Tagged momentum and the memory of a third body
An equal-mass three-body spring chain requires receiver information at a cut even when the tagged body’s position and momentum are retained. Eliminating the receiver gives an exact cosine memory kernel and two initial-data terms. Two sufficiently close exact tagged phase observations recover the missing state. This closes R04’s finite-receiver test: memory supplies classical interface information, while the available action scales remain preparation dependent.
Proof status: C064–C065, accepted by written derivation and coordinator review, 2026-09-10. Literature status: established harmonic-bath elimination and observability ingredients; explicit finite-chain formulas are derived specializations. B34 records bounded coverage.
1. One fixed mechanical experiment
Take three particles on the line with equal mass \(m>0\) and two springs of stiffness \(k>0\):
\[ H=\sum_{i=1}^3\frac{p_i^2}{2m} +\frac{k}{2}\big[(x_1-x_2)^2+(x_2-x_3)^2\big]. \]
Fix centre position and total momentum to zero. Write
\[x=x_1,\qquad y=x_2-x_3,\qquad x_2=\frac{-x+y}{2},\quad x_3=\frac{-x-y}{2}.\]
The restricted kinetic energy is \(3m\dot x^2/4+m\dot y^2/4\). Consequently the canonical momenta are \(P=\mu\dot x\), \(Q=\nu\dot y\), where \(\mu=3m/2\) and \(\nu=m/2\). The physical tagged momentum is \(p_1=(2/3)P\). Retaining \((x,P)\) therefore retains precisely the same information as \((x_1,p_1)\); the rescaling accounts for centre reduction.
Put
\[a=\frac{9k}{4},\quad g=\frac{3k}{4},\quad d=\frac{5k}{4}. \qquad H=\frac{P^2}{2\mu}+\frac{Q^2}{2\nu} +\frac a2x^2-gxy+\frac d2y^2.\]
Here \(ad-g^2=9k^2/4>0\), so the reduced quadratic Hamiltonian is positive definite. Its phase domain is \(\mathbb R^4\) and all trajectories are global and smooth. Fix \(E>0\) and the normalized microcanonical Liouville measure \(\delta(H-E)\,dx\,dP\,dy\,dQ\) throughout the cut experiment. This invariant preparation has compact support. Energy conservation gives \(|\dot x_i|\le\sqrt{2E/m}\); choosing \(E<mc^2/2\) supplies a hard particle speed ceiling in this frame. As in A10, this is a Newtonian spring model, with a speed-bounded preparation rather than a relativistic signal law.
2. Exact elimination at any cut
Hamilton’s equations give
\[\mu\ddot x=-ax+gy,\qquad \nu\ddot y=gx-dy.\]
Let \(\Omega^2=d/\nu=5k/(2m)\). Starting a segment at time zero,
\[ y(t)=y_0\cos\Omega t+\frac{Q_0}{\nu\Omega}\sin\Omega t +\frac{g}{\nu\Omega}\int_0^t\sin\Omega(t-s)x(s)\,ds. \]
Substitution gives the exact reduced equation
\[ \mu\ddot x(t)+ax(t) -\frac{g^2}{\nu\Omega}\int_0^t\sin\Omega(t-s)x(s)\,ds =g y_0\cos\Omega t+\frac{gQ_0}{\nu\Omega}\sin\Omega t. \]
Integration by parts rewrites it as
\[ \boxed{\mu\ddot x(t)+a_{\rm eff}x(t) +\int_0^t\Gamma(t-s)\dot x(s)\,ds=F_0(t)}, \]
where
\[ a_{\rm eff}=a-\frac{g^2}{d}=\frac{9k}{5},\qquad \Gamma(t)=\frac{g^2}{d}\cos\Omega t=\frac{9k}{20}\cos\Omega t, \]
\[ F_0(t)=g\left(y_0-\frac gd x_0\right)\cos\Omega t +\frac{gQ_0}{\nu\Omega}\sin\Omega t. \]
The kernel has stiffness units \(M/T^2\); its velocity convolution and \(F_0\) have force units \(ML/T^2\). Under the chosen preparation \(F_0\) is a random function determined by initial data, with its two quadratures correlated through the energy constraint. No independent noise is introduced at later cuts. The oscillatory memory is reversible, rather than a positive friction coefficient describing irreversible energy loss.
At a cut \(\tau\), the same formulas hold with \(t\) replaced by \(t-\tau\) and \((x_0,y_0,Q_0)\) by \((x_\tau,y_\tau,Q_\tau)\). The inherited \((y_\tau,Q_\tau)\) encodes everything needed from earlier history. One may carry this pair or carry the equivalent memory and its initial quadratures.
3. What fails when only tagged phase is carried
Write \(u=(x,P)\), \(w=(y,Q)\) and block the exact linear flow as
\[\binom{u_t}{w_t} =\Phi_t\binom{u_0}{w_0},\qquad \Phi_t=\begin{pmatrix}A_t&B_t\\C_t&D_t\end{pmatrix}.\]
Then
\[u_{s+t}=A_tu_s+B_tw_s,\qquad A_{s+t}=A_tA_s+B_tC_s.\]
Thus composing full segments and then projecting retains the excursion into the receiver and back. Multiplying only retained blocks discards \(B_tC_s\). At fixed energy, the states \(x=P=Q=0\), \(y=\pm\sqrt{2E/d}\) have identical tagged phase but opposite tagged accelerations \(gy/\mu\). They establish that tagged phase alone cannot determine a unique future over the preparation’s support.
For conditional laws, let \(L(u)\) be a version of the microcanonical conditional receiver law given \(u\), and \(\mathcal P\) the tagged projection. The exact one-segment conditional kernel is \(K_t=\mathcal P_\#(\Phi_t)_\#L\). Composing such kernels inserts \(L\mathcal P_\#\) at a cut: it refreshes the receiver from its stationary conditional law instead of carrying the law conditioned on the full observed history. The deterministic block identity identifies the lost channel; no particular reset-refinement limit is claimed here. Marginalizing the true joint path law instead preserves every coarse observation on every partition.
4. Finite-history recovery
The differential history gives a direct completion:
\[y(t)=\frac{\mu\ddot x(t)+ax(t)}g,\qquad Q(t)=\frac\nu g\big[\mu x^{(3)}(t)+a\dot x(t)\big].\]
There is also an exact two-time phase completion. For a separation \(\delta>0\),
\[u_t=A_\delta u_{t-\delta}+B_\delta w_{t-\delta}.\]
If \(B_\delta\) is invertible, solve this equation for \(w_{t-\delta}\) and propagate to \(w_t=C_\delta u_{t-\delta}+D_\delta w_{t-\delta}\). Taylor expansion of the displayed Hamilton equations gives, in the row order \((x,P)\) and column order \((y,Q)\),
\[ B_\delta= \begin{pmatrix} \dfrac{g}{2\mu}\delta^2+O(\delta^4)& \dfrac{g}{6\mu\nu}\delta^3+O(\delta^5)\\ g\delta+O(\delta^3)& \dfrac{g}{2\nu}\delta^2+O(\delta^4) \end{pmatrix}, \]
\[\det B_\delta=\frac{g^2}{12\mu\nu}\delta^4+O(\delta^6).\]
The leading coefficient is positive, so every sufficiently small positive \(\delta\) permits exact recovery. Since \(\Phi_\delta\) is analytic, exceptional sampling separations are isolated zeros of this determinant on finite intervals away from zero. For the local coefficients, differentiate \(\ddot x=(-ax+gy)/\mu\) and \(\dot P=-ax+gy\): dependence on initial \(y\) first enters \(x\) at order two and \(P\) at order one, while dependence on initial \(Q\) enters one order later. This also verifies the remainder parities.
Exact recovery becomes poorly conditioned as \(\delta\downarrow0\): the inverse block has entries growing as high as \(\delta^{-3}\) in fixed physical units. The observed difference must be known with correspondingly increasing precision. This is an observability requirement, not a derived minimum action or minimum time. It gives R05 a concrete physical question: what receiver and back-reaction are required to obtain these observations at fixed precision?
5. What survives refinement, and what sets its scale
Observational refinement leaves the full flow and the kernel \(\Gamma\) unchanged. The surviving interface object is a receiver phase pair or its equivalent memory, with position and momentum units. Its existence follows from the nonzero spring coupling.
For comparison with action, the chain Laplacian has eigenvalues \(0,k,3k\), with eigenvectors proportional to \((1,1,1)\), \((1,0,-1)\) and \((1,-2,1)\). The two internal normal frequencies are therefore \(\omega_1=\sqrt{k/m}\) and \(\omega_2=\sqrt{3k/m}\). In canonical normal coordinates each modal action, normalized by \(1/(2\pi)\) around its own cycle on the invariant torus, is \(J_j=E_j/\omega_j\). Hence at total energy \(E\),
\[\frac E{\omega_2}\le J_1+J_2\le\frac E{\omega_1}.\]
The lower bound is a bound at supplied positive energy. Across preparations, scale every coordinate and momentum by \(b>0\) while keeping \(m,k\) fixed. Energy and both actions scale by \(b^2\); the flow, memory kernel and sampling invertibility conditions remain the same. The force and speed amplitudes decrease with \(b\). Thus genuine receiver memory survives as a structural law while all these prepared action scales approach zero.
The next selection test should address physical acquisition of interface information, rather than infer an action floor from memory alone. Specify a mechanical readout and its disturbance, then test whether arbitrary refinement at fixed useful accuracy remains possible under its independently stated resources. The present finite receiver supplies the exact baseline for that experiment.
Source route
Zwanzig (1973), printed p. 219, equations (21)–(24), supplies the established harmonic-bath elimination route; B23’s companion records the cached original and coverage. R03 supplies the projection/reset distinction. The finite-history reconstruction is a linear observability calculation, with the explicit block determinant derived above. Hermann–Krener (1977), printed p. 733, Theorem 3.1 supplies the general local weak observability framework, rather than the sampled determinant formula. B34’s companion records three primary pages, and the coordinator review checks their scope and the derivation.