IDOS Energy, Coherence, and Supporting Dynamics — Part 11

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Clock-rate modulation also needs its actual phase spectrum. If $\dot\theta=\Omega_0+a\cos\nu t$, then $\theta=\Omega_0t+(a/\nu)\sin\nu t$ up to a constant. The real quadrature observable encoded by $e^{i\theta}$ has sharp sidebands $\Omega_0+n\nu$ with powers $J_n(a/\nu)^2$. Its nonzero instantaneous rate variance $a^2/2$ equals the second moment of the complete line comb, not the intrinsic width of an individual line. Sideband power depends on $a/\nu$, not merely the fractional rate RMS. A finite observation adds window broadening, and a physical detector requires its own coupling map. Precision of a fitted mass value is not automatically an intrinsic-width bound. Deterministic modulation can still fail a specified coherence or sideband test; such a failure must be calculated rather than inferred from the rate RMS alone.

For an explicitly added Gaussian effective rate noise with covariance $C(t)=\sigma_\Omega^2e^{-|t|/\tau_c}$, direct integration gives

$$\operatorname{Var}[\theta(t)-\Omega_0t] =2\sigma_\Omega^2[\tau_ct-\tau_c^2(1-e^{-t/\tau_c})].$$

The fast-correlation regime $\sigma_\Omega\tau_c\ll1$ has approximate Lorentzian FWHM $2\sigma_\Omega^2\tau_c$. The slow regime has Gaussian ensemble broadening; a short individual record can instead show an almost constant frequency offset. Neither regime turns particle survival directly into a phase-coherence bound without an additional physical bridge. This conditional noise comparison is not a probability law for electron position.

The strong flat balanced target remains a sufficient route to constant local participating energy and one coherent carrier. Deterministic nonflat controls must be judged by their complete same-field energy, integrated phase and observable sidebands. Independent mathematical audits are preserved in thermal-band-dynamics-audit and clock-linewidth-audit under the current run; they retain the useful numerical comparison while narrowing its necessity and exclusion claims.

A Uniform Clock And Its Ambient Boundary

The strict constant-density local-clock branch has a simple global compatibility condition. Let $h\ge0$ be continuous and integrable on ordinary connected $\mathbb R^{10}$. If one nonzero frequency satisfies $h(x)=\sigma_0|\Omega|$ at every point where $h>0$, then $h$ takes only the values zero and one positive constant. Continuity on connected space forces a constant function; integrability excludes the positive choice. Thus the only such full ambient density is zero. This statement concerns the simultaneous strict assumptions, not every possible local clock.

An exactly clocked core with a differently governed boundary is a distinct admissible branch. If its volume is $V_C$, complete energy accounting gives

$$\frac{E_{\rm total}}{|\Omega|} =\sigma_0V_C+\frac{E_{\rm other}}{|\Omega|}.$$

Here $E_{\rm other}$ includes every exterior and support contribution not already in the core density. Setting $\sigma_0V_C=A$ and then adding positive boundary energy does not preserve $E_{\rm total}=A|\Omega|$. A common derived correction may instead enter the action budget explicitly. Intrinsic surface density, a smooth ambient thickness completion and background-subtracted energy are also different quantities requiring their own action and measure. A homogeneous surface is not the only possible local-clock repair: the canonical current branch derives $h=\omega\rho(x)$ with a uniform actual $\omega$ and variable local action density $\rho$. It changes the constant-$\sigma_0$ premise.

A nonuniform exterior does not automatically destroy the weighted signal within one cycle. For a normalized nonnegative classical signal weight $w$, let an exactly clocked core carry fraction $1-\epsilon$ and define $C_w(t)=\int w e^{i[\Omega(x)-\Omega_*]t}dx$. Triangle inequality gives $|C_w(t)|\ge\max(0,1-2\epsilon)$ for all times. With relative core rate error at most $\delta$, the mean squared quadrature mismatch obeys

$$\int w|e^{i[\Omega(x)-\Omega_*]t}-1|^2dx \le(1-\epsilon)\min\{4,(|\Omega_*|T\delta)^2\}+4\epsilon, \qquad |t|\le T.$$

This is a phase diagnostic with a declared weight, not a position probability law, linewidth or lifetime. A detector may weight the boundary quite differently from its share of total energy.

Thin transition regions need complete gradient accounting. For a radial amplitude falling from $a$ at radius $R$ to zero at $R+\ell$, suppose the energy contains $\kappa|\nabla f|^2/2$ and other transition terms are nonnegative. Cauchy--Schwarz gives

$$\frac{E_T}{E_C}\ge\frac{n\xi^2}{R^nI_n},\qquad I_n=\int_R^{R+\ell}r^{1-n}dr,\qquad \xi^2=\frac{\kappa a^2}{2h_0}.$$

The ratio grows as $n\xi^2/(R\ell)$ for a thin layer. For full ambient localization use $n=10$ at every active depth. At $\xi/R=0.01$ and $\ell/R=0.001$, the exterior energy fraction is at least 0.50112 under these assumptions. The bound does not apply to the amplitude-gradient degeneracy of the selected current action or automatically to its zero-energy support squares. Their complete laws need separate estimates. The exact theorem, four repair branches and 268 bounded checks are preserved in uniform-clock-boundary/report.md under the current research run.

Shape Stiffness Also Changes Disturbance Propagation

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