Fixed coupling permits precise calibrated readout
R06’s four-record apparatus admits arbitrarily accurate reconstruction at one fixed sufficiently small nonzero coupling. Use its exact nonlinear calibration map, then contract the incoming apparatus preparation and final record errors. Masses, pulse widths, clock mean energy and observation duration remain fixed. The receiver disturbance stays bounded; multiplying it by the vanishing reconstruction error gives a vanishing action-valued product.
R16, 2026-09-11. This tests whether fixing coupling closes R06’s counterclass. The resources still supplied are arbitrarily concentrated classical preparations, improving access to four final momenta, and exact knowledge of the Hamiltonian. This is delayed deterministic reconstruction, not a complete-history observation oracle.
1. Fixed apparatus, finite error vector
Use R06’s Hamiltonian H_lambda, clock and four probes, with its positive m,k,E,T and fixed smooth pulse geometry. Initial receiver states z lie on the compact shell S={H_s(z)=E}. Write a for the ten apparatus initial coordinates in fixed dimensionless component units, centred on the nominal clock and zero probe phase; ||a||_infinity<=b. Each b>0 permits the same full rectangular support as R06. Denote the four final probe momenta, expressed in fixed momentum units, by
\[G_\lambda(z,a)\in\mathbb R^4,\qquad F_\lambda(z)=G_\lambda(z,0).\]
The actual observation is the finite vector Y=G_lambda(z,a)+e with ||e||_infinity<=rho. Neither a nor e is known individually. The exact map F_lambda, including clock reaction and receiver back-reaction, is supplied as the calibration. Norms below use the declared component units; fixed constants restore dimensions when action products are formed.
2. The nominal record map is uniformly invertible
On a fixed convex neighbourhood of the receiver energy ball, R06’s invertible signal matrix A obeys
\[F_\lambda(z)=-\lambda\mathcal A z+\mathcal R_\lambda(z), \qquad \|\mathcal R_\lambda\|_{C^1}\le C\lambda^3.\]
Here is the derivative step in addition to R06’s value estimate. At nominal apparatus data, probe coordinates and momenta are O(lambda), and receiver and clock deviations from their uncoupled paths are O(lambda^2). The same orders hold for their derivatives with respect to initial receiver z: differentiate the smooth finite-time Hamilton equations. Probe derivative forcing is O(lambda); receiver and clock reaction derivatives contain lambda times a probe or its derivative, and are O(lambda^2). Variation of constants and finite-time Gronwall bounds preserve these orders uniformly on the larger compact energy neighbourhood. The integrated record equation
\[F_{\lambda,j}(z)=-\lambda K\int_0^T f_j(s(t;z))x(t;z)\,dt\]
therefore differs from -lambda A z by O(lambda^3) in both value and first derivative. Fixed cutoff margins keep the exact equations in R06’s linear cutoff regions for all those trajectories. Smooth dependence on initial data justifies differentiation; no numerical calibration experiment is used.
Let alpha>0 satisfy ||A v||>=alpha||v|| in the chosen norms. Integrate the remainder derivative along the line segment between z and w inside the convex neighbourhood. For all sufficiently small positive lambda,
\[\|F_\lambda(z)-F_\lambda(w)\| \ge (\lambda\alpha-C\lambda^3)\|z-w\| \ge\tfrac12\lambda\alpha\|z-w\|.\]
Choose such a lambda once and hold it fixed from here on. This proves a global lower Lipschitz bound on S, rather than inferring global injectivity from a pointwise rank condition. The inverse exists on the image F_lambda(S).
3. Robust exact calibration
Smooth finite-time dependence on a on the compact preparation class gives
\[\|G_\lambda(z,a)-F_\lambda(z)\|\le C_\lambda b.\]
Use the sup norm for records, so the effective finite record error is at most delta=C_lambda b+rho. Choose any minimum-residual estimate
\[\widehat z\in\operatorname*{argmin}_{w\in S} \|F_\lambda(w)-Y\|_\infty.\]
The minimum exists by compactness. Since the true z is a competitor, the residual at the estimate is at most delta. The triangle inequality and the lower Lipschitz bound yield
\[\boxed{\|\widehat z-z\|\le \frac{4}{\lambda\alpha}(C_\lambda b+\rho).}\]
At fixed coupling this tends uniformly to zero as b,rho tend to zero. In contrast, R06’s first-order inverse -A^(-1)Y/lambda retains an O(lambda^2) calibration remainder at fixed lambda. Exact calibration accounts for that known systematic response; it does not treat it as arbitrary record noise. The minimum-residual rule is an existence construction, with no assertion about numerical cost or robustness to an uncertain Hamiltonian.
4. Action products and the limit being taken
Let epsilon_x be the worst initial canonical-position reconstruction error, and D_P the worst canonical momentum disturbance relative to the unmeasured reference over [0,T], as in R06. Fixed length and momentum units L_,P_ give
\[\epsilon_x\le C_\lambda L_*(b+\rho),\qquad D_P\le C P_*(\lambda b+\lambda^2),\]
\[0\le\mathcal U=\epsilon_xD_P \le C_\lambda L_*P_*(b+\rho)(\lambda b+\lambda^2) \longrightarrow0.\]
The product of the two initial canonical coordinate reconstruction errors likewise obeys H_rec<=C_lambda L_P_ (b+rho)^2. Both have action units without a 2 pi factor. Disturbance itself is not required to vanish in this fixed-coupling limit. Known reference trajectories can be reconstructed from the recovered initial state, uniformly on the fixed horizon. That statement differs from weakly disturbing fresh measurements along an unknown motion.
The finite apparent record-error set is contained in a shrinking four-dimensional ball. This proves a sufficient upper error bound, not equality with an independently supplied norm ball of the R14/R15 oracle models. Any imposed positive rho or lower preparation volume is a further resource premise. Fixed apparatus masses, coupling, duration and upper energy/force ceilings alone permit the preparation/record-access sequence above.
5. Next test: a fixed incoming preparation width
R17 should hold b>0 fixed and use exact final record access. Test whether unknown incoming probe momenta can compensate changes in the receiver state, producing identical four-record vectors under the full nonlinear calibration. Use an explicit implicit-function argument with the preparation box margins, and quantify the resulting initial-state ambiguity. This isolates what an independently justified lower preparation width would contribute, rather than identifying a calibration remainder with a physical gap.
C088–C089 are accepted by written review. B47 supplies the bounded source/assumption audit; its ODE precedent supports smooth dependence, while the uniform remainder and inverse bound are derived here.