One positive spatial covariance identity supplies a changed inverse action with exact count-clock states and their complete relativistic dispersion. The smoothed count construction supplies the explicit common potential V_count, its finite-band cutoff chi and its Gaussian Hessian. Its supplied depth units and count function remain inputs. This action removes the independent kinetic coefficient and current-speed saturation of the bounded-current action. It has a C1 energy on H1, but its general evolution theorem remains open. The earlier action's global-history theorem is not transferred to it.
All positions are in ordinary shared R10. For a rank-d orthogonal projector P and Q=I-P, the supplied branches have filtered curvature C*=b P/(k+2)+b Q, smoothing variance sigma^2=1/(4b), and physical Gaussian width matrix B=2b P/(2k+3)+2b Q. Their clock is omega=mu_d binomial(k+d,d). The maintained fourteen labels are (2,75), (2,80), (2,94), (3,0), (3,3), (4,2), (4,19), (4,71), (5,9), (5,14), (5,21), (6,12), (6,34), (10,22). The width label k is not a spatial Hermite excitation number. Classical field profiles and normalized spatial integrals do not replace literal six-dimensional electron trajectories or supply a Born probability rule.
A Positive Geometric Remainder
Let psi be a nonzero H1 classical field on ordinary R10. Define
rho=|psi|^2, u=G_sigma*rho, C=-D^2 log(u)/2,
t=(G_sigma*|grad psi|^2)/u, r=grad log(u),
tau=t-|r|^2/4,
v=(G_sigma*Im(conjugate(psi) grad psi))/u.
At each convolution point x, use the normalized integration weight G_sigma(x-y)rho(y)/u(x). This is an estimate of the classical spatial field, not an electron-position probability postulate. Its mean m and positive-definite covariance Sigma obey
m=x+sigma^2 grad log(u),
Sigma=sigma^2 I-2sigma^4 C.
The covariance is positive definite because a nonzero integrable density cannot be supported on an affine hyperplane. Hence every physical C has eigenvalues strictly less than 1/(2sigma^2), although negative eigenvalues are possible. Define
k(C)=sigma^2 tr[C^2(I-2sigma^2 C)^(-1)].
Write a(y)=grad psi(y)/psi(y) where psi is nonzero, taking zero on the zero set. In the weighted integrals, Re a=grad log(rho)/2 and Im a=j/rho. Integration by parts in y gives
E[Re a]=r/2,
Cov(y,Re a)=-sigma^2 C.
Subtract the best affine part of Re a:
e(y)=Re a-r/2+sigma^2 C Sigma^(-1)(y-m).
It has zero mean and zero covariance with y. Orthogonal expansion of the ordinary spatial integrals then gives the exact identity
tau-k(C)=|v|^2+E|e|^2+E|Im a-v|^2 >= |v|^2. (1)
This is a covariance/completed-square identity, not an assumed uncertainty principle. Its integrations by parts are legitimate for H1 fields after Gaussian cutoffs: grad rho is L1 and Gaussian polynomial moments are finite. In particular k(C)<=tau<=t pointwise.
For a Gaussian psi=exp(i p.x) phi_B with B positive, C=B(I+2sigma^2 B)^(-1) and both residual terms vanish. Therefore
tau-k(C)=|p|^2 (2)
at every x, not only after integration. The conditional variance term is what distinguishes a coherent moving Gaussian from a rapidly oscillating two-phase superposition.
One Smooth Finite-Band Energy
Use the existing compact matrix cutoff chi, equal to one near every target C*, with support strictly below the inverse pole. Define the smooth bounded functions
k_tilde(C)=chi(C) k(C),
V_tilde(C)=V0+chi(C)[V_count(C)-V0], V0>0.
Outside the supported descriptor domain the definitions are k_tilde=0 and V_tilde=V0. The count potential is positive on the compact support, so V_tilde has a strictly positive lower bound and bounded derivatives. For every actual field, (1) implies
s=tau-k_tilde(C) >= 0.
Take the new canonical first-order-in-time energy
E_cv[psi]=integral u sqrt(V_tilde(C)^2+c^2 s) dx. (3)
It is finite for every H1 field:
V_min N <= E_cv <= V_max N+c sqrt(N)||grad psi||_2.
No small-b condition is needed to keep the radicand positive: its positivity comes from (1), not from separately requiring V^2-c^2 k_tilde>0. There is no independent alpha kinetic term and no phase-speed saturation. The count potential and finite-band cutoff remain explicit inverse constitutive inputs.
Full Variation
Put f=sqrt(V_tilde^2+c^2(tau-k_tilde)),
a_f=c^2/(2f),
f_C=(2 V_tilde V_tilde,C-c^2 k_tilde,C)/(2f),
B_f=f-a_f tau+(a_f/4)|r|^2.
Direct differentiation gives
delta E_cv=integral B_f delta u
-(a_f/2) r dot grad(delta u)
+a_f delta(G*|grad psi|^2)+u f_C:delta C.
The full source is consequently
delta E_cv/delta conjugate(psi)
=-div[(G*a_f) grad psi]+psi T_f, (4)
T_f=G*[B_f+(1/2)div(a_f r)
-(1/(2u))partial_ij(u f_Cij)].
As in the bounded-current proof, the last two derivative expressions should be interpreted through their direct Gaussian kernel form on general H1 fields. The derivatives then act on kernels, not on the quotient t. The coefficients a_f and f_C are bounded because f>=V_min, and the only new unbounded density factor grows like sqrt(tau). Its weighted square integral is controlled by integral u tau<=||grad psi||^2.
The continuity and tail argument below establishes an H1-to-H-minus-one Euler derivative and a real C1 energy using strong H1 convergence. The ordinary positive alpha kinetic coercivity of the prior action is absent, so its global H1 evolution proof does not transfer.
Exact Complete Moving Gaussian Branches
At a target, V_count=omega and V_count,C=0. Let Omega=sqrt(omega^2+c^2|p|^2). Then f=Omega, a_f=c^2/(2Omega) and f_C=-a_f k_C are constants along the Gaussian. The identity
integral u tau-integral u k(C)
=||grad psi||^2-(1/2)integral u tr[C(I-2sigma^2 C)^(-1)]
follows from integral u |grad log(u)|^2/4=(1/2)integral u tr C. Its right side is exactly the old kinetic-plus-Gaussian-compensation functional with coefficient one. At a real Gaussian its complete Euler derivative vanishes; at exp(i p.x) phi_B it is p^2 psi-2i exp(i p.x) p dot grad(phi_B).
Using this in the first variation of (3) yields the exact history
psi(t,x)=exp(i p.x-i Omega t) phi_B(x-X0-c^2 p t/Omega),
E_cv=N Omega, P=Np,
speed=c^2|p|/Omega<c,
M_inertial at p=0=N omega/c^2. (5)
Thus E_cv^2=c^2|P|^2+(N omega)^2 on every one of these complete branches. This holds for every supplied count branch under one action, without a low-momentum cutoff or particle-dependent amplitude choice. The overall norm N remains free. An exact relativistic branch dispersion is not a proof of Lorentz invariance of the full equation: the Gaussian spatial kernel selects a time slicing, and no Lorentz contraction or finite-speed propagation theorem for arbitrary disturbances has been derived.
Complete Energy--Momentum Inequality
The covariance identity also controls every H1 field, beyond the exact Gaussian family. Let V_min be the positive global lower bound of V_tilde. Since s>=|v|^2, the triangle inequality for vectors (V_tilde,c v) gives
E_cv >= integral u sqrt(V_tilde^2+c^2 |v|^2)
>= sqrt[(integral u V_tilde)^2+c^2 |integral u v|^2]
>= sqrt[(N V_min)^2+c^2 |P|^2],
P=integral Im(conjugate(psi) grad psi)=integral u v.
The momentum identity uses the normalized Gaussian kernel. Thus the complete energy bounds canonical momentum for arbitrary finite-energy H1 fields; this is not restricted to fitting the dispersion of selected branches. It still does not control the entire H1 norm.
If a target has omega=V_min, its moving Gaussian realizes equality and is a global energy minimizer at prescribed N and P. That last identification requires the stated global potential inequality; it is not inferred from local stationarity of an arbitrary supplied target. Equality in the first step removes the two covariance residuals and selects a uniform-phase- gradient Gaussian. Higher-clock branches have no such ground-state claim.
Rest Hessian Calculation
At a real Gaussian, s=0 pointwise and s>=0 for every nearby field, while the cutoff is identically one near its constant C*. Therefore its first variation delta s vanishes pointwise for compact field variations. Also delta V=0 at the target. The negative square-root second derivative is multiplied by these vanishing first variations and contributes nothing.
The complete rest Hessian is consequently the old smoothed count Hessian with alpha replaced, on that branch, by c^2/(2omega). Its density form is nonnegative with the same symmetry kernel, and its phase Hessian is 2[c^2/(2omega)] integral rho |grad theta|^2. The completed-square identity proves the pointwise first-variation statement; the checks below corroborate it. The Hessian is interpreted in the same closed Gaussian form domain as the parent construction.
Verification and Regularity
Completed First Checks
The independent symbolic calculation checks delta(tau-k)=0 for all 28 two-variable monomials through degree six at an anisotropic Gaussian. The full variation (4), including its variable kinetic coefficient, passes directional finite differences on positive, nodal and rapidly oscillating periodic profiles at two resolutions. The worst relative variation error is 1.17e-10 and the worst density-flux residual is 5.56e-13. The covariance remainder is nonnegative in all controls. The fourteen supplied count branches pass the exact energy-momentum and speed inequalities at four momentum ratios each. See the variation and branch checker and its dated report. The periodic check tests the algebra with a smooth positive toy V and the untruncated k on its sampled domain; it is not an R10 evolution experiment.
C1 Energy on H1
The regularity statement follows from the following estimates. Split the scalar multiplier in (4) into the bounded-coefficient Gaussian-score terms and T0=G*(f-a_f tau). For a fixed constitutive law,
|f-a_f tau| <= K(1+sqrt(tau)),
integral rho |T0|^2 <= K(N+||grad psi||^2).
The other multiplier terms have all finite weighted score moments exactly as before. The principal derivative coefficient D=G*a_f is bounded, so
||DE_cv(psi)||_(H-minus-one) <= K||psi||_H1.
To prove continuity, let psi_n->psi strongly in H1, psi nonzero. Both u_n and e_n=G*|grad psi_n|^2 converge strongly in L1 and locally with every fixed Gaussian derivative. Therefore all normalized coefficients converge locally. The scalar integrand f-a_f tau grows at most like 1+sqrt(e_n/u_n); a common positive mass ball gives an exponential upper bound exp(|x|^2/(4sigma^2)+K|x|), integrable against translated Gaussian kernels on compact target sets. Thus T0_n converges locally uniformly.
The necessary tail control does not require higher kinetic moments. For a tail set A in the physical y variable, Jensen and Fubini give
integral_A rho_n |T0_n|^2
<= K integral (u_n+e_n) w_(n,A),
w_(n,A)(x)=[G*(rho_n 1_A)](x)/u_n(x), 0<=w_(n,A)<=1.
Outside a fixed x ball, tightness of both u_n and e_n makes this small. Inside it, u_n has a common positive lower bound and the numerator is at most ||G||_infinity integral_A rho_n, uniformly small as A moves outward. This proves strong L2 convergence of psi_n T0_n. The remaining scalar source terms use their stronger p>2 score bounds. Bounded local convergence of D_n and strong convergence of grad psi_n give D_n grad psi_n->D grad psi in L2, hence the divergence terms converge in H-minus-one. At zero the displayed derivative bound suffices.
The direct variation along an H1 line has integrable density, curvature and kinetic terms bounded by K||psi||_H1||h||_H1. Local differentiation, spatial truncation and this bound give its line-integral identity. The continuous derivative then establishes real Frechet C1 regularity on H1. This is the natural variational domain of (3), including physical nodes. It is not a Cauchy theorem.
Why the Previous Global Proof Does Not Transfer
For a unit-norm target phi, set psi_p=p^(-1/2)exp(i p x1)phi. Its exact branch energy tends to c while ||grad psi_p||^2 grows like p. Thus bounded energy and bounded norm do not control H1 for this relativistic-dispersion action. This is a concrete loss of the old coercive estimate, not merely a missing technical citation. The appropriate energy topology or an additional evolution estimate must be derived. No global evolution or uniqueness result is claimed for this changed action.
Related Gaussian Mathematics
The Gaussian weighted-mean identity is the multivariate version of the classical Tweedie identity; see Bradley Efron, Tweedie's Formula and Selection Bias (2011), Stanford report record. Its covariance form follows by a second differentiation, explicitly given above. The residual identity (1) is derived here by completing a covariance square. This use of normalized classical field integrals does not import the source's statistical physical interpretation into IDWT. The linked record and indexed formula were consulted; the full paper was not retrieved.
Equality Selects a Gaussian Shape
For the uncut covariance defect R=tau-k(C), the completed-square identity is
R=|v|^2+E|e|^2+E|Im(grad psi/psi)-v|^2.
If R=0 at one convolution point, every nonnegative term vanishes. Since the Gaussian kernel is strictly positive throughout ordinary space, the physical phase gradient is zero almost everywhere on the field's support, and its amplitude logarithmic gradient is affine in y. Equivalently the field obeys the distributional first-order system
grad psi=(b-B y)psi
with real b and symmetric B. Its nonzero integrable solutions are precisely constant-phase Gaussians with B positive definite. This formulation also excludes separate arbitrary phase choices across nodal components: the H1 field itself solves the global first-order system, whose integrating factor is constant on connected ordinary space. A nonzero solution cannot acquire an internal nodal boundary.
More generally R=|v|^2 with zero residual squares gives a Gaussian with constant spatial phase gradient, exp(i p.y) phi_B. Thus the coherent moving Gaussian is the equality class of a positive spatial integral identity, not merely a shape guessed for an ansatz calculation.
For the cut defect s=tau-chi k=R+(1-chi)k, s=0 additionally requires chi=1 on a nonzero Gaussian, since k>0 there. If a common positive potential is chosen with V_tilde>=omega_min and equality on selected lowest-clock target matrices, then E_cv>=N omega_min. Equality forces exactly those rest Gaussian shapes. This last potential inequality is an additional global constitutive condition; the present note does not assume that arbitrary penalty choices already satisfy it. Higher-frequency stationary branches need not be global or local energy minima.
Small Angular-Velocity Response
At a rest Gaussian the covariance action has the same phase quadratic energy as the smoothed action with alpha_*=c^2/(2omega):
E_phase=alpha_* integral rho |grad theta|^2.
Consider rotation in a physical coordinate plane (i,j), with density widths B_i and B_j. Define angular velocity Omega by x_i'=-Omega x_j, x_j'=Omega x_i. The rotating Gaussian density has
rho_t/rho=-2 Omega(B_i-B_j) x_i x_j.
The linearized continuity equation is rho_t=-2alpha_* div(rho grad theta). The phase
theta=-Omega(B_i-B_j)x_i x_j/[2alpha_*(B_i+B_j)]
solves this equation. Its angular momentum and energy are
L_ij=integral rho(x_i partial_j theta-x_j partial_i theta)
=I_ij Omega,
E_phase=(1/2) I_ij Omega^2,
I_ij=N(B_i-B_j)^2/[4alpha_* B_i B_j(B_i+B_j)]
=N omega(B_i-B_j)^2/[2c^2 B_i B_j(B_i+B_j)].
This is a positive complete quadratic response for reorienting an anisotropic density. Equal widths give zero because rotating an isotropic density does not change it. It is not an independent negative-mass or spin calculation. The phase is the minimum kinetic-energy solution for the prescribed density derivative, modulo a constant, by the weighted gradient variational principle.
For a broad/complement pair of directions in a retained branch,
B_P=2b/(2k+3), B_Q=2b,
I_PQ/M_translation=(k+1)^2/[2b(k+2)],
M_translation=N omega/c^2.
Thus the same internal geometry and selected clock fix both translation inertia and this orientation response, without a second particle-specific inertial coefficient. There are d(10-d) such orientation tangent directions. At depth ten the Gaussian is isotropic and these tangents are absent; this does not establish an absence of angular-momentum-carrying deformations or identify a tau candidate's observed spin.
These are linearized continuity and quadratic-energy results. Finite rotation generates shape changes at higher order, and an exact rotating relative equilibrium has not been constructed. No quantization, electric charge or electron orbital trajectory follows merely from this inertia formula.
Universal Integrated Transport Bounds
Use rho=|psi|^2, u=G*rho, j=Im(conjugate(psi) grad psi), s=tau-chi k, f=sqrt(V_tilde^2+c^2 s), and N=integral rho. The completed-square identity implies the stronger pointwise inequality
s >= E_x |Im(grad psi/psi)|^2,
because its remaining amplitude-residual square and cutoff remainder are nonnegative. Define h(x) as the expectation on the right. Then G*|j| <= u sqrt(h) and f^2>=V_min^2+c^2 h.
Density Flux and Its Complete Bound
The full Euler equation has real principal coefficient D=G*[c^2/(2f)] and real scalar multiplier. Therefore its exact density balance, whenever the equation is realized with sufficient regularity, is
rho_t+div F=0, F=2D j=c^2 j [G*(1/f)].
The normalized Gaussian is symmetric and positive. Fubini and the preceding pointwise inequalities give
integral |F| <= c^2 integral (G*|j|)/f
<= c^2 integral u sqrt(h)/f
<= c integral u sqrt(1-V_min^2/f^2)
<= c N sqrt[1-(N V_min/E_cv)^2] < cN. (1)
For the last step, first use concavity of the square root with weight u/N, then Jensen's inequality for 1/f^2 and E_cv=integral u f. The strict bound uses finite energy and V_min>0. The inequalities themselves hold for every H1 field, independently of whether a generic evolution theorem is known.
For any one-Lipschitz spatial test function g, a history satisfying the density balance consequently obeys
|d/dt integral g rho| <= integral |F| <= cN.
With a finite first moment, its center X=(integral x rho)/N therefore has |X_dot|<c, and the sharper energy-dependent bound in (1) applies as well. This is a universal transport consequence, not only a property of the selected Gaussian branch. If a moving target has omega=V_min, it saturates the sharper bound. The common count potential still supplies that omega.
Spatial Moments
For M2=integral |y|^2 rho(y), weighted conditional Cauchy--Schwarz gives
integral |y| |F(y)| dy
<= c^2 integral u E_x[|y| |Im(grad psi/psi)|]/f
<= c integral u sqrt(E_x|y|^2)
<= c sqrt(N M2).
Thus |M2_dot|<=2c sqrt(N M2), and the root-mean-square distance from any fixed origin grows no faster than c. Truncated moment functions justify the bound before assuming the full moment exists throughout a history. Likewise spatial cutoffs give a tail-mass estimate with rate cN/R. These estimates can assist a future compactness construction; they do not supply the missing global control of spatial gradients.
There is no pointwise assertion |F(y)|<=c rho(y). The convolution and the local phase gradient are evaluated at different points. The integral transport estimate therefore does not prove a light cone for disturbances, finite propagation, or Lorentz invariance. A general Cauchy theorem remains open. The exact Gaussian histories already constructed satisfy these laws.
The transport checker verifies the energy--momentum and integrated-flux inequalities on four periodic controls at two resolutions. These finite controls corroborate the analytic argument; they do not establish a general Cauchy theorem. No special depth-five suppression follows from this common law.