Bound motion: transport variance and a canonical action estimator
A phase-uniform circular orbit has vanishing short- and long-window transport response, while twice its projected canonical covariance area equals its orbital action at every time. This supplies an action estimator for bound mechanics and separates conservation of a prepared scale from its selection. A15, 2026-09-09; C054–C055, written proof and sequential B29 source review.
1. One orbit, one preparation
Take any circular solution of the planar external-potential Hamiltonian \(H=\sqrt{m^2c^4+c^2|p|^2}+V(|q|)\) with radius \(R>0\), momentum magnitude \(P>0\) and positive angular frequency \(\omega\). Put
\[ q(t)=R(\cos\theta,\sin\theta),\quad p(t)=P(-\sin\theta,\cos\theta),\quad \theta=\omega t+\Phi,\quad \Phi\sim\operatorname{Unif}[0,2\pi). \]
Hamilton’s velocity law gives \(P=\gamma mR\omega\), where \(\gamma=(1-R^2\omega^2/c^2)^{-1/2}\). The normalized orbital action is \(\ell=(2\pi)^{-1}\oint p\cdot dq=RP\). Use the canonical Cartesian projection \(x=q_1\), \(p_x=p_1\); the physical coordinate is bounded and the preparation is invariant. For the singular Kepler circles C052 gives \(\ell>k/c\); C053 supplies softened circles with arbitrarily small positive \(\ell\). Newtonian circles obey the same formulas with \(P=mR\omega\) and \(\gamma=1\).
2. What a finite observation window measures
Let \(z=\omega\Delta>0\). Uniform-phase trigonometric averaging gives \(\mathbb E[x(t)x(t+\Delta)]=(R^2/2)\cos z\). Consequently
\[ \mathsf h_x(\Delta) =\frac{m}{\Delta}\operatorname{Var}(x(t+\Delta)-x(t)) =\frac{mR^2}{\Delta}(1-\cos z) =\frac{\ell}{\gamma}\frac{1-\cos z}{z}. \]
At a known initial position \(x\in(-R,R)\), the two velocity signs have equal conditional weights. Their next positions are \(x\cos z\pm\sqrt{R^2-x^2}\sin z\). This specifies the conditional version, with the continuous degenerate extension at the endpoints, and gives
\[ \operatorname{Var}(x(t+\Delta)\mid x(t)=x) =(R^2-x^2)\sin^2z. \]
Its phase-averaged action coefficient is \(\overline{\mathsf h}_{\rm cond}=\ell\sin^2z/(2\gamma z)\). Conditioning on the full canonical state makes the future deterministic. The conditional variance here measures the unobserved direction at a cut.
Both coefficients vanish as \(\Delta\downarrow0\) and as \(\Delta\to\infty\). There is no parametrically broad positive plateau of orbital-action size in this one-frequency family. Indeed \(1-\cos z\le\min(z^2/2,2)\), so \(\mathsf h_x\ge\eta\ell\) for fixed \(\eta>0\) requires
\[2\gamma\eta\le z\le\frac{2}{\gamma\eta}.\]
Every interval satisfying this lower bound therefore has endpoint ratio at most \(1/(\gamma^2\eta^2)\). The statement keeps the positive fraction \(\eta\) fixed; a broad intermediate plateau in another model requires additional time scales and a separate estimate.
3. A canonical covariance area
For \(Z=(x,p_x)\) define the centered covariance matrix \(\Sigma\) and
\[ \mathcal A_{\rm cov}=2\sqrt{\det\Sigma}. \]
Uniform phase yields \(\operatorname{Var}x=R^2/2\), \(\operatorname{Var}p_x=P^2/2\) and \(\operatorname{Cov}(x,p_x)=0\). Thus \(\mathcal A_{\rm cov}=RP=\ell\), with units of action. The factor two is chosen for this uniform orbit: the projected ellipse encloses area \(\pi RP\), half the full planar canonical orbit integral.
For a linear canonical change \(Z'=SZ+b\), \(\Sigma'=S\Sigma S^T\) and \(\det S=1\), hence \(\mathcal A_{\rm cov}\) is unchanged. This invariance is for the selected canonical plane and affine symplectic maps; arbitrary nonlinear canonical transformations need a different argument. The determinant is the familiar rms-emittance construction; B29 audits that literature connection and the normalization.
On this fixed circular family the projected evolution itself is linear:
\[ S_t=\begin{pmatrix} \cos\omega t & (R/P)\sin\omega t\\ -(P/R)\sin\omega t & \cos\omega t \end{pmatrix},\qquad \det S_t=1. \]
It preserves the determinant even for a nonuniform initial phase law. A point-phase preparation has determinant zero despite orbital action \(\ell>0\). Uniform phase is sufficient for the equality with \(\ell\); other phase laws with the same first and second moments also give equality. The point-phase example shows the dependence on preparation. Hamiltonian phase rotation preserves the prepared estimator instead of attracting all preparations to a common value.
4. Selection consequence and next test
For the uniform singular-Kepler circular family, \(\inf\mathcal A_{\rm cov}=k/c\) by C052. For a fixed softened core it is zero by C053. The covariance estimator transfers the known admissibility threshold to an operational ensemble quantity, keeping its singular-core and preparation premises visible. It gives a constant-in-time classical action observable; the remaining selection problem is a physical mechanism fixing its value and its preparation across systems.
A16 now proves the conditional dilation obstruction, including its effect on coupling and preparation. A17 tests a fixed potential and force ceiling. A future receiver extension of A15 would need two separated time scales before an intermediate-window plateau is proposed. The B29 audit and coordinator review complete this single-orbit test.