Fixed Count Clocks Leave a Testable Curvature Constraint

The continuous and locally affine count interactions agree in value and first derivative at every eligible count node. They therefore produce the same stationary fields and frequencies. Their curvature differs, and the full spatial index test can distinguish that difference without changing the mass inputs.

1. One Common Local Interpolation

Let f_0 be the locally affine envelope and f_1 the continuous count function, both with the same diagnostic sector scale. On their common open constitutive domain define

$$f_\gamma=(1-\gamma)f_0+\gamma f_1,\qquad 0\le\gamma\le1. \tag{1}$$

At each eligible x_l, f_gamma and its first derivative agree with f_1. Its second derivative is gamma f_1 double prime. Hence all count clocks and stationary fields remain unchanged, while their exact real source derivative becomes

$$B_\gamma=H+\gamma N_1. \tag{2}$$

The complete positive-wave spatial stiffness is B_gamma squared minus omega squared, with the gyroscopic velocity term retained. Squaring a projected B_gamma is not substituted for that operator.

Equation (1) specifies one constitutive interpolation, not a separately adjusted correction for each particle. Its global extension and its physical derivation remain additional questions.

2. The Same Linear Response Has an Explicit Forward Form

The existing analytic count-energy correction has a common interpolation with precisely the same jets at every node. Put F_1(chi)=epsilon P_d(chi), c_1=F_1 prime and define

$$K_\gamma(\chi)= \left[\frac{\gamma}{F_1''(\chi)}+ \frac{(1-\gamma)\chi}{c_1(1-c_1^2)}\right]^{-1},$$ $$\widetilde F_\gamma(\chi)=F_1(\chi)+ \frac{K_\gamma(\chi)-F_1''(\chi)}{2\pi^2} \sin^2[\pi(\chi-d/2)]. \tag{3}$$

Work on chi>0, 0<c_1<1 and F_1 double prime positive. The denominator is positive there. At every integer shell h_l,

$$\widetilde F_\gamma(h_l)=F_1(h_l),\quad \widetilde F_\gamma'(h_l)=c_l,\quad \widetilde F_\gamma''(h_l)=K_\gamma(h_l)>0.$$

Eliminate chi locally from the preferred-time auxiliary potential rho F_gamma minus chi sqrt(rho squared-s squared). Its scalar curvature obeys

$$f_\gamma''(x_l)= \frac{h_l}{c_l^3}- \frac{x_l^2}{c_l^2K_\gamma(h_l)} =\gamma f_1''(x_l). \tag{4}$$

Thus (3) has the same stationary field and linearized operator (2). Gamma=1 returns the original continuous count function. Gamma=0 returns the existing explicit zero-curvature correction. These actions can differ beyond their second derivatives at the nodes. Global convexity of (3) between integer shells is not asserted.

3. A Pair Test Misses the Spatial Constraint

For the restricted polarization pair, the real derivative is omega-2lambda+gamma q_1 and the imaginary derivative is omega-2lambda. Its sufficient non-growth interval includes

$$0\le\gamma\le\gamma_{\rm pair},\qquad \gamma_{\rm pair}=\frac{2(\lambda-\omega)}{q_1}. \tag{5}$$

Every one of the fourteen maintained assignments has gamma_pair>1 under the shared diagnostic scales. Thus all pass the restricted pair test throughout the interpolation. The full spatial source test nevertheless proves:

The same outward interval method samples 1/16, 1/8, 1/4, 1/2, 3/4 and 1 at both fields. A failed sufficient test is not stability. Because the positive source increases monotonically with gamma, its positive Schur complement also increases. The negative-block gap has already been bounded at the largest gamma=1. Consequently the strange certificate proves growth throughout [1/2,1], and the up certificate proves it throughout [3/4,1], including the points between the samples.

These are sufficient growth intervals, not exact transition values. The index proof gives existence, with 0<sigma<=omega, rather than a precise multiplicity, lifetime or outgoing decay channel.

4. What This Constrains

The spatial test supplies information that the fitted frequencies and isolated pair do not supply. It identifies an allowed constitutive degree of freedom and places a mathematical constraint on it under the stated spatial preparation.

The zero-curvature envelope endpoint retains its separate nonlinear persistence theorem. The analytic zero-curvature endpoint in (3) has the same complete linearized positive-wave dynamics, but lacks the open affine chart needed to transfer that nonlinear proof. Neither endpoint selects occupied depths or derives physical support, common normalization or complete translational inertia.

All six occupied depths enter the unchanged-clock and pair comparison. Only the displayed depth-three and depth-four full-matrix fields receive full spatial certificates here. The result gives no special depth-five preference and is not a result for minimum-column carriers.

Reproduction

count-curvature-response.py and its report pass 1,098 conditions using the exact source-projection checker as a source-pinned dependency. All twelve interval certificates, all fourteen exact pair limits, rational Gram matrices and rational congruence transforms are retained. Equations (3) and (4) are the explicit algebraic relation to the earlier forward correction.