New information behind a delayed record
A delayed observer of a particle with an unknown bounded force has a sharp prediction error, even with perfect records of its earlier state. The joint position–momentum uncertainty is a curved reachable region with a finite phase-space area. Both scales come from the force budget and the delay; refining records strictly before the delay boundary leaves them unchanged.
R07, 2026-09-10: C070–C071 accepted by written proof and coordinator review. B38 identifies established bounded-input double-integrator machinery; these fixed-time formulas are derived specializations, with exact prior publication unassessed.
1. Experiment and information boundary
Take a particle on the line with mass \(m>0\), on an observation window \([0,\ell]\), \(\ell>0\). The state at zero, \((q_0,p_0)\), and the complete earlier history are known exactly. The observer receives no information about the force or state after zero before making a prediction at time \(\ell\). This is a delay assumption about the accessible records, not a minimum time axiom. The future force lies in the fixed admissible class
\[\mathcal F=\{f\in L^\infty([0,\ell]):|f|\le F\text{ a.e.}\},\qquad F>0.\]
Both signs and independent future profiles are allowed by this information model. The external force is an unresolved input, rather than a previously reconstructed deterministic receiver. No probability law is assigned to it. For each \(f\) the classical dynamics is
\[\dot q=p/m,\qquad \dot p=f,\]
with unique absolutely continuous momentum and continuously differentiable position. All experiments share the same accessible past. Constant or piecewise constant forces are permitted; they create no momentum impulses. If desired, set \(p_0=0\) and choose \(F\ell/m<c\) to keep every admitted particle speed strictly below \(c\). A finite propagation law for the recording apparatus would need its own model.
The exact endpoint increments from inertial prediction are
\[v:=p(\ell)-p_0=\int_0^\ell f(s)ds,\qquad u:=q(\ell)-q_0-\frac{p_0\ell}{m} =\frac1m\int_0^\ell(\ell-s)f(s)ds.\]
2. Sharp causal prediction bounds
For a deterministic predictor based only on the accessible records, define separate worst-case absolute errors \(R_q=\sup_{f\in\mathcal F}|\widehat q-q_f(\ell)|\) and \(R_p=\sup_{f\in\mathcal F}|\widehat p-p_f(\ell)|\). The two forces \(f=+F\) and \(f=-F\) give indistinguishable input records and endpoint separations \(F\ell^2/m\) in position and \(2F\ell\) in momentum. The triangle inequality therefore gives
\[R_q\ge\frac{F\ell^2}{2m},\qquad R_p\ge F\ell.\]
The inertial predictor \((\widehat q,\widehat p)=(q_0+p_0\ell/m,p_0)\) attains both bounds, since the endpoint integrals above have these absolute maxima. Thus the coordinatewise minimax errors and their product are
\[r_q=\frac{F\ell^2}{2m},\qquad r_p=F\ell,\qquad r_qr_p=\frac{F^2\ell^3}{2m}.\]
The product has action units \(ML^2/T\). It is a product of prediction risks for one information task, not a variance product or a measurement-disturbance relation. Its lower bound concerns estimators; individual admitted motions include the undisturbed inertial trajectory.
3. The joint uncertainty region
Independent coordinate error bars obscure correlations between the two increments. Put \(d=v/(F\ell)\) and \(z=mu/(F\ell^2)\). Their exact reachable region is
\[\boxed{-1\le d\le1,\qquad \left|z-\frac d2\right|\le\frac{1-d^2}{4}.}\]
To prove the upper boundary, fix \(v\) and write \(f=2Fh-F\) with \(0\le h\le1\). Then \(\int_0^\ell h(s)ds=a:=\ell(1+d)/2\). The decreasing weight \(\ell-s\) makes its weighted integral largest when \(h=1\) on \([0,a]\) and zero afterwards. Indeed the integral of \([(\ell-s)-(\ell-a)]\,[h(s)-\mathbf1_{[0,a]}(s)]\) is nonpositive on both sides of \(a\), while the constant-weight integral vanishes. Evaluating the weighted integral gives \(z=d/2+(1-d^2)/4\). Putting the positive-force portion last gives the lower boundary \(z=d/2-(1-d^2)/4\). Convex combinations of these two force profiles give every intermediate \(z\) at the same \(d\). This proves necessity and sufficiency without a numerical reachability calculation.
With canonical area measure \(dq\,dp=du\,dv\), the region has area
\[\mathcal A_{\rm reach} =\frac{F^2\ell^3}{m}\int_{-1}^{1}\frac{1-d^2}{2}\,dd =\frac{2F^2\ell^3}{3m}.\]
There is no \(2\pi\) normalization: this is ordinary canonical area of the reachable set, not an orbital action. The extremal controls are the familiar one-switch bounded-force profiles of the double integrator.
4. What the result selects
At fixed \(F,m,\ell\) the unresolved endpoint region persists despite arbitrarily fine observation of the accessible past. This is genuine missing information relative to the stated observer, whereas R06 reconstructs a known deterministic flow from sufficient stored records. It does not require ontic randomness.
The positive scale depends on the allowed force range and the observation delay. Taking \(F\downarrow0\) at fixed \(m,\ell\), or \(\ell\downarrow0\) at fixed \(m,F\), closes both action-valued quantities. A force ceiling alone specifies a worst-case class; it supplies no irreducible force fluctuation in every trajectory. A derivation of a universal action scale must explain what fixes the relevant force–delay combination and why the observer’s information boundary is physically unavoidable.
5. Completed comparison: insert a cut inside the hidden window
R08 proves exact joint composition across an unobserved cut and derives the momentum fibre and terminal area after an observed position. It retains the same bounded-force class and total horizon. This connects the ancient-cut compatibility question to an explicit classical information region. R09 now tests finite-precision position records and the joint precision/delay limit.
The present formulas are elementary bounded-input reachability and deterministic minimax specializations. Liberzon §4.4.1 supplies the classical double-integrator and one-switch control precedent. His task minimizes arrival time; our fixed-time lens is proved directly above. B38’s companion records the one readable primary passage and two failed source routes.