An unknown clock hides an exact fixed-energy receiver phase
For an admissible fixed pulse design, eight exact final pointer coordinates and known incoming probe momenta leave a smooth family of receiver states on one exact energy shell when the initial clock data are hidden. Both initial canonical coordinates vary along this family. Consequently their worst-case reconstruction-error product is positive at fixed clock preparation width.
R20, 2026-09-11. Use R19’s apparatus and common compact cutoff margins, with incoming probe momenta fixed and revealed at zero. The observation time T and Hamiltonian are known. Initial probe positions and clock position/momentum range independently over small positive boxes; the receiver ranges over H_s=E>0. The family constructed below lies inside this Cartesian support. This is a design-existence result and a preparation-dependent reconstruction bound. Literature status is in B51.
1. Reduce identical records to one energy equation
Write the clock data as c=(s,v), with p_s=M_c v and v>0. Let M generate the free receiver flow Phi_t, H_s(z)=z^T Gz/2 with G positive definite, and
\[A_{s,v}=K\int_0^T f_j(s+vt)e_x\Phi_t\,dt\quad\hbox{(row j)}.\]
R19’s scaled eight-record map extends smoothly to lambda=0:
\[F_{\lambda,c}(z,q)=(-A_cz,q)+O_{C^1}(\lambda). \tag{1}\]
The extension and estimate hold also for clock derivatives: divide the integrated momentum equation by lambda before taking the limit, then use smooth finite-time flow dependence on all initial data. All pulses remain interior for clocks in the chosen box. Fix nominal c_0=(s_0,v_0), an interior probe position q_=0, and a shell point z_ to be selected below. Set Y_lambda=F_{lambda,c_0}(z_*,0).
Since the derivative in (z,q) at lambda=0 is diag(-A_c,I), the parameter implicit-function theorem gives a smooth local solution
\[F_{\lambda,c}(z_\lambda(c),q_\lambda(c))=Y_\lambda,\] \[z_0(c)=A_c^{-1}A_{c_0}z_*,\qquad q_0(c)=0. \tag{2}\]
Neighbourhoods can be chosen uniformly for sufficiently small lambda, including zero. This follows by a contraction with the fixed invertible derivative on a small product neighbourhood. At positive fixed lambda these are identical unscaled final records as well. The remaining physical constraint is the scalar equation
\[e_\lambda(c):=H_s(z_\lambda(c))-E=0. \tag{3}\]
Thus clock dimension counting is replaced by a specific shell equation.
2. Clock offset changes receiver phase at zero coupling
Interior pulse support permits integration by parts, giving the exact identity
\[\partial_s A_{s,v}=-A_{s,v}M/v.\]
Equation (2) therefore gives at c_0
\[\partial_s z_0=Mz_*/v_0,\qquad \partial_s e_0=0. \tag{4}\]
The last equality uses G M+M^T G=0: free receiver motion preserves its quadratic energy. A change of clock offset is, to leading order, a change of receiver phase. A transverse clock-speed derivative will keep this phase freedom on the exact interacting energy shell.
3. A fixed pulse design with transverse speed response
Choose four distinct positive dimensionless numbers a_j and a nonnegative smooth bump phi of positive integral, supported in (-1,1). Fix eta>0 small enough that the intervals a_j+eta(-1,1) are positive and disjoint. For a positive design time epsilon set
\[f_j(s)=\phi\left(\frac{s-s_0-v_0\epsilon a_j} {v_0\epsilon\eta}\right).\]
For small epsilon all pulses finish before T. With r=v_0/v and t=r epsilon u,
\[A_{s_0,v,j}=K r\epsilon\int\phi((u-a_j)/\eta) e_x\exp(Mr\epsilon u)\,du.\]
The first four Taylor rows e_x M^n, n=0,1,2,3, are independent by R06. The moment matrix W_{jn}=integral phi((u-a_j)/eta)u^n/n! du is invertible for small fixed eta: its row-normalized limit is the Vandermonde matrix a_j^n/n!. Expanding the determinant by multilinearity, the first nonzero power uses distinct n=0,1,2,3. Hence, uniformly with one v derivative near v_0,
\[\det A_{s_0,v}=D(r\epsilon)^{10}(1+O(\epsilon)),\qquad D\ne0. \tag{5}\]
The exponent is four integration factors plus 0+1+2+3. Analyticity of the matrix exponential gives the differentiated remainder after factoring the first nonzero coefficient. Fix epsilon sufficiently small once and for all. Then A is invertible and, putting B=-A^{-1}partial_v A at c_0,
\[\operatorname{tr}B=-\partial_v\log|\det A| =10/v_0+O(\epsilon)>0. \tag{6}\]
This design limit is used only to prove existence. Pulse widths, masses and all derivative bounds subsequently stay fixed; reducing lambda_0 meets the apparatus force ceilings and cutoff margins.
Let S_B=(GB+B^T G)/2. Its G-normalized trace is tr(B)>0, so it has a positive quadratic direction. Choose z_* on H_s=E with z_^T S_B z_>0. This is an open subset of the shell. Within it choose also P_^{state} nonzero and -a x_+g y_* nonzero: the excluded hyperplanes have empty relative interior. These conditions imply
\[\partial_v e_0=z_*^T GBz_*>0,\quad (Mz_*)_x=P_*^{state}/\mu\ne0,\quad (Mz_*)_P=-a x_*+g y_*\ne0. \tag{7}\]
The symbol P_^{state} here is the chosen state’s momentum, distinct from the fixed momentum unit P_ used in risk bounds.
4. Exact interacting shell family and preparation margins
By (7) and smooth dependence, partial_v e_lambda remains positive near (s_0,v_0) for all sufficiently small lambda. Since e_lambda(c_0)=0, solve (3) as v=v_lambda(s), with v_lambda(s_0)=v_0. Equations (4) and (7) give
\[v'_\lambda(s_0)=O(\lambda),\qquad \frac{d}{ds}z_\lambda(s,v_\lambda(s))\bigg|_{s_0} =Mz_*/v_0+O(\lambda). \tag{8}\]
Both canonical components therefore have nonzero derivatives. Choose a fixed small offset interval |s-s_0|<=sigma within the implicit-function domains. Continuity, first in s and then in lambda, gives physical constants k_x,k_P>0 such that along the whole interval the two derivatives retain their signs and have magnitudes at least k_x,k_P. The constants have units respectively one and momentum/length, since s is a length coordinate.
At lambda=0 equation (4) holds along the family at fixed v_0, so that v_0(s)=v_0 and q_0(c)=0. Uniform smoothness gives v_lambda(s)-v_0=O(lambda |s-s_0|) and q_lambda(s,v_lambda(s))=O(lambda |s-s_0|). Choose sigma smaller than half the physical clock-position half-width. For fixed positive clock-momentum and probe-position half-widths, reduce lambda_0 if necessary to leave at least half their margins. Receiver energy is exactly E throughout. Known incoming probe momenta and all eight final records agree exactly. The apparatus upper resource bounds persist on the inherited compact domain.
5. A phase uncertainty gives a canonical risk product
The endpoints s_0-sigma and s_0+sigma have canonical separations at least 2 k_x sigma and 2 k_P sigma and one common record. Every deterministic estimator consequently has worst-case initial-coordinate errors
\[\epsilon_x\ge k_x\sigma,\qquad \epsilon_P\ge k_P\sigma,\qquad \mathcal H_{\rm rec}\ge k_x k_P\sigma^2>0. \tag{9}\]
The product has action units. This curve need not enclose positive canonical area; positive coordinate-risk products and positive phase-space area are different assertions. Equation (9) survives sufficiently small positive coupling with one fixed sufficiently small sigma and fixed preparation widths. It closes as the admitted clock-offset interval contracts. The known-clock result R19 recovers zero exact-record risk by removing precisely this preparation freedom.
Next R21: reveal only the initial clock position, leaving its momentum unknown. The transverse derivative (7) suggests local recovery after imposing the receiver energy shell. Test uniform local uniqueness and record stability, and separate them from recovery on the entire shell.