These results use the smoothed count action and its bounded-current completion. They concern classical fields in shared ordinary R10, with supplied count and depth assignments.
Energy Clustering from the Bounded Constitutive Functions
Let phi and chi be any two H1 fields. Translate chi by a vector R whose length tends to infinity, and allow any fixed relative phase. Write psi_R=phi+exp(i theta) chi(.-R). Then
H[psi_R] -> H[phi]+H[chi],
||psi_R||_2^2 -> ||phi||_2^2+||chi||_2^2.
The kinetic and norm cross terms tend to zero by the weak convergence of translations in L2. The nonlinear proof is especially transparent for the bounded-current action. First approximate both fields strongly in H1 by smooth compactly supported fields. The energy is continuous on H1, but continuity alone is not uniform over all R, so this approximation needs an additional estimate. The source growth bound gives
|H_nl[f]-H_nl[g]|
<= K (||f||_H1+||g||_H1) ||f-g||_H1.
It follows by integrating the bounded derivative along their connecting line. The constant is global for the fixed constitutive functions. Thus the compact approximation error is uniformly controlled as R varies.
For compact disjoint supports, rho_R is exactly the sum of the two densities and j_R is exactly the sum of their currents. Around either occupied region, the translated Gaussian-smoothed density and current tend to zero with all derivatives. The normalized local coefficients converge to those of that isolated pattern, since its own smoothed density is positive.
To control the intervening space, choose a fixed radius L around each support. Outside these two regions, the total convolved density has integral at most the sum of the two isolated Gaussian tails beyond L. Both nonlinear integrands are bounded by a constant times this density:
|u Z(C)|+|u gamma(C) F(|J/u|^2)| <= K u.
Dominated convergence on the two fixed translated neighborhoods, followed by L tending to infinity, proves nonlinear energy additivity. No relative frequency locking or special orientation is required. For the original density-only action the same proof uses its L2 derivative bound.
This is a limit theorem, not a finite-distance rate or a force calibration. It does not make the Gaussian interaction finite range. It does establish that infinitely separated finite-energy patterns do not retain a spurious global-moment interaction in these actions.
The Complete Nonlinear Source Also Clusters
Let B(psi)=F_0(psi)+F_I(psi) denote the non-Laplacian source, with F_I=0 for the original action. Then, for arbitrary fixed H1 profiles,
||B(phi+exp(i theta) chi(.-R))
-B(phi)-exp(i theta) B(chi)(.-R)||_2 -> 0.
Around either center, the other profile tends to zero locally in H1 and its Gaussian-smoothed density and current tend to zero with all fixed derivatives. Cross density and cross current terms tend to zero globally in L1: each is a product of two fixed L2 functions translated apart. Thus the normalized coefficients converge locally to their isolated values. A common positive mass ball centered on the chosen profile supplies the polynomial score bound needed for the multiplier's convolution. This argument is repeated in coordinates centered on the other profile.
Outside a radius L around both centers, the density is bounded by twice the sum of the two isolated densities. The weighted p>2 multiplier estimates make its scalar source tail arbitrarily small as L grows, uniformly in R. The transport tail is controlled by bounded A times the sum of the two gradient tails, and bounded div A times the density tails. Local convergence and these uniform tails prove the displayed L2 limit. At a zero isolated profile the assertion is immediate.
The estimates are uniform in the relative phase. Consequently the sum of two fixed rest Gaussians, each evolving at its own isolated frequency, has an L2 equation residual tending to zero with separation, uniformly on each bounded time interval. This verifies the unlike-frequency baseline without prescribing synchronization. A small residual is not yet a quantitative closeness theorem for an exact pair history: that would need an appropriate stability or continuous-dependence estimate for the flow.
A Constraint on Interpreting the Positive Gaussian Hessian
The action is homogeneous of degree two in the overall field amplitude. Suppose normalized rest profiles phi_h and phi_l have energies omega_h and omega_l with omega_l<omega_h. The common action contains both branches. For 0<epsilon<1, consider
psi_(epsilon,R)=sqrt(1-epsilon^2) phi_h
+epsilon phi_l(.-R),
and rescale by a scalar tending to one as R tends to infinity to keep the norm exactly one. By clustering,
H[psi_(epsilon,R)]-omega_h
-> epsilon^2(omega_l-omega_h)<0.
For each epsilon choose R sufficiently large that the finite-separation error is smaller than half this strict decrease. Meanwhile the H1 distance to phi_h is bounded by O(epsilon), uniformly in R, because the translated lower branch has fixed H1 norm. Hence a higher-frequency Gaussian cannot be a constrained local minimum in the unrestricted H1 topology whenever the same action admits a lower-frequency branch.
This does not contradict its nonnegative closed Gaussian relative-density Hessian. That form controls its own weighted perturbation domain; it is not a proved uniform second-order expansion in the ordinary H1 topology. Adding a tiny remote profile probes a different tail limit. The nonlinear energy is C1, and a Frechet C2 theorem in H1 was never established.
Nor does this argument prove dynamical decay or even nonlinear orbital instability. It constructs lower-energy nearby states, not a trajectory connecting them, a transfer rate, or a lifetime. Finite-lived branches remain permitted. The useful next dynamical question is whether the actual equation can move field norm into those remote or outgoing components on a relevant timescale while conserving complete energy and momentum.
At rest the added current energy vanishes on each real isolated Gaussian, so these statements apply to both smoothed actions. The result is common to all maintained depths and does not by itself single out depth five or identify an observed particle. Counts and positive sector units retain their status as supplied inverse-construction inputs.
Equal-Profile Separation in a Physical Direction
Let phi_B be a real target Gaussian with norm one, and let n be an ordinary unit vector in R10. Normalize the sum phi_B(x-hn)+phi_B(x+hn) to norm one. The centers are separated by 2h. At h=0 it is the single normalized branch. The relative-density perturbation is
delta rho/rho=h^2[(n^T B x)^2-(n^T B n)/2]+O(h^4).
Gaussian filtering maps this to the analogous quadratic with B replaced by C, and therefore
delta C=-h^2(Cn)(Cn)^T+O(h^4).
The real field has zero current. Thus its current correction is exactly zero, and the result applies to both the original smoothed action and the saturated-current action. The kinetic-plus-compensation density Hessian vanishes for quadratic relative-density disturbances. Only the positive count Hessian T contributes at this order:
H_pair/N=omega+(h^4/2) T[(Cn)(Cn)^T,(Cn)(Cn)^T]+O(h^6).
The count-frequency term itself has zero first and second descriptor jets; its third jet can enter the remainder. For high counts the range where the quartic term dominates may be small. This expansion is at fixed one-profile norm and is not the distant sum of two unit-norm particles.
Broad and Narrow Directions Enter the Same Formula
Write C=aP+bQ, rank P=d, a=b/(k+2). For d<10 put q=|Qn|^2 in [0,1]. The broad-direction fraction is 1-q. Define
A=2 Lambda pi^2 b^2/(d^2 a^4),
Z=8 Lambda pi^2/(a-b)^2, R=2 eta(a-b)^2,
U=R a^4(1-1/d), W=[Z+R(1+1/d)] b^4, L=a^2+b^2.
The exact quartic coefficient is
K4(q)=(1/2){A[a^2-Lq]^2+U(1-q)^2+Wq^2}. (1)
It is strictly positive. The omitted cross projector block is a rotation zero direction of the count Hessian, rather than an omitted elastic penalty. Equation (1) includes all block-diagonal contributions of the rank-one matrix (Cn)(Cn)^T.
The softest direction is generally slightly tilted out of the broad plane:
q_min=(A L a^2+U)/(A L^2+U+W), (2)
K4_min=(1/2)[A a^4+U
-(A L a^2+U)^2/(A L^2+U+W)]. (3)
For positive penalties, 0<q_min<1. The small tilt partly cancels the change in the width-index descriptor, leaving the depth and residual-matrix penalties. This is a consequence of the common geometric action, not an orientation acceptance factor inserted afterwards.
At fixed d and large k, q_min is asymptotic to a^2/b^2=1/(k+2)^2, so the angle out of the broad plane is asymptotic to 1/(k+2) radians. The ratio of the minimum curvature to the purely broad-direction curvature tends to zero like a^4, with constants depending on the common penalties and d. At d=10 there is no Q direction, so only q=0 is physically available. One must not minimize over a nonexistent complement at depth ten.
The pure-direction values are K4(0)=[A a^4+U]/2 and K4(1)=[A b^4+W]/2. Consequently the action already predicts a substantial contrast between response along a pattern's broad and narrow directions. Depth five follows the same formula as its neighbors; this does not derive the neutrino's weak electromagnetic response. Likewise, quartic energy curvature does not by itself supply an observed cross section or an electric charge. It is a concrete orientation-dependent consequence to test further.
Checks and Scope
The accompanying exact rational comparison fixes b=eta=1 and Lambda pi^2=1 for all branches. These are diagnostic shared units, not a particle-dependent adjustment. Equations (1)–(3) are checked for all fourteen maintained count labels at depths 2,3,4,5,6,10, respecting depth ten's absent complement. The mathematical input is the complete count Hessian, already given in the smoothed count action. This derivation does not add an occupied-branch selection mechanism or derive the supplied sector units.
The exact orientation checker passes 66 rational assertions across the fourteen maintained labels. With the shared diagnostic units, the electron and muon controls have minimum-to-broad-direction quartic curvature ratios about 0.00587175 and 0.000131004. Their minimizing tilts are about 0.0711213 and 0.0277691 radians. These are profile-energy coefficients, not measured interaction rates. At depth ten the complementary directions are absent and no such tilt is available.