Calibrated two-energy records still hide both canonical coordinates
A fixed smooth pulse design admits an exact common-record receiver curve with nonzero variation in both initial x and P, despite a known nonzero q_1=c, exact initial energies E,H_0, and all ten final apparatus coordinates at known T. Its two endpoints give a positive canonical risk product, uniformly for sufficiently weak positive coupling at fixed preparation width. The result is an existence construction within R27’s apparatus class; arbitrary pulse designs require the rank test below.
R28, 2026-09-12. Retain R06’s Hamiltonian, K, positive masses, coupled receiver g>0, cutoff regions and endpoint pulse margins. Retain R27’s preparation: H_s=E>0, H_app=H_0=M_c v_0^2/2, q_1=c nonzero, |c|/L_1<b/4, and every remaining apparatus coordinate allowed in the fixed box of half-width b. Choose the fixed pulse design as in section 2 before decreasing lambda. All coordinate norms used for smoothness and margins use fixed component units. The canonical errors below retain physical length and momentum units.
1. Two scaled constraints on the exact compensator
Let eta_c have q_1=c, all other probe coordinates and momenta zero, and clock (s_0,M_c v_0). For a receiver shell point z_bar to be chosen below set Y_lambda=G_lambda(z_bar,eta_c). R25’s inverse-shear contraction, centred now at eta_c, gives a smooth compensator eta_lambda(w) on a fixed neighbourhood of z_bar with
\[G_\lambda(w,\eta_\lambda(w))=Y_\lambda,\qquad \|\eta_\lambda(w)-\eta_c\|\le 2L\lambda\|w-z_{\rm bar}\|. \tag{1}\]
The chart may include an ambient receiver neighbourhood when taking derivatives. Reducing lambda preserves a b/2 margin from the original box boundary: eta_c is already within b/4 of its centre. Smoothness follows from the invertible apparatus derivative and smooth finite-time flow. In particular eta_0(w)=eta_c and eta_lambda(z_bar)=eta_c exactly.
Write h(t)=f_1(s_0+v_0t), x_w(t)=e_x Phi_t w and define linear functionals
\[A(w)=\int_0^T t h(t)x_w(t)\,dt,\qquad B(w)=\int_0^T h(t)\dot x_w(t)\,dt. \tag{2}\]
On the compensator the additional calibration and energy conditions have smooth extensions at lambda=0 after division by lambda:
\[U_\lambda(w)=\frac{q_1(\eta_\lambda(w))-c}{\lambda},\qquad V_\lambda(w)=\frac{H_{\rm app}(\eta_\lambda(w))-H_0}{\lambda},\] \[U_0(w)=-\frac K{M_1}A(w-z_{\rm bar}),\qquad V_0(w)=-Kc B(w-z_{\rm bar}). \tag{3}\]
To verify signs and weights, Hamilton’s equations give exactly q_1(T)-T pi_1(T)/M_1=q_1(0)+(lambda K/M_1) integral t f_1(s(t))x(t) dt. Matching the final record and differentiating at zero coupling gives U_0. At eta_c only the first probe has a nonzero free displacement, so the first clock impulse is -lambda Kc integral f’_1(s_0+v_0t)x_w(t) dt. Matching final clock momentum requires initial momentum difference +lambda Kc integral f’_1(x_w-x_bar) dt. Initial apparatus energy has first variation v_0 times this difference; initial probe kinetic energies have zero first variation. Integration by parts, with h zero at both endpoints, gives V_0 in (3). Changes of the clock trajectory and receiver forcing enter only higher orders in this first variation. Smooth divisibility follows by writing each numerator as lambda times the integral of its coupling derivative. Explicitly, for either smooth numerator N with N(0,w)=0, its extension is integral from 0 to 1 of partial_lambda N(theta lambda,w) d theta. This is jointly smooth in (lambda,w), including all receiver derivatives. The smooth Hamiltonian flow also exists on a small signed coupling neighbourhood for this analytic argument; physical states use only positive lambda. Thus the three exact equations to solve are
\[H_s(w)=E,\qquad U_\lambda(w)=0,\qquad V_\lambda(w)=0. \tag{4}\]
2. A fixed pulse with a canonical direction in its kernel
There is a smooth nonnegative first pulse for which A,B are independent and W=ker A intersect ker B contains a vector v with v_x v_P nonzero. Here v is a receiver displacement per unit dimensionless curve parameter, not clock speed.
Choose a small time tau strictly between 0 and T. In the design limit of a pulse concentrated near tau, writing I=integral h>0, the normalized rows A/(tau I), B/I tend to e_x Phi_tau and e_x D Phi_tau, where D is the free receiver generator. These rows specify x(tau) and P(tau)/mu and have rank two. The vector
\[v^{(0)}=\Phi_{-\tau}(0,0,Y,0),\qquad Y>0 \tag{5}\]
lies in their kernel. The receiver equations give, by Taylor expansion,
\[v_x^{(0)}=\frac{gY}{2\mu}\tau^2+O(\tau^4),\qquad v_P^{(0)}=-gY\tau+O(\tau^3). \tag{6}\]
Both are nonzero for sufficiently small fixed tau. Fix that tau. A sufficiently narrow but positive smooth pulse width preserves rank two and a nearby kernel vector with both components nonzero. For example, project v^(0) onto the perturbed kernel using the continuous orthogonal projection in fixed units. The positive mass and g>0 are used in (6); a decoupled receiver need not pass this test. After this design choice, neither tau nor width varies with lambda.
The other three R06 pulses can retain an invertible four-momentum signal matrix. The evaluation row at tau extends to a basis of four evaluation rows: analytic observability spans all four dimensions even after excluding small neighbourhoods of previously chosen times. Choose those times, then all four positive widths sufficiently small with disjoint supports to preserve the basis and the first pulse kernel property. There is therefore no conflict between this construction and the original apparatus’s calibrated readout rank. Indeed, a vector annihilated by the evaluation rows on any remaining open interval has analytic output identically zero. Its first four derivatives at zero recover x,P,y,Q with nonzero diagonal coefficients 1,1/mu,g/mu,g/(mu nu), as shown in R06 section 2. Such a vector must vanish.
3. Continue a shell curve to positive coupling
Let H_s(w)=w^T J w/2 with J positive definite. The two-dimensional plane W contains v above. Choose a nonzero z_bar in W with z_bar^T J v=0, and rescale it so H_s(z_bar)=E. This is possible because J restricts to a positive inner product on W. The rows
\[dH_s(z_{\rm bar}),\quad A,\quad B \tag{7}\]
are independent: A,B are independent and vanish on W, whereas dH_s(z_bar)[z_bar]=2E. Their common kernel is exactly span(v). The nonzero constants K/M_1 and Kc in (3) preserve this rank.
Choose a linear parameter functional ell with ell(v)=1. Apply the implicit function theorem to (4) and ell(w-z_bar)=t, at (lambda,t,w)=(0,0,z_bar). The four by four w derivative is invertible by (7). It supplies a smooth curve w_lambda(t) for |t|<=delta and 0<=lambda<=lambda_1, with delta,lambda_1 positive and fixed, such that
\[w_\lambda(0)=z_{\rm bar},\qquad \partial_t w_0(0)=v. \tag{8}\]
For an explicit uniform neighbourhood, set F_lambda(w)=(H_s(w)-E,U_lambda(w),V_lambda(w),ell(w-z_bar)), scaling each output by a fixed positive unit. Let D_=D_w F_0(z_bar). Joint smoothness and invertibility allow a receiver ball of radius rho and a coupling interval on which ||D_^{-1}(D_w F_lambda-D_)||<=1/2. Since F_lambda(z_bar)=0, the map w -> w-D_^{-1}(F_lambda(w)-(0,0,0,t)) is a contraction and maps that ball to itself whenever ||D_*^{-1}(0,0,0,t)||<=rho/2. This supplies one fixed positive t interval and the same joint smooth branch as the IFT.
For positive lambda every point has the exact two energies and calibrated q_1=c, and (1) makes all ten final records exactly Y_lambda. Choose a compact subrectangle of the implicit-function neighbourhood and reduce lambda_1 so (1) leaves the b/2 box margin there. The same compact flow neighbourhood preserves positive clock speed, linear cutoffs and pulse-free endpoints. This proves actual admissible states, including all nonlinear reaction terms.
By continuity, shrink delta and lambda_1 once more so both canonical derivatives keep the signs of v_x,v_P and magnitudes at least |v_x|/2, |v_P|/2 throughout that rectangle. Integrating from -delta to delta gives
\[|w_\lambda(\delta)_x-w_\lambda(-\delta)_x|\ge\delta|v_x|, \qquad |w_\lambda(\delta)_P-w_\lambda(-\delta)_P|\ge\delta|v_P|. \tag{9}\]
4. Canonical risk, scope and next question
For any deterministic estimator from the complete record and supplied calibration/energies, let epsilon_x,epsilon_P be its separate worst-case absolute errors on the admitted constrained preparation class. Applying the triangle inequality at the two common-record endpoints in (9) proves
\[\epsilon_x\ge\frac{\delta|v_x|}{2},\qquad \epsilon_P\ge\frac{\delta|v_P|}{2},\qquad \epsilon_x\epsilon_P\ge\frac{\delta^2|v_xv_P|}{4}>0. \tag{10}\]
The product has units ML^2/T with no 2 pi normalization. The same fixed bound holds for all 0<lambda<=lambda_1 and hence for the liminf of the optimal risk product as lambda decreases. Its constants depend on E,c,b, masses and pulse design. No uniform assertion is made under cooling, shrinking preparation, c tending to zero, or varying the design. A curve and its two endpoints do not establish positive projected area or exact minimax constants. This settles the existence branch of R28 and advances the exclusion-of-zero test for this prepared information class; universal scale selection and quantum identification remain open.
Source capsule: R25/B56’s parameter-dependent compensator and inherited smooth-flow/inverse methods -> divide the two residual constraints by coupling -> test the rank on an actual receiver energy shell. R27/B58’s topological result motivated locating ambiguity in canonical coordinates; its theorem is not used in this proof. B59 records the bounded method audit; review separates written proof acceptance from literature status.
Next R29: reveal a second initial probe displacement q_2=0, retaining the same two energies and full final record. Its additional scaled row is integral t f_2(s_0+v_0t)x_w(t) dt. Test whether the resulting three linear constraints plus receiver energy give a local inverse and whether distinct global common-record pairs remain. An isolated solution is not by itself global recovery.