Two position records recover the phase state
Two bounded-error position records give a causal momentum estimate with exact worst-case error for this estimator. Improving precision with a suitable sample separation recovers both coordinates, strengthening R09’s area-only closure.
R10, 2026-09-10. C076–C077 accepted by written review. B41 identifies the finite-difference bound and optimum as established results: Seeber–Haimovich, Lemma 4.3 and Theorem 4.1, with \(N=\varepsilon\), \(L=F/m\). The model retains fixed \(m,F,T>0\), known initial \((q_0,p_0)\), and all measurable forces \(|f|\le F\). Records are non-disturbing and available by prediction time \(T\); their timing and precision are protocol inputs. The estimates concern deterministic errors, with no probability law.
1. Exact compatible-state propagation
Let \(t_2=T-b\), \(t_1=t_2-\delta\ge0\), with \(\delta>0\) and \(b\ge0\). The two records are \(y_i=q(t_i)+e_i\), \(|e_i|\le\varepsilon_i\). Let \(H_i=\{(q,p):|q-y_i|\le\varepsilon_i\}\). With the shear \(S_t\) and force lens \(K_t\) of R08, the exact sets are
\[C_1=(S_{t_1}\{(q_0,p_0)\}+K_{t_1})\cap H_1,\qquad C_2=(S_\delta C_1+K_\delta)\cap H_2,\qquad E_T=S_bC_2+K_b.\]
An empty set rejects inconsistent records. Each intersection retains all and only record-compatible states; unrestricted concatenation of bounded segment forces proves each propagation equality. This preserves position–momentum correlations and uses only two records, however small their separation.
2. A causal estimator and its exact worst-case errors
Define
\[\widehat p=\frac{m(y_2-y_1)}\delta,\qquad \widehat q_T=y_2+\frac b m\widehat p,\qquad B=\frac{m(\varepsilon_1+\varepsilon_2)}\delta+\frac{F\delta}2.\]
Integration of momentum backwards from \(t_2\) gives
\[p(t_2)-\frac{m[q(t_2)-q(t_1)]}\delta =\frac1\delta\int_{t_1}^{t_2}(s-t_1)f(s)ds.\]
The weight integrates to \(\delta^2/2\). Consequently every compatible motion satisfies
\[|p(T)-\widehat p|\le R_p:=B+Fb,\qquad |q(T)-\widehat q_T|\le R_q:=\varepsilon_2+\frac b m B+\frac{Fb^2}{2m}.\]
The second bound uses \(q(T)=q(t_2)+bp(t_2)/m+ m^{-1}\int_{t_2}^T(T-s)f(s)ds\). Both errors are simultaneously attained by \(f=F\) on the two final intervals, \(e_1=+\varepsilon_1\) and \(e_2=-\varepsilon_2\). Any allowed earlier input joins this example to the fixed initial state. Thus these are the exact worst-case errors of the stated estimator over all allowed records and motions. They are upper bounds for the best possible estimator; extra initial-state information may improve them.
Every conditional terminal set lies in the rectangle of half-widths \(R_q,R_p\) about \((\widehat q_T,\widehat p)\). It has coordinate diameters at most \(2R_q,2R_p\), canonical area at most \(4R_qR_p\), and an error product \(R_qR_p\) in action units. The area bound is an outer approximation, not an exact area.
3. Precision, separation and delay
At fixed \(m,F,T\), the sufficient conditions
\[\delta\to0,\qquad \frac{\varepsilon_1+\varepsilon_2}{\delta}\to0,\qquad b\to0\]
give \(R_p,R_q\to0\), uniformly over the force class and errors. The records approach the terminal time while their accuracy improves faster than their separation. Unlike one position strip, the joint set now shrinks in both coordinates. No proliferation of probes is assumed in this information model.
For equal errors \(\varepsilon>0\), minimizing \(B\) over available separations gives
\[\delta_*=2\sqrt{\frac{m\varepsilon}{F}},\qquad B_*=2\sqrt{mF\varepsilon},\]
provided \(\delta_*\le T-b\). This follows from equality of the two positive terms in \(2m\varepsilon/\delta+F\delta/2\). If that separation is unavailable, the minimization must use the allowed timing range. At \(b=0\) the corresponding error product is
\[R_qR_p=2\sqrt{mF}\,\varepsilon^{3/2}\longrightarrow0.\]
For a genuinely delayed sequence, taking \(b=\delta_*\) gives \(R_p=4\sqrt{mF\varepsilon}\), \(R_q=7\varepsilon\), and product \(28\sqrt{mF}\varepsilon^{3/2}\); \(t_1=T-2\delta_*\ge0\) for sufficiently small \(\varepsilon\). This supplies positive delay at every finite stage. All constants above follow by substitution into the written bounds.
4. Premise exposed by the calculation
The force ceiling controls the bias from unresolved acceleration. Record precision controls the amplified position errors. Their optimized balance is a preparation/protocol scale, with mass and force dependence, closing as precision improves. To derive a fundamental positive action constant, the next test must supply a physically justified restriction on attainable precision and timing and then establish a lower bound for every estimator.
R11 should first settle the lower-bound half inside this model: construct two bounded-force trajectories with identical allowed records and shared initial state. Compare their terminal momentum separation with R10’s upper bound. The resulting minimax benchmark will distinguish an unavoidable information loss at fixed precision from merely a chosen estimator’s error.