A finite mechanical clock and four persistent readout records
A fixed finite-mass classical apparatus can reconstruct the initial state of R04’s receiver with vanishing error and disturbance over a fixed positive observation time. Five apparatus coordinates suffice: one clock and four free probe pointers. The interaction is a smooth coordinate potential, all kinetic terms are positive, and force and coupling ceilings are uniform. The closing limit weakens the coupling and concentrates the incoming apparatus preparation; it retains exact access to four final momentum records. Thus these upper resource bounds alone do not select a positive accuracy–disturbance action.
This is a delayed reconstruction of a known deterministic trajectory. It supplies four physical records from which arbitrarily fine samples may be computed, with one fixed observation latency. It does not perform a fresh measurement at every refined cut. Preparation precision, final record access, and knowledge of the receiver dynamics are explicit resources of the result.
Proof status: C068–C069 accepted by written derivation and coordinator review, 2026-09-10. Literature status: derived consequences, novelty unassessed; B37 and review record the bounded audit.
1. Fixed class and autonomous Hamiltonian
Use the reduced three-body receiver of R04, with \(z=(x,P,y,Q)\), positive masses \(\mu=3m/2\), \(\nu=m/2\), and
\[H_s(z)=\frac{P^2}{2\mu}+\frac{Q^2}{2\nu} +\frac a2x^2-gxy+\frac d2y^2, \qquad (a,g,d)=(9k/4,3k/4,5k/4).\]
Fix \(m,k,E,T>0\) and initial receiver support \(H_s=E\). The reference motion is \(z^0(t)=\Phi_tz_0\) on \(0\le t\le T\). The physical tagged momentum is \(2P/3\); all products below use canonical \(P\). Add a clock \((s,p_s)\) of mass \(M_c>0\) and four probe pairs \((q_j,\pi_j)\) of fixed masses \(M_j>0\). All coordinates have length units and all conjugate momenta momentum units; the phase domain is \(\mathbb R^{14}\). The added total mass \(M_c+\sum_j M_j\) is fixed.
Let \(X(x)\) and \(R(q)\) be smooth bounded functions with bounded derivatives, equal to \(x\) and \(q\) respectively on fixed neighbourhoods containing the reference receiver trajectory and zero probe position. They have length units. For fixed dimensionless smooth compactly supported functions \(f_j(s)\), put
\[H_\lambda=H_s+\frac{p_s^2}{2M_c} +\sum_{j=1}^4\frac{\pi_j^2}{2M_j} +\lambda K\sum_{j=1}^4 f_j(s)X(x)R(q_j), \qquad 0<\lambda\le\lambda_0,\]
where \(K>0\) is a fixed stiffness and \(\lambda\) is dimensionless. There is no external time dependence. A fixed additive constant can make the Hamiltonian nonnegative, since the interaction is uniformly bounded below. Its kinetic quadratic form is positive definite. The reduced receiver coupling is an internal-coordinate interaction; no spatial locality or finite signal speed is assumed for this finite-dimensional potential model.
Fix a nominal clock trajectory \(s^0(t)=s_0+vt\), \(v>0\). Place all supports of \(f_j\) strictly inside \((s_0,s_0+vT)\) with a fixed endpoint margin. Their shapes, widths and derivatives are fixed before taking any limit. Preparation has full rectangular support of dimensionless half-width \(b>0\) about \(q_j=\pi_j=0\), \(s=s_0\), \(p_s=M_cv\), using fixed length and momentum units. An independent uniform density on this rectangle is admissible. Each \(b>0\) has positive phase volume. The class permits \(b\downarrow0\); there is no lower phase-volume or temperature constraint. Energy, coordinate and momentum upper bounds are fixed across this class. In particular the clock has fixed positive nominal energy \(M_cv^2/2\), which is never cooled to zero.
The interaction forces are bounded globally by constants independent of \(\lambda,b\), including the clock force involving \(f'_j\). The harmonic receiver forces are uniformly bounded on the admitted trajectories: conservation of \(H_\lambda\), bounded interaction and positive receiver quadratic energy bound all receiver positions and momenta. Thus a common finite total force ceiling and speed ceiling hold on this preparation class. A global ceiling outside the admitted energy class is not asserted for the harmonic springs. Any prescribed positive apparatus-force ceiling can be met by reducing \(\lambda_0\) after fixing the shapes. The constants may depend on \(E,T\), all masses, \(K\), the cutoffs and the chosen pulse shapes.
2. Four fixed-duration signals determine the receiver state
The scalar history \(x^0(t)=e_x\Phi_tz_0\) observes all four receiver coordinates. At \(t=0\) the first four derivative rows are
\[ x=x_0,\qquad \dot x=P_0/\mu,\qquad \ddot x=(-a x_0+g y_0)/\mu,\qquad x^{(3)}=-aP_0/\mu^2+gQ_0/(\mu\nu). \]
The map from \(z_0\) to these four derivatives is triangular with diagonal \(1,1/\mu,g/\mu,g/(\mu\nu)\) and is invertible since \(g>0\). Consequently the analytic row functions \(e_x\Phi_t\) span four dimensions on any open time interval: a vector annihilated by every row on an interval has an analytic output identically zero and hence all four initial coordinates zero. Four distinct times \(t_j\in(0,T)\) can therefore be selected with independent evaluation rows.
Choose nonnegative smooth bumps \(f_j(s_0+vt)\) near those times, with disjoint supports and positive time integrals \(c_j\). Their widths can be chosen positive and sufficiently small that the matrix
\[\mathcal A_{j\cdot}=K\int_0^T f_j(s_0+vt)e_x\Phi_t\,dt\]
is invertible: after dividing row \(j\) by \(Kc_j\), its limit as the design width shrinks is the independent evaluation row. Continuity of the determinant then gives a fixed nonzero width. This existence argument selects the design once; no pulse width shrinks with \(\lambda\), \(b\) or the output mesh. Small signal gains or poor conditioning enter the fixed constant \(\|\mathcal A^{-1}\|\); there is no uniform bound as \(T\downarrow0\). The rows carry the appropriate units so that \(\mathcal A z_0\) is momentum. All following norms use the fixed component units of section 1.
3. Finite-duration motion, clock reaction and stored records
Uniformly over receiver data and allowed apparatus preparation, the probes, receiver disturbance and clock deviation obey
\[\sup_{[0,T]}(|q_j|+|\pi_j|)=O(b+\lambda),\] \[\sup_{[0,T]}\|z(t)-z^0(t)\|=O(\lambda b+\lambda^2),\] \[\sup_{[0,T]}\bigl(|s(t)-s^0(t)|+|p_s(t)-M_cv|\bigr) =O(b+\lambda^2).\]
Here and below component sums mean dimensionless values. To prove these bounds, the global interaction-force bound gives \(\dot\pi_j=O(\lambda)\); integrating it and \(\dot q_j=\pi_j/M_j\) gives the first estimate. For sufficiently small \(\lambda,b\), the cutoffs are therefore linear along every admitted trajectory. Hamilton’s equations then give exactly
\[\dot\pi_j=-\lambda K f_j(s)x,\qquad \dot P=-ax+gy-\lambda K\sum_j f_j(s)q_j,\] \[\dot p_s=-\lambda K\sum_j f'_j(s)xq_j.\]
Variation of constants in the receiver equations gives the second estimate. The clock equation has force \(O(\lambda b+\lambda^2)\), which together with its initial uncertainty gives \(O(b+\lambda^2)\) over fixed \(T\). Uniform energy bounds and cutoff margins justify these estimates by continuation. This proves the bounds without treating the clock as an unaffected external parameter. In particular \(p_s>M_cv/2\) for small enough parameters, so the clock traverses every pulse and leaves their supports before \(T\).
The four final momenta satisfy
\[\pi_j(T)=\pi_j(0)-\lambda(\mathcal A z_0)_j +O(\lambda b+\lambda^3).\]
Indeed replace \(s(t)\) and \(x(t)\) in the integrated force by \(s^0(t)\) and \(x^0(t)\); bounded derivatives of \(f_j\) give integrand error \(O(b+\lambda^2)\), and multiplication by \(\lambda\) gives the stated remainder. Once the clock has passed all supports, each \(\pi_j\) and \(p_s\) is exactly conserved. The clock continues forward, so there is no later recoupling. These momenta are persistent mechanical records although their conjugate positions subsequently drift. The bounded-coordinate resource claim concerns the fixed observation horizon, not unlimited storage space.
4. Reconstruction, action product and output refinement
Assume the final four momentum records can be accessed exactly. Define the calibrated estimate
\[\widehat z_0=-\mathcal A^{-1}\frac{\boldsymbol\pi(T)}{\lambda}.\]
The incoming momenta are unknown but bounded by \(O(b)\), so
\[\sup\|\widehat z_0-z_0\| =O(b/\lambda+b+\lambda^2).\]
Choose \(b=\lambda^3\). Then reconstructed initial-state error and receiver trajectory disturbance are both \(O(\lambda^2)\). This choice keeps all masses, the receiver’s initial preparation, observation duration, clock mean energy, pulse geometry and upper resource ceilings fixed. Probe incoming phase-volume and clock uncertainty shrink; coupling strength decreases within one fixed upper bound. A fixed nonzero coupling or a lower preparation-width constraint is a different class.
For a precise action-valued observable put
\[\epsilon_x=\sup|\widehat x_0-x_0|,\qquad D_P=\sup_{0\le t\le T}|P(t)-P^0(t)|,\qquad \mathcal U=\epsilon_xD_P.\]
Suprema include all permitted receiver and apparatus initial data. This is an accuracy–disturbance product for reconstruction of initial canonical position and finite-horizon canonical momentum disturbance, in units \(ML^2/T\), without a \(2\pi\) factor. It is not R05’s intrinsic single-shear support product. For fixed length and momentum units \(L_*,P_*\),
\[0\le\mathcal U\le C L_*P_*\lambda^4\longrightarrow0.\]
Thus no uniform positive bound on this specified observable follows from fixed finite mass, finite duration, bounded forces and positive kinetic energy within this preparation and record-access class. The fixed nonzero clock energy does not change the conclusion.
For any partition of \([0,T]\), report \(\widehat z(t_i)=\Phi_{t_i}\widehat z_0\). Boundedness of \(\Phi_t\) gives uniform \(O(\lambda^2)\) error relative to both the unmeasured reference and the actual receiver. Refining only the reporting partition uses the same four records and leaves all old estimated nodes unchanged. Taking also \(\lambda\downarrow0\) changes the experiment but improves that common bound independently of the mesh. This is reconstruction after latency \(T\) with known equations; causal repeated interventions, unknown forcing and model error are not covered by the theorem.
If final record errors have component support width at most \(r\) in fixed momentum units, the same estimator instead has bound \(O((b+r)/\lambda+b+\lambda^2)\). For this scheme \(r=O(\lambda^3)\) suffices to retain the stated rate. A fixed nonzero \(r\) prevents this particular weak-coupling estimate from proving convergence. That upper-bound observation is not an impossibility theorem for every estimator or apparatus.
5. Source connection and next research obligation
Theurel’s existing B36 audit supplies the conserved-momentum record and isolates thermal preparation as an action-scale input; its delta interaction does not supply this finite-duration clock. Hermann–Krener’s observability construction, routed through B34, motivates using four output histories instead of shrinking R04’s two-sample baseline. The explicit derivative-rank argument, autonomous equations and bounds above are written derivations for this apparatus.
R06’s bounded result advances the exclusion-of-zero test by identifying a finite apparatus counterclass with delayed records. The next task R07 is to fix a physically justified preparation or record-resolution restriction and require causal new information at shrinking latency. Specify that information task before seeking a lower bound; distinguish a latency or precision cost from a uniform positive action product. Theorem acceptance and the bounded prior-literature comparison are recorded separately in the ledger and review.