IDOS Energy, Coherence, and Supporting Dynamics — Part 14

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$$F(Ra)-F(a),\quad R=\left(\frac{1+\delta}{1-\delta}\right)^{1/8},\quad a=\frac{R^{3/4}-1}{R^{7/4}-1}, \quad F(z)=35z^3-105z^4+126z^5-70z^6+15z^7.$$

At one-percent tolerance this fraction is only $0.00220323408$. This deterministic energy weighting is not a position probability or an observed linewidth. It shows that the rational support does not approximate the strict uniform-density clock on most of its energy. The coherent variable-action-density branch remains mathematically distinct and viable within its proved scope. A flatter core requires its own complete support, boundary and dynamical analysis.

Derivations and reproducible checks are preserved in local-support-geometry, advected-support-parent, covariant-count-support and covariant-support-cone under the current research run, together with principal/support-clock-uniformity.md. They distinguish inverse construction, exact branch identities, local linear consistency and unresolved selection.

Flatter Cores And The Cost Of Approximate Uniformity

The support density can be changed to $\rho=\rho_{\rm peak}/(1+t^k)^2$, $t=r^2/L^2$, with integer $k\ge4$. Retain the first-order support residuals with $v=(1+t)^{-k}$, $z_i=(x_i/L)(1+t)^{-k-1}$ and replace the density residual by $\rho-\rho_{\rm peak}F_k(v)$, where

$$F_k(v)=\left[\frac{v}{v+(1-v^{1/k})^k}\right]^2,\qquad 0<v\le1.$$

An even continuation through zero and constant continuation above $|v|=1$ is $C^2$ for every finite $k$. All residuals still vanish with their full first variation. Thus the canonical advected-support action retains its exact coherent carrier, complete energy and inertia, within its earlier positive-density and rest-cone limitations. This statement is about that action with variable local action density. Merely evaluating $h/\sigma_0$ does not make it a solution of the different constant-$\sigma_0$ clock law.

The complete norm, including the transition and exterior, is

$$S=\frac{\rho_{\rm peak}\pi^5L^{10}}{24k} B(5/k,2-5/k),\qquad \frac{\langle r^2\rangle}{L^2} =\frac{B(6/k,2-6/k)}{B(5/k,2-5/k)}.$$

One common $k,L,S$ fixes one common peak density across all active depths. Under the normalized energy integration measure, $y=\sqrt{\rho/\rho_{\rm peak}}$ has beta parameters $(2-a,a)$, $a=5/k$. For $k\ge5$, the optimal fraction within relative local-rate tolerance $\delta$ is the central-core fraction

$$f=I_x(a,2-a),\qquad x=1-\sqrt{\frac{1-\delta}{1+\delta}}.$$

This includes the inner transition region in the excluded fraction, not only the geometric exterior $r>L$. At one-percent tolerance, $k=64$ gives $f=0.74860322$. The minimum integer exponents for 90%, 99% and 99.9% are 174, 1803 and 18093, respectively. The neighboring threshold values were checked independently at high precision. For fixed tolerance, $1-f=-(5/k)(\log x+1-x)+O(k^{-2})$.

Sharpness has a dynamical cost in these variables: at $r=L$, $v=2^{-k}$ and $F_k'(v)=2^{k-1}$, giving exponentially large lower-order density-restoring coefficients. Bounding the principal cone alone misses this stiffness. A changed representation keeps the auxiliary exponent four and uses $F_k(v)=[1+(v^{-1/4}-1)^k]^{-2}$. Its boundary derivative is only $2k$, but fixed auxiliary kinetic coefficients no longer satisfy the old tail bound. Density-weighted coefficients proportional to $F_k(v)/v^2$ repair that product bound while becoming vacuum-degenerate. The canonical momentum energy then has a singular inverse kinetic coefficient. A full covariant, lower-order and vacuum analysis is still required.

These controls show how approximate uniformity can be approached without discarding boundary energy. They do not supply exact uniformity, occupied band selection or a persistent solution of the original local-clock law. The derivation and 25,246 checks are preserved in flat-core-support.

Exact Phase Integration For Prescribed Local Energy

For real standing modes at a point, define $U_{ij}=A_iA_j\omega_i\omega_j e_ie_j$ and $V_{ij}=A_iA_j\nabla e_i\cdot\nabla e_j$, with $\alpha_i=\omega_it+\phi_i$. The scalar quadratic energy is

$$h=\frac14\sum_{ij}[(U_{ij}+V_{ij})\cos(\alpha_i-\alpha_j) +(V_{ij}-U_{ij})\cos(\alpha_i+\alpha_j)].$$

A shared balanced real plane removes the sum-frequency terms but generally retains the difference beats. Grouping degenerate frequencies gives $h=h_\infty+\operatorname{Re}\sum_\nu C_\nu e^{i\nu t}$ and the exact prescribed local-clock phase

$$\theta(t)-\theta(0)=\frac1{\sigma_0} \left[h_\infty t+\operatorname{Re}\sum_{\nu\ne0} \frac{C_\nu(e^{i\nu t}-1)}{i\nu}\right].$$

The modulation indices are $|C_\nu|/(\sigma_0|\nu|)$. An independent 20-mode scalar control reproduces the substantial sidebands: with the tested clock-to-mode ratios one and ten, about 0.08787 and 0.009972 of windowed signal power lies within one percent of the prescribed center. Exact phase integration and time-step halving retain that scoped result. They do not certify every larger-band calculation or an observed linewidth.

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