Count Clocks and Spatial Response in One Explicit Constitutive Formula

The forward variational form also makes the response adjustment explicit. It is possible to preserve the count clocks and stationary spatial fields while changing the constitutive curvature with one common analytic expression. This provides an alternative to evaluating a correction through an implicit inverse function.

1. Correct the Count Energy Directly

Retain h_l=l+d/2, epsilon=mu_d R and

$$F_0(\chi)=\epsilon P_d(\chi),\qquad c_0(\chi)=F_0'(\chi).$$

Define, on chi>0,

$$A(\chi)=\frac{c_0(\chi)[1-c_0(\chi)^2]}{\chi}-F_0''(\chi),$$ $$\boxed{F(\chi)=F_0(\chi)+\frac{A(\chi)}{2\pi^2} \sin^2[\pi(\chi-d/2)].} \tag{1}$$

At every integer shell in 0<c_l=F_0'(h_l)<1,

$$F(h_l)=F_0(h_l),\quad F'(h_l)=c_l,\quad F''(h_l)=\frac{c_l(1-c_l^2)}{h_l}>0. \tag{2}$$

The sine factor and its first derivative vanish there; its second derivative equals 2 pi squared. Equations (2) follow immediately. The positive second derivative establishes a convex elimination chart in an open neighborhood of each maintained shell. Global convexity between all shells is not asserted.

Use the explicit auxiliary potential

$$V=\frac1R\left[qF(\chi)-\chi\sqrt{q^2-s^2}\right]. \tag{3}$$

Here q can be the preferred-time density rho, the proper current norm n, or the responding-background contraction u dot j, as specified below. These choices define different actions. The auxiliary equation is F'(chi)=sqrt(q squared-s squared)/q. On each stationary balanced spatial pair, variation still sets chi=h_l, so the clock, scalar-current ratio, complete source and stationary energy retain their original values. The correction is an explicit constitutive choice, not a new derivation of the counting interaction's physical origin.

2. The Complete Eliminated Invariant Hessian

Set x=s/q, c=sqrt(1-x squared), and eliminate chi locally. Write R V=q f(x). The resulting scalar constitutive curvature is

$$f''(x)=\frac{\chi}{c^3}-\frac{x^2}{c^2F''(\chi)}. \tag{4}$$

Substituting (2) gives f''(x_l)=0. This also follows without explicitly constructing f. At the stationary point the direct potential derivatives in invariant coordinates (q,s) are

$$R V_{(q,s)(q,s)}=\frac{h_l}{q c_l^3} \begin{pmatrix}x_l^2&-x_l\\-x_l&1\end{pmatrix},$$ $$R V_{(q,s)\chi}=\begin{pmatrix}-x_l^2/c_l\\x_l/c_l\end{pmatrix}, \qquad R V_{\chi\chi}=qF''(h_l).$$

The Schur complement subtracts V_(q,s)chi times V_chichi inverse times V_chi(q,s). With the curvature in (2), it cancels the entire displayed two-by-two Hessian. Thus the eliminated interaction has the same value, gradient and invariant Hessian as the affine expression m_l s+eta_l q at each selected shell. This statement concerns the interaction's second derivatives at the state, rather than an open affine neighborhood.

3. What Each Density Realization Establishes

For q=rho, the complete co-rotating field linearization is

$$\mathcal A_l=D_R+\frac{h_lx_l}{Rc_l}\beta -\frac{h_l}{Rc_l}I.$$

It is self-adjoint on the full ordinary sphere Dirac domain. Its spectrum is

$$\frac1R\left[\pm\sqrt{(m+d/2)^2+(h_lx_l/c_l)^2}-h_l/c_l\right], \qquad m=0,1,2,\ldots.$$

The full linear evolution is unitary for every maintained assignment. This applies to all spatial angular representations and matrix-column disturbances; the operator argument is not limited to sampled harmonics. The price of this realization is its chosen time density, as in the existing persistence construction.

For q=n, the proper-current derivatives remain and add vector-current terms. The preceding unitary conclusion does not transfer. For q=u dot j with u supplied by a retained same-field timelike matrix block, the complete local Hessian is the responding-clock Hessian in the covariant calculation. Its existing ground linear results apply only with their stated load and background preparations. No full excited-shell persistence is established here.

4. Linear Persistence and an Affine Neighborhood Are Different

Formula (1) is analytic away from chi=0 and has nonzero cubic response after elimination at every tested assignment. It therefore is not the open affine patch used in the earlier nonlinear persistence proof. The stationary fields and their linear Hessians agree, while their nonlinear disturbance equations differ. The reduced pair disturbance frequency is 2 h_l/(R c_l), agreeing with the massive pair gap; this is the frequency of disturbances to the internal shape, distinct from the particle's unchanged mass-frequency mu_d S_l.

Verification

forward-count-response.py and its report pass 281 evaluations across all fourteen assignments. Independent 110-digit derivatives of the complete three-variable potential verify the eliminated Schur Hessian, rather than assuming the desired inverse curvature. Separate massive-Dirac pair diagonalizations agree with the full spectral formula. The script retains a nonzero cubic derivative to prevent an unsupported transfer of the nonlinear affine-patch result.

The calculation joins exact clocks and a chosen spatial response in one explicit action. Its mathematical result is conditional on that action, the ordinary sphere and the stated density realization. It supplies no new preferred sector assignment, localized particle, observed charge or complete translational inertia.