One uncertain calibration leaves an exact receiver curve

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Keeping three supplied constraints exact and allowing the fourth to vary leaves a one-dimensional common-record receiver family. Its leading direction is the corresponding column of the inverse calibration-response matrix. Thus the location of the uncertainty matters: its canonical effects are read from that column, rather than from the number of missing constraints.

R32, completed with an explicit pulse subfamily, 2026-09-12. Retain R30’s fixed pulse design, nonzero c, small fixed apparatus box, full exact final record and receiver energy ball Z={H_s<=E_max}, E_max>0. Receiver energy is not supplied. Coordinates use R30’s fixed component units, with canonical conversion factors L_* and P_. Write C=(q_1,q_2,q_3,H_app) and C_=C(eta_c). Only component j has report error epsilon; all others equal C_* exactly. For apparatus-energy uncertainty j=4, epsilon is measured in the fixed energy unit. Sections 1–3 prove the curve and conditional canonical bound for any R30 design. Section 4 supplies an explicit admissible pulse selection making all four column tests positive; an arbitrary fixed design retains its own test.

1. Construct the fibre through the exact coupled reference

Use R31’s exact reference Y_lambda=G_lambda(0,eta_c), with a_lambda=S^{-1}Y_lambda, and its inverse apparatus chart eta_lambda(w,a). Define

\[T_\lambda(w)=\frac{C(\eta_\lambda(w,a_\lambda))-C_*}{\lambda}.\]

The numerator vanishes at lambda=0 for every w, and at w=0 for every lambda. R30’s smooth divided-map argument gives

\[T_\lambda(0)=0,\qquad T_0(w)=Lw,\qquad D_wT_\lambda(0)=J_\lambda=L+O(\lambda). \tag{1}\]

Indeed T_lambda=D_lambda(w,a_lambda)-D_lambda(0,a_lambda), and a_lambda tends to eta_c. This cancellation includes the displacement of the nominal coupled apparatus; replacing Y_lambda by a free record would change the reference preparation.

Since L is invertible, the parameter-dependent inverse theorem yields fixed positive s_0 and lambda_0 and a smooth family

\[w_{\lambda,j}(s)=T_\lambda^{-1}(s e_j),\quad |s|\le s_0, \quad w_{\lambda,j}(0)=0. \tag{2}\]

Here the inverse is on a neighbourhood of zero contained in the interior of Z. Choose a smaller common neighbourhood first and then a fixed coupling ceiling, so its compensating apparatus remains inside the original box. The derivative is invertible uniformly near (lambda,w)=(0,0); the usual contraction proof supplies a common image neighbourhood, making s_0 independent of lambda. These restrictions shrink the constructed family, not the physical preparation box.

Every member has exactly the same ten final apparatus coordinates and

\[C(\eta_\lambda(w_{\lambda,j}(s),a_\lambda)) =C_*+\lambda s e_j. \tag{3}\]

Consequently the allowed common-record segment includes every

\[|s|\le s_\epsilon:=\min\{s_0,\epsilon/\lambda\}. \tag{4}\]

This is an exact nonlinear construction with all apparatus reaction retained. At positive epsilon it proves full receiver-state ambiguity for each of the four choices j. At epsilon=0 R30 recovers the unique reference state.

2. Which canonical quantities move?

Differentiation gives the explicit test

\[\partial_s w_{\lambda,j}(0)=J_\lambda^{-1}e_j =d_j+O(\lambda),\qquad d_j=L^{-1}e_j. \tag{5}\]

For j=4 this is the unique vector annihilated by A_1,A_2,A_3 and normalized by -Kc B_1(d_4)=1, with the fixed output-unit factors of R30 understood. The response in position is L_(d_j)x; in momentum it is P(d_j)_P. The nonzero vector d_j guarantees state ambiguity. A positive product of canonical risks follows when both displayed canonical components are nonzero.

Under this additional test, reduce s_0 and lambda_0 so each of the two canonical derivatives keeps its sign and has magnitude at least half its limiting magnitude. The endpoints s=+/-s_epsilon then have separations at least L_|(d_j)x|s_epsilon and P|(d_j)_P|s_epsilon. Every estimator on the admitted class therefore obeys

\[\epsilon_x\epsilon_P\ge \frac{L_*P_*}{4}|(d_j)_x(d_j)_P|\,s_\epsilon^2. \tag{6}\]

R31’s feasible-estimator upper bound remains valid with three zero tolerances. Together with bounded-domain zero estimates, (6) gives optimal canonical product order min{1,(epsilon/lambda)^2}, up to fixed positive action-unit constants, whenever the two-component test holds. A vanishing leading component requires examining the exact derivative or higher-order curve; it does not establish that the corresponding observable is recovered.

3. A positive risk product can live on a curve

At fixed positive epsilon, weak coupling retains this whole local curve segment. Unlike R31’s four-error construction, it does not fill an energy ball. R30’s derivative margin makes T_lambda a smooth injective local diffeomorphism on the receiver domain. Its intersection with a coordinate line is locally one-dimensional. The canonical projection of each smooth curve patch has zero planar area even when both coordinate diameters are positive. Thus positive worst-case position and momentum errors do not require a positive symplectic area of compatible states.

The scale in (6) is set by calibration tolerance, pulse response and the admitted preparation neighbourhood. Improved supplied precision at fixed coupling closes it. Section 4 evaluates all four columns for an explicitly selected R30 subfamily; positivity for arbitrary admissible pulse designs remains a separate question.

4. An explicit pulse subfamily makes all four canonical products positive

Choose the first three nominal pulse times to be tau, 2 tau, 3 tau, with 0<3 tau<T and tau sufficiently small. Then choose disjoint nonnegative smooth pulses with sufficiently small positive widths, and fix those widths. This is an explicit additional selection within R30’s admissible designs; its rank conditions alone were not a test of individual inverse entries. For this selection every d_j has nonzero x and P components. Thus (6) holds for each of the four single uncertain constraints, including apparatus energy.

Here is a written small-time proof, using physical receiver coordinates to make the scalings transparent. Put alpha=a/mu+d/nu and beta=(ad-g^2)/(mu nu). The free receiver output satisfies

\[x^{(4)}+\alpha\ddot x+\beta x=0.\]

Since g>0, the jet (x(0),xdot(0),xddot(0),x’’‘(0)) determines the full receiver state: P=mu xdot, y=(mu xddot+a x)/g, and Q=(mu nu x’’’+a nu xdot)/g. For fixed positive tau use coordinates

\[b=(x(0),\tau\dot x(0),\tau^2\ddot x(0)/2, \tau^3x'''(0)/6).\]

In these coordinates the four functionals

\[\mathcal H_\tau x=(x(\tau),x(2\tau),x(3\tau),\tau\dot x(\tau))\]

converge, with matrix error O(tau squared), to evaluation of p(u)=b_0+b_1 u+b_2 u^2+b_3 u^3 at u=1,2,3 and p’(1). To verify the error despite the scaled coordinates, set X(u)=x(tau u). Its equation is X’’’‘+alpha tau^2 X’’+beta tau^4 X=0, with initial jet (b_0,b_1,2b_2,6b_3). Integrating this equation on the fixed interval [0,3] gives the asserted uniform linear-map error. This is a Taylor/ODE estimate, not a numerical check.

The limiting Hermite map is invertible: its homogeneous cubic has zeros at 1,2,3 and a double zero at 1, so is zero. Its inverse columns are:

Direct substitution verifies the four interpolation conditions for each column; differentiating the displayed products gives the last column. Continuity of inversion gives initial x and P components p_j(0)+O(tau squared) and (mu/tau)(p’_j(0)+O(tau squared)) for H_tau^{-1}e_j. Every displayed leading coefficient is nonzero.

To transfer to L, write I_j=integral f_j(s_0+v_0 t) dt>0. In the narrow-pulse limit its rows are nonzero multiples of the H_tau rows, with multipliers

\[\sigma_j=-K I_j j\tau/M_j\ (j=1,2,3),\qquad \sigma_4=-Kc I_1/\tau.\]

Normalize by these factors before taking the width limit; no lower bound on pulse integrals in that auxiliary limit is needed. In physical input and output units the inverse column has limiting canonical components

\[ (d_j)_x=\sigma_j^{-1}[p_j(0)+O(\tau^2)],\qquad (d_j)_P=\frac{\mu}{\tau\sigma_j}[p'_j(0)+O(\tau^2)]. \tag{7}\]

These formulae describe the zero-width limit at fixed tau; for actual pulses there are additional errors tending to zero after row normalization as widths decrease. Here is the finite-width estimate. For fixed tau, let e(t) be the row mapping b to x(t), and place each pulse support within h of j tau, with h<tau/4. The normalized position row is the positive weighted average of (t/(j tau))e(t); the normalized fourth row is the average of tau e’(t) under pulse 1. By the mean-value bound their distances from e(j tau) and tau e’(tau) are at most h times, respectively, sup ||(t e(t))‘||/(j tau) and tau sup ||e’’(t)|| over these supports. These constants are finite at the chosen tau. Thus the normalized four-row matrix N satisfies ||N-H_tau||<=C_tau h in b coordinates. If ||H_tau^{-1}|| C_tau h<1/2, the inverse identity gives ||N^{-1}-H_tau{-1}||<=2||H_tau^{-1}||2 C_tau h. Choose h still smaller so this is below half the smallest absolute entry in the first two rows of H_tau^{-1}. All eight entries then stay nonzero; these rows recover x and tau P/mu. This also proves the required uniform operator estimate independently of the pulse amplitudes or integrals. Select tau first, then positive widths so all eight entries remain nonzero. Conversion to R30’s fixed component units multiplies inverse entries by fixed nonzero factors and preserves this conclusion. Formula (6) uses the dimensionless entries and the factors L_* and P_* as before.

The first two times are exactly R29’s construction. Invertibility of H_tau makes the third row nonzero on their three-row kernel, as R30 requires. The first three position rows are independent. Analytic observability then selects a fourth disjoint time completing the ordinary four-position signal matrix. Choose all four widths small enough to preserve both determinants and the eight inverse-entry margins. After this choice all pulses, tau, masses and c stay fixed. R30 selects its small fixed apparatus box for this design; R32 shrinks only the constructed curve and coupling ceiling thereafter.

For any other already fixed R30 design, the exact algebraic criterion is that both cofactors cof_{j,1}(L) and cof_{j,2}(L) are nonzero, since (L^{-1}){i,j}=cof{j,i}(L)/det L in the ordering (x,P,y,Q). The construction above proves simultaneous feasibility, not positivity for every R30 design. It introduces no shrinking-pulse physical limit into the risk theorem. At fixed design its optimal product order remains min(1,(epsilon/lambda)^2), with action units and preparation-dependent constants.

C121 accepts this pulse-selection result after the B64 review. The interpolation methods and bounded source coverage are recorded in B64; exact model matches and novelty remain unassessed.