# Wider timing sketch: finite counts and temporal closure Development — no registry entry. Physical promotion 0; independent review pending. This is a diagnostic of two already known synthetic PR195 records. It is not independent discovery, a new acquisition, or an extension of the phase adapter. ## 1. Waveform span and shift closure are different For preparation a, set s=(-1)^a and t_ak=o_a+(1+s mu)k+s j0(-1)^k. Origins are preparation-specific constants. For each of the 52 positive reference phases theta_j, let \[ \omega_{js}=(1+s\mu)\theta_j,\qquad g_{js}(k)=e^{i\omega_{js}k}e^{is\theta_jj_0(-1)^k}. \] The factor e^(i theta_j o_a) rotates that preparation's coefficients and does not add a temporal function. With two rate classes, the real waveform family has at most one stationary column and two real functions per pair per class: \[ \operatorname{rank}D_{\rm model}\le1+52\cdot2\cdot2=209. \] The sideband decomposition is \[ g_{js}(k)=\cos(\theta_jj_0)e^{i\omega_{js}k} +is\sin(\theta_jj_0)e^{i(\omega_{js}+\pi)k}. \] The two sideband amplitudes are tied. Counting them independently would wrongly identify the observation-column bound as 417. One-step shifts can separate the tied components, giving a shift-invariant closure bound of at most 1+52*2*4=417. Both are upper bounds: collisions, absent amplitudes or coefficient dependence may lower them. No exact dimension 209 or 417 is asserted. Two-step shifts instead preserve the original waveform family exactly: \[ g_{js}(k+2)=e^{2i\omega_{js}}g_{js}(k). \] This supplies a possible future stride-two adapter. It does not extend PR189 automatically: sampled collisions modulo pi, excitation, finite conditioning, subspace truncation, actual solver residuals and phase emission all need checks. ## 2. Nested sketches and two threshold rules The integer sketch matrix is A=[2W_old | v | 2W_new], with denominator 128. W_old is regenerated in its ORIGINAL 2294-by-160 shape with seed172026. W_new is a separately frozen 2294-by-160 sign matrix with seed184052. Widths 161,257,321 select nested column prefixes. Generating a larger matrix with the old seed and selecting its first160 columns would not preserve the old sketch because its row stride changes RNG consumption. For S_m=A_m/128, its exact squared Frobenius norm s_m^2 is recomputed. Let tau be the exact positive dyadic cutoff from the parent SKETCH_RULE.json. The two frozen diagnostic thresholds are \[ t_{m,\rm common}^2=\tau^2,\qquad t_{m,\rm scaled}^2=\tau^2s_m^2/s_{161}^2. \] Both squared thresholds are exact rationals. Their comparisons do not use rounded displayed decimals. For a declared per-preparation62-vector error delta, ||E S_m||2 <=sqrt(37*8191)*delta*||S_m||F. The scaled rule preserves the parent's threshold/noise-envelope ratio, using delta=2^-20 for that diagnostic comparison. The stress record is NOT asserted to satisfy an additive-to-old-clean error ball. Neither rule creates acquisition admission or a new theory constant. Nested prefixes imply Z_next Z_next^T-Z_old Z_old^T is positive semidefinite. Therefore counts above a COMMON threshold cannot decrease. Scaled-threshold counts have no such monotonicity requirement. ## 3. Exact small Grams from every encoded reading The raw little-endian int64 records have shape(37,8191,62), quantum2^-48. Reshape to X of size8191-by2294 with coordinate index c*37+a. Before int64 matrix multiplication, verify max|X|*max_column_l1(A)<2^63, which bounds every partial sum. Hence N=X A is exact, and the mathematical sketch is Z=N/2^55. \[ G=Z^TZ=N^TN/2^{110}. \] The integer numerator Gram is formed exactly with FLINT and stored losslessly as decimal integers. Its dimension is321, with smaller widths using principal prefixes. Hashes bind the complete raw record before/after projection. A second complete raw projection reproduces every Gram integer. No8191-by8191 temporal Gram or full Hankel array is stored. This certifies the exact mathematical sketch of the encoded bytes. It does not silently equate that sketch with a separately computed floating BLAS product. Floating eigensystems below are untrusted certificate proposals. ## 4. Validated eigenvalue enclosure Let Q be an untrusted square dyadic eigenvector proposal, Lambda a sorted real dyadic diagonal proposal, and compute exact residual norms with validated roots: \[ \eta\ge\|Q^TQ-I\|_2<1,\quad r\ge\|GQ-Q\Lambda\|_2,\quad g\ge\|G\|_2. \] Frobenius norms bound each operator norm; all residual numerators and scales are exact integers. Set qnorm=sqrt(1+eta) and d=eta/(1+sqrt(1-eta)). The polar factor O of Q is orthogonal with ||Q-O||<=d. The identities and triangle inequality give \[ \|G-Q\Lambda Q^T\|\le g\eta+r\,qnorm, \] \[ \|Q\Lambda Q^T-O\Lambda O^T\| \le\|\Lambda\|d(qnorm+1). \] Thus each ordered eigenvalue of symmetric G lies within R=g*eta+r*qnorm+||Lambda||*d*(qnorm+1) of the corresponding Lambda entry. The checker uses512-bit interval arithmetic and outward rational rounding. Strict lower>threshold_squared certifies above; strict upper=205. Thus this return places the comparison-model rank in[205,209], without proving its exact value. The scaled rule instead certifies204 retained values for its larger cutoff. ## 6. Limits These are source-aware finite constructor diagnostics. The PR195 stress proposer's HOLD remains immutable; no new phase outputs are certified. The nominal747-dimensional dynamical observer kernel is unchanged. Source/metric selection, SI clocks, independent review and the full acquisition contract remain open. O03 thresholds and seeds are protocol controls.