Classical readout under cut refinement
Two classical canonical probes can read the tagged phase of R04 with arbitrarily small error and disturbance if their initial phase-space widths may shrink. For a fixed observation horizon, widths of order the fourth power of the mesh suffice for uniform recovery of the hidden receiver state despite the inverse observation map’s cubic sensitivity. This settles the quiet-preparation test for an externally switched impulsive Hamiltonian instrument. Finite-duration, autonomous implementation is the next mechanical obligation.
1. Instrument, state and supplied resources
Use the fixed reduced three-body Hamiltonian and flow of the R04 note: \(z=(u,w)\), \(u=(x,P)\), \(w=(y,Q)\), \(z_t=\Phi_tz_0\), with masses and stiffness fixed and \(z_0\) on the compact positive-energy shell \(H=E\). The physical tagged momentum is \(p_1=2P/3\). Attach two fresh probes with canonical pairs \((q,\pi)\) and \((r,\rho)\), positions in length units and momenta in momentum units. Let \(\alpha>0\) be dimensionless and \(\beta>0\) have units \(T/M\). All phase domains are Euclidean.
At each cut apply, in order, the Hamiltonian kicks generated by the integrated Hamiltonians
\[G_x=\alpha x\pi,\qquad G_P=\beta P\rho.\]
Both \(G\) have action units. They denote unit-parameter canonical flows, or ideal zero-duration pulses with those integrals; receiver drift and probe free drift during a pulse are omitted by definition. The coupling is externally switched and can depend on momentum. It is a classical Hamiltonian instrument, not yet a passive spring apparatus or a finite-speed measurement mechanism. On the unreduced chain it measures the internal coordinate \(x_1-X\) and its conjugate \(P=p_1-(p_2+p_3)/2\) on the zero-total-momentum subspace. Its reaction is distributed over the chain, preserving centre reduction; it is not a local interaction with particle 1 alone.
The final probe positions are recorded exactly in this instrument model. Any additional recording error must be included in the position widths below. No energetic or entropic cost of preparing or recording the probes is assumed. These are explicit resources left open for the implementation test.
2. Exact readout and back-reaction
Hamilton’s equations for \(G_x\) give \(q^+=q+\alpha x\) and \(P^+=P-\alpha\pi\), with \(x,\pi\) unchanged. The second flow gives \(r^+=r+\beta P^+\) and \(x^+=x+\beta\rho\), with \(P^+,\rho\) unchanged. The hidden pair \(w\) is unchanged by both kicks. Thus, in terms of pre-cut values,
\[\widehat x=\frac{q^+}{\alpha}=x+\frac q\alpha, \qquad \widehat P=\frac{r^+}{\beta}=P+\frac r\beta-\alpha\pi,\]
\[d_x=x^+-x=\beta\rho,\qquad d_P=P^+-P=-\alpha\pi.\]
In particular, the second measurement includes the disturbance from the first; its error is relative to the pre-cut momentum. Each shear is symplectic and invertible on system plus probes. Recording and discarding probes is a separate operation from that reversible evolution.
Assume the incoming preparation has full rectangular support \(|q|\le s_q\), \(|\pi|\le t_q\), \(|r|\le s_r\), \(|\rho|\le t_r\), with positive widths. An independent uniform product distribution is one ordinary, nonsingular classical example. The exact worst-case bounds are
\[\epsilon_x=\frac{s_q}{\alpha},\quad \epsilon_P=\frac{s_r}{\beta}+\alpha t_q,\quad D_x=\beta t_r,\quad D_P=\alpha t_q.\]
The two intrinsic accuracy-disturbance products are
\[\epsilon_xD_P=s_qt_q,\qquad \left(\frac{s_r}{\beta}\right)D_x=s_rt_r.\]
They have action units, with support half-width normalization and no \(2\pi\) factor. These are properties of a prepared instrument, not orbital actions. At fixed incoming widths changing gain exchanges accuracy and disturbance. Across incoming classical preparations, both products have infimum zero: replace all four widths by \(b\) times themselves, \(b\downarrow0\). Each finite \(b\) has positive phase-space volume. Liouville volume preservation constrains evolution of a fixed preparation; it does not prohibit choosing these different initial ensembles. The singular quiet limit \(q=\pi=r=\rho=0\) gives exact, undisturbing readout in this ideal model and is not needed for the infimum.
3. Propagation through the inverse observation map
Fix once and for all length and momentum units for each state component and a time unit \(t_*>0\). All norms and big-\(O\) constants in this section refer to these fixed dimensionless coordinates. Constants may depend on \(m,k,T,t_*\), the units and the fixed gains, but not on the mesh. In R04 notation,
\[u_{i+1}=A_\delta(u_i+d_i)+B_\delta w_i,\]
where \(z_i\) is the actual pre-kick state at \(t_i\), and \(d_i=(d_x,d_P)\). Let \(\widehat u_i=u_i+e_i\). For sufficiently small \(\delta>0\), define
\[\widehat w_i=B_\delta^{-1} (\widehat u_{i+1}-A_\delta\widehat u_i).\]
Subtracting the true relation proves the exact error identity
\[\widehat w_i-w_i=B_\delta^{-1} (e_{i+1}-A_\delta e_i+A_\delta d_i).\]
R04 gives \(\|B_\delta^{-1}\|\le C(\delta/t_*)^{-3}\) for small positive \(\delta\). Hence, if each incoming width is \(O(\eta^p)\), where \(\eta=\delta/t_*\) and gains are fixed,
\[\max_i\|\widehat w_i-w_i\|=O(\eta^{p-3}).\]
This is a deterministic support estimate: neither independent noise nor cancellation is required. It includes the ignored kick between samples. For the last sample, propagate the preceding estimate using \(w_{i+1}=C_\delta(u_i+d_i)+D_\delta w_i\); omitting the unknown \(d_i\) adds only \(O(\eta^p)\) because the flow blocks are uniformly bounded. Recovery has one sampling interval of latency, or uses the preceding pair at a current cut. The initial hidden state is recovered after the second sample.
The inverse’s leading entries in physical units are
\[B_\delta^{-1}= \begin{pmatrix} \dfrac{6\mu}{g\delta^2}+O(1)& -\dfrac{2}{g\delta}+O(\delta)\\ -\dfrac{12\mu\nu}{g\delta^3}+O(\delta^{-1})& \dfrac{6\nu}{g\delta^2}+O(1) \end{pmatrix}.\]
This follows by swapping the diagonal entries of R04’s \(B_\delta\), negating the off-diagonal entries and dividing by its determinant. It displays how a position error enters the recovered \(Q\) with cubic amplification. Fixed nonzero rectangular error widths therefore give an unbounded worst-case sensitivity for this linear inversion as \(\delta\) shrinks. It is not a lower bound for every estimator: known energy support can bound estimates, and a longer history can change conditioning.
4. Uniform finite-horizon refinement and resource ledger
Take equal cuts \(t_i=iT/N\), \(0\le i\le N\), at fixed \(T>0\), and fresh probes at each cut. Choose all four widths proportional to \(\eta^4\). Let \(z^0(t)=\Phi_tz_0\) denote the unmeasured trajectory with the same initial state. At pre-cut times the exact disturbance sum is
\[z_i-z^0(t_i)=\sum_{j<i}\Phi_{(i-j)\delta}(d_j,0).\]
Since \(\sup_{|t|\le T}\|\Phi_t\|<\infty\) and \(N=O(\eta^{-1})\), this gives \(O(\eta^3)\) uniformly, also between cuts and immediately after kicks. The reconstructed full state differs from the actual pre-cut state by \(O(\eta)\) and thus from \(z^0(t_i)\) by \(O(\eta)\). The conclusion holds uniformly over the original energy shell and all allowed probe data. The actual system energy is disturbed at finite mesh; quadratic continuity on a fixed compact neighbourhood gives \(H(z_i)-E=O(\eta^3)\). The preparation of the original system is held fixed throughout this limit.
| Resource or observable | Scaling at fixed \(T,m,k,\alpha,\beta\) |
|---|---|
| Incoming probe position and momentum half-widths | \(O(\eta^4)\) |
| Single-readout errors and state kicks | \(O(\eta^4)\) |
| Hidden-state reconstruction error | \(O(\eta)\) |
| Maximum deviation from unmeasured trajectory | \(O(\eta^3)\) |
| Each intrinsic support action product | \(O(\eta^8)\) |
| Probe count, two per cut | \(2(N+1)=O(\eta^{-1})\) |
| Total mass for fixed positive probe masses | \(O(\eta^{-1})\) |
| Sum of initial free-probe kinetic energies | \(O(\eta^7)\) |
For the last row assign fixed masses \(M_q,M_r>0\) and free kinetic energies \(\pi^2/(2M_q)+\rho^2/(2M_r)\) outside the ideal kicks; momenta are unchanged by the kicks. The row bounds only those energies at incoming readout times, not switching, trapping, preparation, storage or recording costs. Probe positions after readout contain an \(O(1)\) signal and need not remain narrow. Arbitrarily many increasingly well-prepared probes and an externally timed impulse sequence are permitted. The limit is not fixed-apparatus refinement. It also changes the probe preparation and intervention schedule with \(N\); finite-mesh experiments are not exact restrictions of one common measured law. Their common limit is the unmeasured classical trajectory, whereas sampling that trajectory alone has exact restriction compatibility at every mesh.
5. Selection consequence and next experiment
This explicit instrument shows that R04’s inverse sensitivity plus classical canonical back-reaction does not by itself force a positive action floor. At fixed probe phase widths there is a supplied accuracy-disturbance scale; quiet and increasingly concentrated preparations close it. A positive lower bound requires an additional restriction on admissible preparations or on the apparatus and its operation. The derivation advances the exclusion-of-zero test; it neither establishes a universal field nor its quantum phase role.
R06 should replace the two ideal kicks by finite-duration interactions with specified positive kinetic energy and mechanical couplings, and include a clock and record. Hold total apparatus mass, force or coupling ceilings, preparation class and observation horizon explicitly fixed. First test whether accurate, weakly disturbing refinement remains possible; if it fails, identify whether the result is a time/resource bound or an action-valued bound and which supplied premise makes it positive. Reusing a probe requires a preparation or reset account before it can remove the growing-mass cost above.
Source route
Katagiri, Measurement theory in classical mechanics, v2 §5, supplies the classical von Neumann interaction and explicit pointer/target translations. The two-channel map and all estimates above follow directly from the stated Hamilton equations. Theurel’s 2024 APS abstract motivates an apparatus-specific thermal restriction; its full-model assumptions remain R06 reading. B35 and its companion give the bounded coverage and source-sign caveat. C066–C067 have coordinator proof review; their literature status is derived consequences with novelty unassessed.