normalized squared amplitudes obey $\langle |z_1|^2|z_2|^2\rangle=1/[r(r+1)]$, and uniform relative phase gives $\langle\sin^2\varphi\rangle=1/2$. For the two rotations above,

$$\langle\mathcal G\rangle=0,\qquad \langle\mathcal G^2\rangle= \frac{8(\Delta u)^2}{(d+1)(d+2)}. \tag{9b}$$

Direct amplitude and phase quadrature verifies these moments. The depth-five variance is $4/3$ of the depth-six variance at common $\Delta u$, so this uniform preparation provides no neutrino suppression. That conditional comparison identifies the missing selection of physical polarization; it does not exclude other depth-dependent spatial patterns.

The probe's own scalar term

The complete degree-one self-projection is

$$V_{1\to1}(C)=b\left[\operatorname{Tr}C\,I+C\right], \qquad b=\frac{d+1}{d+2}. \tag{10}$$

At the zero-gap stationary preparation $C=qI$, this adds an identity insertion. A variation adds $b\delta C$ in each off-diagonal direction. The resulting self-consistent response is

$$\delta(gC)_{ij}= \frac{2g^2\Omega(u_j-u_i)} {\Omega^2-g^2(r_i-r_j)^2-2gb\Omega(u_j-u_i)} \delta(R_N)_{ij}. \tag{11}$$

Its finite-frequency form differs from (4), but the leading term (5) and common external coefficient (9) survive. Thus the cancellation does not depend on omitting the probe self-interaction.

A finite normal kinetic gap preserves the balanced cancellation

For $H(R)=\epsilon\tau_x\otimes I+g\tau_z\otimes R$, let $P_+(R)$ be the complete positive projector. The stationary density is now

$$\boxed{\mathcal D=I_2\otimes U+qP_+(R)}, \qquad [R,U]=0, \qquad E_i=\sqrt{\epsilon^2+g^2r_i^2}. \tag{12}$$

In each coefficient eigendirection its positive and negative occupations are $u_i+q$ and $u_i$. The contribution $I_2\otimes U$ commutes with the full Hamiltonian and has zero scalar current and zero energy, including the normal kinetic term. Its norm remains $2\operatorname{Tr}U$. The stationary scalar eigenvalues therefore are

$$c_i=q\frac{gr_i}{E_i}, \tag{13}$$

independent of the balanced occupation.

The unbalanced $q$ population has the even susceptibility $\chi^q_{ij}$ derived in the linked mass-source note. The balanced contribution is

$$\begin{aligned} \chi^U_{ij}(\Omega)&=2g^2\Omega(u_j-u_i) \left[\frac{1-T_{ij}}{\Omega^2-(E_i-E_j)^2} +\frac{T_{ij}}{\Omega^2-(E_i+E_j)^2}\right],\\ T_{ij}&=\frac{E_iE_j+\epsilon^2-g^2r_ir_j}{2E_iE_j}. \end{aligned} \tag{14}$$

The first fraction comes from transitions within the same Dirac sign, and the second from transitions between the signs. Both are required. For distinct positive $r_i,r_j$, elementary substitution gives

$$\boxed{g^2\left[ \frac{1-T_{ij}}{(E_i-E_j)^2} +\frac{T_{ij}}{(E_i+E_j)^2}\right] =\frac1{(r_i-r_j)^2}.} \tag{15}$$

Thus the leading balanced geometric response is exactly (5), even at finite $\epsilon$. The formula presumes an adiabatic spectral separation; it must not be extended through unresolved degeneracies by assigning values to zero divided by zero.

Finite-gap scalar self-consistency

For the coherent target (6), the complete stationary insertion and normal scalar current share the same two eigenspaces. Their eigenvalues satisfy

$$r_i=r_i^{\rm ext}+b\left[\sum_jc_j+c_i\right], \qquad c_i=q\frac{gr_i}{\sqrt{\epsilon^2+g^2r_i^2}}, \tag{16}$$

where $r_\parallel^{\rm ext}=a_NJ+B_N$ and $r_\perp^{\rm ext}=a_NJ$. The perpendicular value occurs $d$ times. All scalar self-terms are retained, including the trace.

For positive external values, monotone iteration from $r^{\rm ext}$ is bounded above by $r^{\rm ext}+bq(r+1)$ in each entry, so a positive solution exists. It is unique in the positive cone: the map is monotone, and strict sublinearity under a common factor greater than one follows from the positive external term and concavity of $r/\sqrt{\epsilon^2+g^2r^2}$ for $r>0$. If two positive fixed points differed, compare them at the largest component ratio; strict sublinearity would contradict the component attaining that ratio.

Let $\Delta r=r_\parallel-r_\perp$ and $\Delta c=c_\parallel-c_\perp$. The gap equation and static off-diagonal susceptibility obey

$$\Delta r=B_N+b\Delta c, \qquad \chi^q_{\parallel\perp}(0)=g\frac{\Delta c}{\Delta r}, \qquad 1-\frac bg\chi^q_{\parallel\perp}(0)=\frac{B_N}{\Delta r}. \tag{17}$$

For an off-diagonal external drive, put $\chi=\chi^q+\chi^U$. Full self-consistency replaces it by

$$\chi_{\rm full}=\frac{\chi}{1-b\chi/g}. \tag{18}$$

Its leading odd part is $\chi^U/(1-b\chi^q(0)/g)^2$. Equation (15) supplies $1/(\Delta r)^2$; equation (17) supplies the factor $(\Delta r/B_N)^2$. The two external vertices again supply $B_N^2$. Hence (9) survives the finite-gap complete scalar square. Diagonal responses additionally couple through the trace term in (16); the executable full Liouvillian retains that coupling as well.

At the assigned depth-five, depth-six and depth-ten degrees, independent finite matrix calculations with $\epsilon=0.1,1,5$ reproduce the same coefficient $4\Delta u=2.4$ for the declared $\Delta u=0.6$. No physical coupling value or normal kinetic gap is fitted here.

Reciprocal motion under the same complete interaction

A coherent target and a degree-one normal probe form an invariant zero-spatial-kinetic control. Keep the entire probe rank $r=d+1$, its complex occupied matrices $Y_+,Y_-$ and the target's unit ray $z$. Define $\mathcal D_\pm=Y_\pm Y_\pm^\dagger$, $C=\mathcal D_+-\mathcal D_-$ and $W=\mathcal D_++\mathcal D_-$. The varying scalar energy is

$$E=g a_NJ\left[\operatorname{Tr}C+N z^\dagger Cz\right] +\frac{gb}{2}\left[(\operatorname{Tr}C)^2+\operatorname{Tr}C^2\right]. \tag{19}$$

The coherent target's own term $gJ^2\lambda_N/2$ is constant on this fixed-norm ray manifold, but its full scalar phase is retained when reconstructing the target field. Its induced ray kinetic coefficient is $JN$, obtained from the canonical field $\sqrt J\,z^{\otimes N}$. The resulting equations are

$$\begin{aligned} i\dot z&=g a_N C z,\\ i\dot Y_\pm&=\pm g L Y_\pm,\\ L&=\left[a_NJ+b\operatorname{Tr}C\right]I+B_NP_z+bC. \end{aligned} \tag{20}$$

The target's additional scalar phase rate is $g a_N\operatorname{Tr}C+gJ\lambda_N$. The scalar identity terms in $L$ remain in the full complex-field evolution; they are not removed merely because they commute with densities. Direct variation of the complete symmetric tensor verifies that the degree-one insertion maps this coherent ray manifold into its tangent. The source contains all $N$ index insertions, plus that scalar phase.

Both objects respond. In particular,

$$\boxed{JN P_z+W\ \text{is conserved}},\qquad Y_\pm^\dagger Y_\pm\ \text{are conserved},\qquad \|z\|=1. \tag{21}$$

The first is the common coefficient-rotation moment. It follows directly by adding the projector equation to $i\dot W=g[L,C]$. The full energy is conserved as well.

If initially $C=qI$ but the balanced population $U$ is misaligned with $P_z$, the target initially has only a scalar phase. Its subsequent interaction is not zero:

$$\dot C(0)=-2igB_N[P_z,U]. \tag{22}$$

The scalar current becomes anisotropic and then moves the target ray. This is an explicit reciprocal mechanism for changing compatibility, derived from the same interaction instead of an imposed alignment rule.

For an aligned stationary preparation, the active orientation block at fixed total rotation moment has signed frequencies

$$\boxed{\Omega_\pm=-gb\Delta u \pm g\sqrt{b^2(\Delta u)^2+a_N^2JN(JN+2\Delta u)}.} \tag{23}$$

One obtains them by linearizing the full density equations for $(P_z,C,W)$ and using $JN\delta P_z+\delta W=0$. They are real for the stated $\Delta u>0$. This checks the local orientation block; it is not a theorem of complete spatial stability.

A nonlinear control at depth six uses the same $J=1$, $g=0.7$, $q=0.4$, $u_\perp=0.2$, $\Delta u=0.6$ and initial complex misalignment at $N=12,34$. Initially both normal polarizations have target overlap $0.671823$. During $0\le t\le60$, the positive population's overlap decreases to $0.642635$ and $0.640150$, respectively; the negative population's overlap increases to $0.743467$ and $0.718852$. Their subsequent motion oscillates. There is no automatic common alignment or imposed dissipation.

Independent time refinement changes the full complex fields by less than $4.1\times10^{-7}$. In the refined runs, energy drift is below $3.5\times10^{-10}$, each occupied Gram drift below $9.4\times10^{-11}$, and total rotation-moment drift below $1.4\times10^{-10}$. These trajectories retain both polarizations and their phases. They remain finite mode controls with zero spatial kinetic operator, rather than localized encounters between identified particles.

What this establishes and what still needs selection

This supplies a common geometric response across distinct retained grades under the actual scalar projection, with canonical normalization and self-consistent normal dynamics. A stationary scalar excess can still carry the complementary mode count described in the linked mass-source note. The geometric cancellation therefore permits different stationary sources and a shared moving-polarization response within one interaction.

It is only one response component. Scalar density vertices and the full interaction energy retain their occupation and band dependence. No electric current, measured charge, encounter rate or neutrino suppression has been identified by equation (9). The chosen populations are additional physical preparation data; common $\Delta u$ is not derived by the scalar kernel. At zero gap their occupied density spectra are conserved by (3), so this closed mode model cannot dynamically choose those spectra from arbitrary initial data.

Physical spatial geometry must supply the normal modes, their preparation and their overlap with other nested sectors. The sectors are nested; one particle is not a constituent inside another. The full spatial Dirac residual and finite-norm binding must be checked when passing from this mode control to particle fields. The positive-current localization condition states one relevant flat-space restriction. Neither a physical mass operator nor a complete coupling law is claimed here.

Verification

The reciprocal stability calculation now retains the complete finite-gap normal density and target ray. Its exact angular quartic and full nonlinear fields distinguish a bounded and a growing preparation at equal total canonical norm and stationary scalar source. The geometric response alone does not select the population's dynamical viability.

balanced_dirac_response_checks.py checks independent two-sign Liouvillians, the exact master projection, coherent tensor tangents, frame covariance, full action gradients, geometric vertices and the complete orientation spectrum. An additional exact rotating-ray control checks both normal density equations directly. The finite laws are in balanced_dirac_response.py.

gapped_balanced_response_checks.py checks both transition families, finite-gap cancellation, the positive scalar fixed point and complete matrix self-feedback; its implementation is gapped_balanced_response.py. balanced_dirac_reciprocal_flow.py stores the entire complex reciprocal trajectories, and dirac_response_trajectory_checks.py independently verifies refinement, energy and the common rotation moment.