Full final apparatus records can hide the entire receiver energy shell

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Unknown incoming apparatus phase permits exact compensation of receiver changes in R06, even when all ten final apparatus coordinates are recorded at known T. A uniform contraction gives a common-record shell patch with an interior preparation margin. At any fixed positive box width, sufficiently weak positive coupling hides the whole shell behind one record: the coordinate minimax risks then equal their no-record values exactly.

R25, 2026-09-12. Use R06’s autonomous Hamiltonian, fixed finite masses, smooth fixed pulses, cutoff margins and observation time. The information includes the Hamiltonian and exact initial receiver energy E>0. Every pair in the Cartesian product of that shell and the full initial apparatus box is admissible; there is no known exact initial apparatus or total energy. Risks below are deterministic worst-case errors for initial canonical (x,P). They have no probability-density or accuracy–disturbance interpretation.

1. Invert the free apparatus shear

Write the physical initial apparatus state as eta=(q_1,…,q_4,pi_1,…,pi_4,s,p_s), with centre eta_=(0,…,0,s_0,M_c v_0), v_0>0. Use fixed positive component length and momentum units for both initial and final records; henceforth these coordinates and all sup norms are dimensionless. The unknown box is ||eta-eta_||<=b, 0<b<=b_0. The receiver domain Z is the convex energy ball H_s<=E, or a fixed slightly larger ball when derivatives need a neighbourhood.

Let G_lambda(z,eta) be the ten unscaled final apparatus coordinates. At zero coupling the receiver decouples and G_0=S eta. In physical coordinates the invertible linear shear S is

\[(q_j,\pi_j,s,p_s)\longmapsto (q_j+T\pi_j/M_j,\pi_j,s+Tp_s/M_c,p_s). \tag{1}\]

In dimensionless coordinates the coefficients include the corresponding fixed momentum-to-length unit ratios. Set F_lambda=S^{-1}G_lambda. Uniform smooth finite-time flow dependence on the common compact trajectory neighbourhood gives constants C,L>0 and lambda_0>0, independent of b and lambda, such that

\[\|D_\eta F_\lambda-I\|\le C\lambda\le\tfrac12, \qquad \|D_zF_\lambda\|\le L\lambda, \quad 0\le\lambda\le\lambda_0. \tag{2}\]

Indeed F_0=eta, its two derivatives are I and zero, and their coupling derivatives are bounded on this compact smooth-flow domain. R06’s conserved energy upper bound, bounded interaction and fixed T supply common existence and trajectory bounds. Reduce b_0 and lambda_0 to preserve all cutoff and pulse margins, including positive clock speed. Unknown initial probe momenta make free probe positions drift; b_0 is chosen to cover that drift throughout T. Equation (2) controls the complete interacting flow, including clock reaction. No inverse signal matrix or pulse-rank assumption is needed for this ambiguity.

Fix any shell point z_* and the final record Y_=G_lambda(z_,eta_). For w in Z define on ||eta-eta_||<=b/2

\[T_w(\eta)=\eta-F_\lambda(w,\eta)+F_\lambda(z_*,\eta_*).\]

If ||w-z_*||<=b/(4L lambda), (2) and the segment in Z imply

Thus T_w is a self-mapping contraction with constant at most 1/2. Its unique fixed point in that ball obeys

\[G_\lambda(w,\eta(w))=Y_*,\qquad \|\eta(w)-\eta_*\|\le2L\lambda\|w-z_*\|\le b/2. \tag{3}\]

Equality holds for every final position and momentum, without dividing a record by lambda. The prepared product support admits these correlated choices of initial data; they need not have positive probability under a density.

2. A quantitative canonical patch at fixed coupling

Write the receiver energy as

\[H_s=\frac{P^2}{2\mu}+\frac{Q^2}{2\nu} +\frac{k_x}{2}x^2+\frac d2(y-gx/d)^2, \qquad k_x=a-g^2/d>0.\]

At z_*=(0,0,sqrt(2E/d),0), use the exact shell chart

\[w(x,P)=\left(x,P,\frac gd x+ \sqrt{\frac{2E-k_xx^2-P^2/\mu}{d}},0\right). \tag{4}\]

Choose fixed canonical units L_,P_ and r_0,C_0>0 so that the square |x|<=L_r_0, |P|<=P_r_0 lies strictly inside the radicand domain and ||w(x,P)-z_||<=C_0 max(|x|/L_,|P|/P_*). These constants are independent of b and lambda. With

\[r=\min\{r_0,b/(4LC_0\lambda)\}>0,\]

the common-record fibre contains this entire square at radius r. For any estimator, taking the two endpoints of each compatible coordinate interval gives

\[\epsilon_x\ge L_*r,\qquad \epsilon_P\ge P_*r,\qquad \epsilon_x\epsilon_P\ge L_*P_*r^2. \tag{5}\]

The projected compatible area is at least 4L_P_r^2. These are lower bounds, not optimal constants in the general fixed-coupling regime.

3. Exact saturation at fixed preparation width

Let D=max_{H_s(w)=E}||w-z_*||>0 in the fixed receiver units. If

\[0<\lambda\le\min\{\lambda_0,b/(4LD)\}, \tag{6}\]

applies to every point of the compact shell with the same Y_* and the same b/2 preparation margin. Consequently the compatible receiver set at Y_* is exactly the entire admitted shell. Its projection is exactly the ellipse

\[k_xx^2+P^2/\mu\le2E: \tag{7}\]

necessity follows from positive residual energy; sufficiency follows by choosing Q=0 and y=gx/d plus a square root as in (4), including the boundary. The coordinate extrema are X_E=sqrt(2E/k_x) and P_E=sqrt(2mu E). Define each minimax risk as the infimum over estimators of the supremum over the admitted shell times apparatus box. Common-record endpoints give lower bounds X_E,P_E; the constant estimator (0,0) attains both bounds over the whole admitted class. Therefore

\[R_x=X_E,\quad R_P=P_E,\quad \inf_{(\widehat x,\widehat P)}\epsilon_x\epsilon_P =X_EP_E=2E\sqrt{\mu/k_x}. \tag{8}\]

The common-record projected area is exactly pi X_E P_E. Products and areas have units ML^2/T; the risk product has no 2 pi normalization. The ellipse area is an ordinary area in the canonical plane, not an area assigned to the three-dimensional shell. The effective stiffness k_x describes an energy projection; sqrt(k_x/mu) is not asserted to be a normal-mode frequency.

At fixed b>0, the weak-coupling regime (6) loses all worst-case coordinate advantage of the ten records. At fixed lambda>0, the patch lower bound closes as b tends to zero; (6) cannot be maintained in that limit. Under joint limits its validity requires the explicit ratio condition (6). The saturated value is set by receiver energy, masses and stiffness; it closes under receiver cooling and establishes no universal positive action scale. This result advances the preparation/access test for exclusion of zero, leaving universal scale selection and the quantum role open.

4. Source connection and next test

B56 audits inherited B48 contraction and common-record methods and B50/B55 smooth-flow coverage. The free-shear preconditioner, exact complete-record compensation and full-shell saturation are model derivations; novelty is unassessed. Proof acceptance is recorded in the review.

R26 should supply exact initial apparatus energy in addition to receiver energy and all final apparatus records. Along the unique compensator eta(w), test the extra scalar constraint H_app(eta(w))=H_app(eta_*), with the initial interaction zero before all pulses. Determine an actual common-record shell family and its preparation margins or a recovery bound. Conservation of total energy constrains final receiver energy but does not alone settle that fibre.