Positive preparation width hides receiver states from exact records
The R06 apparatus has a common exact four-momentum record for a two-dimensional patch of distinct receiver states on its fixed-energy shell. Unknown incoming probe momenta compensate the receiver changes. For every sufficiently small fixed coupling lambda>0 and preparation half-width b>0, this gives positive worst-case position and momentum reconstruction errors. Their product has action units and a lower bound supplied by the preparation box.
R17, 2026-09-11. The information consists of four final probe momenta only. The Hamiltonian, receiver energy and preparation support are known; individual incoming apparatus coordinates are unknown. All receiver/apparatus pairs in the Cartesian product of the shell and box are admissible. These are deterministic support risks, not average errors under a density. Additional records, correlations restricting that product, or other apparatus designs require a new information-fibre test. Literature status is recorded in B48.
1. A uniform compensation estimate
Use R06’s Hamiltonian and R16’s component units and record map. Set the clock and probe positions to their nominal initial values and vary only the four incoming probe momenta u, measured in the same momentum units as their final records. This is an admissible slice of the full ten-coordinate box. Write G_lambda(z,u) for this restricted map, and F_lambda(z)=G_lambda(z,0). All norms in this section are component sup norms and induced operator norms.
There are positive b_0, lambda_0, L, independent of b and lambda, such that on a fixed convex receiver neighbourhood and ||u||<=b_0,
\[\|D_uG_\lambda-I\|\le\tfrac12,\qquad \|D_zF_\lambda\|\le L\lambda, \quad 0\le\lambda\le\lambda_0.\]
At lambda=0, the probes are free and G_0(z,u)=u. Smooth finite-time flow dependence on the compact initial-data class makes D_uG uniformly close to I for small coupling. The second estimate follows from R16’s uniform C1 expansion F_lambda=-lambda A z+O(lambda^3). A common compact trajectory neighbourhood and fixed cutoff/pulse margins follow from the R06 energy and continuation bounds after reducing b_0 and lambda_0. This proof needs no assumption that back-reaction vanishes at the chosen positive coupling.
Fix z_* on the shell, Y_=F_lambda(z_), and 0<b<=b_0. For any shell state w satisfying
\[\|w-z_*\|\le \frac{b}{4L\lambda},\]
consider T_w(u)=u-G_lambda(w,u)+Y_* on ||u||<=b/2. The derivative bound makes T_w a contraction with constant at most 1/2, and
\[\|T_w(0)\|\le L\lambda\|w-z_*\|\le b/4, \qquad \|T_w(u)\|\le b/4+\|u\|/2\le b/2.\]
The contraction theorem gives a unique solution in this ball,
\[G_\lambda(w,u(w))=Y_*,\qquad \|u(w)\|\le2L\lambda\|w-z_*\|\le b/2.\]
This is a quantitative implicit-function construction with a b/2 preparation margin. It establishes exact equality of nonlinear records, including clock reaction. Injectivity of the nominal map F_lambda is compatible with this ambiguity because the incoming u changes.
2. A common-record patch on the energy shell
Use physical canonical coordinates z=(x,P,y,Q) and R06’s
\[H_s=\frac{P^2}{2\mu}+\frac{Q^2}{2\nu} +\frac a2x^2-gxy+\frac d2y^2=E>0.\]
Here a,d,mu,nu>0 and ad-g^2>0. At z_*=(0,0,sqrt(2E/d),0), a shell chart is
\[w(x,P)=\left(x,P,\frac gd x+ \sqrt{\frac{2E-(a-g^2/d)x^2-P^2/\mu}{d}},0\right).\]
Completing the square in the potential verifies H_s(w)=E exactly. Choose fixed length and momentum units L_,P_ and a dimensionless r_0>0 so that the square |x|<=L_r_0, |P|<=P_r_0 lies strictly inside the positive radicand domain and the receiver neighbourhood. Smoothness on this square provides C_0>0 with
\[\|w(x,P)-z_*\|\le C_0 \max(|x|/L_*,|P|/P_*).\]
Set
\[r=\min\left(r_0,\frac{b}{4LC_0\lambda}\right)>0.\]
Every point of the square |x|<=L_r, |P|<=P_r therefore has an admissible incoming momentum u(w), with ||u(w)||<=b/2, yielding the same exact Y_*. Both the initial receiver energy and all fixed apparatus design parameters are preserved. No fixed total energy equality for the entire apparatus was assumed in R06; its uniform upper energy bound still holds.
3. Reconstruction risks and canonical projected area
For any deterministic estimator of initial canonical position and momentum from the four records, define
\[\epsilon_x=\sup_{z\in S,\,\|a\|\le b} |\widehat x(G_\lambda(z,a))-x|,\qquad \epsilon_P=\sup_{z\in S,\,\|a\|\le b} |\widehat P(G_\lambda(z,a))-P|.\]
At Y_*, the estimator has one answer while compatible x and P each span a centred interval of the stated half-width. The triangle inequality between the two endpoints proves, for every estimator,
\[\boxed{\epsilon_x\ge L_*r,\quad \epsilon_P\ge P_*r,\quad \mathcal H_{\rm rec}=\epsilon_x\epsilon_P\ge L_*P_*r^2.}\]
The projection of that common-record compatible receiver fibre onto the (x,P) plane contains the whole square, so its ordinary canonical projected area is at least 4L_P_r^2. This is projected area, not a volume of the three-dimensional energy shell or a symplectic area assigned to that shell. Both products have units ML^2/T, with no 2 pi normalization. The result is an explicit positive lower bound; optimal constants and the full fibre are not calculated.
At fixed lambda the bound scales as b^2 for small b, consistently with R16’s O_fixedlambda(b^2) upper bound on the reconstruction product. At fixed b, weakening coupling does not remove this ambiguity: the displayed lower bound saturates at L_P_r_0^2. Positivity is conditional on the fixed-width product support and the specified record access. The quantity here is a reconstruction error product. No lower bound for R06’s accuracy-disturbance product follows without a separate disturbance argument.
4. Next test: access to the incoming momenta
R18 should reveal the four initial probe momenta as extra records while retaining positive unknown initial probe-position widths. Test the derivative of final momenta with respect to those positions and its rank at small coupling. Determine whether position preparation can still compensate a shell patch, or whether the augmented records recover the receiver. This identifies which apparatus variables must be calibrated before proposing a preparation principle that selects an action scale.
C090–C091 are accepted by written review. The B48 audit records established contraction and indistinguishable-observation methods and bounded literature coverage.