# UD thirty seven preparation instrument error contract **RESULT — Separate preparation, readout, clock and generator bounds jointly preserve the 1893-direction Hankel signal gap.** A common candidate allocation passes all three finite schedules, with retained-signal condition bounds 16657, 3156 and 1755. The bounds are sufficient and conservative; they are not measurements or necessary hardware specifications. Physical promotion remains **0**. This is development work with no registry entry and independent review pending. All tolerances are protocol controls under coefficient-ledger O03. No SI time conversion or physical-clock identification is supplied. ## Declared plant and measurement quantities Keep the frozen nominal generator A, the original 62-channel coordinate observer R and the explicit 37 preparation vectors from the parent certificate. Each nominal preparation has norm `a=sqrt(165/169)`. The candidate actual generator is \[ K=H^{-1}(A(w)+J),\qquad H=H^T>0, \] where the three native incidence rates obey `|w_k-1|<=r`, the metric obeys `||H-I||<=eta<1`, and an additional generator discrepancy obeys `J^T=-J`, `||J||<=sigma`. All matrix norms are Euclidean operator norms. The constant plant is H-skew, giving \[ \|e^{tK}\|\le g=\sqrt{\frac{1+\eta}{1-\eta}} \quad\text{for every real }t. \] Native incidence blocks have pairwise zero products by boundary nilpotency. With the independently established nominal bound `||A||<4`, this gives `||A(w)-A||<=4r` and \[ \|K-A\|\le e= \frac{4r+\sigma+4\eta}{1-\eta}. \] The actual preparation is `x_a+d_a`, with `||d_a||<=p`. At nominal time t, the actual time is `(1+theta)t+b+zeta`, with `|theta|<=chi`, `|b|<=b 0` and `|zeta|<=j`. The readout is `R+E`, `||E||<=rho`, with an additional output-vector error of norm at most nu. Readout errors may vary with preparation and time within their bounds; no independence or zero-mean assumption is used. The metric, rate and clock bounds refer to distinct quantities. Equal numerical allocations do not identify these objects or determine a physical time unit. ## Joint response error theorem Set `tau=b 0+j` and \[ e_{\rm eff}=e+(4+e)\chi. \] Duhamel's formula and H-skew norm control imply, for nominal t at least zero, \[ \|y_{\rm actual}(t)-R e^{tA}x_a\| \le u+vt, \] where \[ u=g\left[(1+\rho)p+\rho a+a(4+e)\tau\right]+\nu, \qquad v=g a e_{\rm eff}. \] Preparation, instantaneous readout and absolute timing errors enter u. Generator error and relative clock drift enter the accumulating term vt. This separation prevents a fixed jitter budget from silently absorbing clock drift over a long record. The formula includes simultaneous errors and their cross terms. It does not sum separately optimized one-error-at-a-time limits. It assumes the declared constant H-skew plant; unrestricted nonnormal or time-varying generators need a different propagator bound. ## Finite Hankel error theorem Use the parent normalized response matrix, with M samples on each Hankel axis and time `(i+j)/2` at block(i,j). The mean and mean-square times under the actual Hankel duplication weights are \[ \overline t=\frac{M-1}{2},\qquad \overline{t^2}=\frac{7M^2-12M+5}{24}. \] These are not the uniform moments of the 4095 or 8191 raw time samples. The matrix repeats raw vectors with triangular weights. Direct counting gives \[ \boxed{\|\Delta H_M\|\le\beta_M= \sqrt{37}\sqrt{u^2+2uv\overline t+v^2\overline{t^2}}.} \] The certificate verifies both time-moment formulas by exact weighted integer sums. No large Hankel matrix is materialized. The simpler uniform raw-vector bound is `u+v(M-1)`; both bounds are checked against the parent error contract. If s is the parent lower bound on the smallest nonzero singular value, and B its upper bound on the largest, then `beta_M3s/4,\qquad \widetilde\sigma_{1894}