Common probes of different self-mass branches

Summary: Reduced currents give exact common-probe responses across different coherent mass grades, with reciprocal shape evolution, same-grade interference and vacuum regularity kept explicit.

Use the self-mass action for the second pattern too

The self-mass construction defines the canonical fields, exact contraction maps $T_{Nk}$, reduced scalar currents $\rho_{N,k}$ and their adjoints $B_{Nk}$. Its common law is

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

All currents are evaluated at the same physical spatial point before integration. Spatial profiles, both polarizations and their phases remain part of the fields. The independent canonical grade decomposition and weights in (1) are inverse-action premises, not a derived master reduction.

The quartic pair kernel

For $\Phi(r)=\kappa r^2/2$, take two coherent fields in distinct grades: $\Psi_N=\chi_a\otimes z^N$ and $\Psi_M=\chi_b\otimes w^M$. Write $A=\chi_a^\dagger\beta\chi_a$, $B=\chi_b^\dagger\beta\chi_b$ and $\tau=|z^\dagger w|^2$. The cross energy is exactly

$$\boxed{V_{NM}=\kappa ABG_K(\tau),\qquad G_K(\tau)=\sum_{k=0}^K\binom{k+d-1}{d-1}\tau^k, \qquad K=\min(N,M).} \tag{2}$$

The same action has isolated self-source $\kappa D_NA$ on the first coherent field. At perfect compatibility, $G_K(1)=D_K$ by Hockey-Stick. At a fixed $\tau<1$, however,

$$G_K(\tau)\longrightarrow(1-\tau)^{-d},\qquad (1-\tau)^dG_K(\tau)=I_{1-\tau}(d,K+1), \tag{3}$$

where $I$ is the regularized incomplete beta function. Equation (3) quantifies a region where the response becomes insensitive to the occupied degrees while the isolated self-source continues to grow as $D_N$.

At depth six,

This is not uniform agreement over orientations. Order zero also gives $G_K(0)=1$, so orthogonal coherent rays do not eliminate the quartic interaction. No electric or weak channel has been identified by this scalar compatibility polynomial.

A fixed probe gives a stronger exact statement

Let a probe have fixed degree $M$, arbitrary complex mode coefficients and every spinor component. Its currents only reach order $M$. Any coherent target of degree $N\ge M$ has

$$\rho_{N,k}=A P_{z,k},\qquad 0\le k\le M. \tag{4}$$

Therefore its full cross energy with that probe, and its diagonal source on the target's occupied ray, are independent of $N$ at the same local $A$ and $z$. This does not require the probe itself to be coherent. The complete target gradient still contains $B_{Nk}$ and depends on $N$: equal diagonal responses do not imply equal changes of shape.

For the root or smooth law $\Phi_\sigma(r)=\mu(\sqrt{r^2+\sigma^2}-\sigma)$, linearize about a nonzero coherent target and a zero probe field. Since the probe current is quadratic in its amplitude, its complete linear source operator is

$$\boxed{\mathcal V_M(x)= \mu\,s_\sigma(A(x))\,\beta \sum_{k=0}^Mh_{k,d}B_{Mk}(P_{z(x),k}),\qquad s_\sigma(A)=\frac{A}{\sqrt{A^2+\sigma^2}}.} \tag{5}$$

At the positive root endpoint, $s_0(A)=1$ for every $A>0$. Equation (5) is then independent of both target degree and target density. Different nonconstant target profiles sharing the same local ray texture give the same full probe Dirac operator, including every probe coefficient and spinor component. This is stronger than equality of one expectation value. For a parallel slack target ray, its transverse probe occupation $j$ has source coefficient

$$\mu\,s_\sigma(A)\frac{D_M}{D_j}\,\beta, \qquad D_j=\binom{j+d}{d}. \tag{6}$$

The contraction identity used in (6) is the first sum in equation (14) of the self-mass note. Both the spatial kinetic operator and these coefficient sectors must be retained when computing a physical response.

The target responds at second order in the small probe amplitude. That backreaction comes from the same energy; fixing the target in the linear probe equation is the stated perturbative limit. It does not remove the reciprocal variation of the full action. Nor does a weak field perturbation automatically represent a separately normalized massive particle.

Root interaction and the vacuum limit

For positive coherent currents in two different grades, the contribution of a shared order is

$$\mu h_{k,d}\left[\sqrt{A^2+B^2+2AB\tau^k+\sigma^2}-\sigma\right]. \tag{7}$$

Unshared orders contain only their own grade's current. At $\sigma=0$, subtracting the two isolated energies gives a nonpositive cross term. At alignment it vanishes; at other orientations it generally remains. In particular, the order-zero root is additive for positive currents. It must not be confused with the quartic action's uniform order-zero cross term in (2).

The endpoint also has a physical limitation. Equation (5) remains nonzero as a positive target density tends to zero, if the probe amplitude is taken to zero first. Reversing those limits gives the isolated probe problem. A globally weak tail cannot simply be assigned the same physical meaning as a finite normalized particle while ignoring this noncommuting vacuum limit.

Ordinary rotations and orientation averages

An average over real spatial directions is different from a Haar average over complex mode rays. For real unit tangent directions in $d$ ordinary dimensions, $\tau=(z\cdot w)^2$ has beta distribution $\operatorname{Beta}(1/2,(d-1)/2)$. Hence

$$\langle\tau^k\rangle_{SO(d)} =\frac{(1/2)_k}{(d/2)_k}. \tag{8}$$

At depth six the means of $G_{12}$ and $G_{34}$ are approximately $59.40644$ and $1023.83613$. Their second moments are approximately $190632.19$ and $649947765.65$. The high-compatibility region matters strongly even though (3) converges away from it.

More generally, any polynomial $F_N(\tau)=\sum_{k=0}^Nc_{N,k}\tau^k$ with nonnegative coefficients summing to $D_N$ satisfies

$$\langle F_N\rangle_{SO(d)} \ge D_N\frac{(1/2)_N}{(d/2)_N}. \tag{9}$$

The moments in (8) decrease with $k$, proving the bound term by term. At fixed $d$, its right side grows as $N^{(d+1)/2}$. Thus fixed-overlap agreement does not imply a degree-independent average over all these ordinary rotations. Equation (9) has the stated positive- polynomial scope; it is not a bound on every geometric interaction.

Same-grade contributions retain interference

If two contributions occupy the same canonical grade, their sum is one field. It is incorrect to replace it by two independently phased carriers. For $\Psi=\chi_a z^N+\chi_b w^N$, define

$$t=\chi_b^\dagger\beta\chi_a,\qquad s=w^\dagger z,\qquad\tau=|s|^2.$$

The exact current is

$$\rho_k=AP_{z,k}+BP_{w,k} +t s^{N-k}|z^k\rangle\langle w^k|+\text{Hermitian conjugate}. \tag{10}$$

Direct multiplication gives

$$\begin{aligned} \operatorname{Tr}\rho_k^2={}&A^2+B^2+2AB\tau^k +2|t|^2\tau^{N-k}\\ &+2\operatorname{Re}(t^2s^{2N}) +4(A+B)\operatorname{Re}(ts^N). \end{aligned} \tag{11}$$

Summing (11) includes both $G_N(\tau)$ and $H_N(\tau)=\sum_kh_{k,d}\tau^{N-k}$, with $H_N(0)=h_{N,d}$. Averaging relative phase removes the explicitly oscillating terms but leaves the $|t|^2$ term. The canonical norm also contains the corresponding spinor Gram interference. These features concern field superposition, not particles nested inside other particles.

Reciprocal evolution and a small scalar self-source

A complete coefficient evolution at $d=6$, $N=12,34$, unit norms, $\kappa=1/D_{12}$ and initial $\tau=0.7$ tests the quartic action. Two active coefficient directions and one positive-$\beta$ spin line form an invariant subspace; every occupation within those directions evolves. It is a coefficient control, not a spatial particle encounter.

Over time $20$, the cross energy changes from $0.0344058623299$ to $0.0405001677154$. The first-current compatibility changes from $0.7$ to $0.7018348509519$, and the second-current compatibility from $0.49$ to $0.4989699515065$. The first-current purities finish at approximately $0.9896030421225$ and $0.9999999338726$: one pattern deforms substantially more than the other. Total energy, both norms and the total rotation moment are conserved to numerical accuracy. Tighter integration tolerances change the complete complex trajectory by at most $4.49\times10^{-8}$.

The root law at the same grades, norms, initial compatibility and duration, with $\mu=1/D_{12}$, gives a different but equally reciprocal evolution. Its first-current purities finish at $0.91769258$ and $0.9999999919$; the first- and second-current compatibilities reach $0.71452485$ and $0.51880053$. The cross energy changes from $-0.56192935$ to $-0.51481218$, with total energy and both norms conserved. Tighter time integration changes the complete complex pair trajectory by at most $4.46\times10^{-8}$. The light pattern again deforms more strongly. The evolution is not restricted to a prescribed changing overlap.

Small self-source requires a separate interpretation. At a scalar-null coherent spinor with positive canonical density $\rho$, the quartic source vanishes, but its occupied real derivative has a nonzero singular value $2\kappa D_N\rho$. Cancellation between spinor components therefore does not suppress every response. A small field norm, a small scalar balance and a small microscopic coefficient are different mechanisms.

The existing odd-depth scalar-null classes also annihilate every current in (1) when all coefficient spinors belong to the same class. They provide no self-binding or selected small nonzero mass, and a general external scalar source can drive complementary components. These checks do not yet identify the neutrino's electric or weak channels, or derive a suppression scale for the assigned depth-five states.

Verification

The later localized response calculation extends the comparison to solved spatial target profiles under one binding and self-mass action. A finite normalized lower-grade probe has identical cross energy, complete probe source and reciprocal translation force for targets with the same profile and different larger grades. Their full shape responses and frequency-proportional inertias still differ. The test does not identify the probe as an observed particle or settle a full encounter.

A different forward calculation uses the actual scalar master projection and balanced normal populations. Their stationary scalar contribution cancels while their moving polarization produces a common geometric response. It retains finite normal kinetic gaps, complete probe self-interaction and ordinary tangent rotations, without replacing the raw projection weights by shell weights.

graded_current_tower_checks.py checks complete reciprocal sources, same-grade interference, ordinary rotations, common probes and the scalar-null derivative. graded_current_actions_checks.py checks the full root and smooth probe operators and pair energies. graded_current_tower_flow.py evolves the stated coefficient control with all active occupations. graded_current_root_flow.py uses the same complete coefficient space for the root or smooth action. graded_current_flow_comparison.py checks both refined trajectories and all stated conserved quantities. The self-source, response and evolution all use the same canonical action.