# Record-wise certificate for archived floating two-step outputs

Development — no registry entry. Physical promotion0. This is conditional numerical verification of named archived outputs, not new floating-estimator execution, independent data/review, uniform convergence, or apparatus qualification.

## 1. Eligibility and approximation quality have separate carriers

The raw record has37 preparations,8191 times and62 channels, with exact integer/2^48 encoding. Every raw reading is range/hash/layout checked against the pinned encoded stress baseline. Eligibility requires each62-vector's squared INTEGER difference≤1; this is vector norm≤2^-48 in model units. An outside-ball record HOLDs before this contract's phase proof. HOLD does not establish phase failure.

The matrix being approximated is the8191×321 exact SKETCH B=YA/2^55. It is not the8191×2294 raw matrix and is not the full Hankel matrix. An actual binary64 basis Ua and untrusted coefficient proposal Ca must satisfy

ra=||B−UaCa||F ≤ rparent+Cparent·2^-48.

The right side is the previously certified available-approximation envelope: ||ES||F≤||E||F||S||2≤Cparentδ. It is a measured output-quality gate; no claim is made that every floating output meets it. G=UaᵀUa, K=UaᵀB and ||B||F² determine ra² by exact integer cancellation. Negative squared residuals reject inconsistent evidence rather than being clipped. Separate full raw projection replays reproduce G,K and the raw sketch norm. Cached-Gram portable replay does not reread omitted raw data.

The source-order theorem at2^-37 covers the smaller2^-48 raw ball, preserving exactly209 encoded-sketch singular values above the frozen2^-24 cutoff. Actual archived discovery metadata must report209; no truncation or forced order is introduced.

## 2. Measured polar alignment

Let U0 denote the pinned saved baseline basis and Ui=Ua the actual archived basis. Put Gi=UiᵀUi and Wi=UiGi^-1/2. Boundsηi≥||Gi−I||2<1 establish their rank and polar normalization. Let O be an untrusted approximate orthogonal alignment withηO≥||OᵀO−I||2<1 and Q=polar(O). Define p(x)=x/(1+sqrt(1−x)). Then

||WaQ−W0||2 ≤ ||UaO−U0||F + sqrt(1+ηa)p(ηO)+p(ηa)+p(η0) = d.

All Gram defects use exact dyadic arithmetic with conservative Frobenius bounds. The alignment residual is evaluated exactly from

||UaO−U0||F²=tr(OᵀGaO)+tr(G0)−2tr(OᵀUaᵀU0).

A floating Procrustes SVD only proposes O; acceptance does not trust its SVD equations. The actual basis Gram and cross-Gram are recomputed from the archived binary64 arrays in portable replay.

## 3. Neither saved shift is assumed exact least squares

For each saved binary64 transition Ti define Tbar_i=Gi^1/2 TiGi^-1/2 and

sigma_i=||Ui_plus−Ui_minus Ti||2/sqrt(1−ηi).

Thus ||Wi_plus−Wi_minus Tbar_i||2≤sigma_i. Both residuals must be charged. The reference deleted-row lower bound is

beta0=sqrt(1−||U0,last2||F²/(1−η0)),

and ||Tbar_0||2≤M0=(1+sigma0)/beta0. Alignment givesσmin((WaQ)_minus)≥beta0−d. Require beta0>d. Directly subtracting the two saved shift equations yields

||QᵀTbar_aQ−Tbar_0||2 ≤ b=[sigma0+sigmaa+d(1+M0)]/(beta0−d).

This proof uses measured arrays and residuals; it makes no exact-SVD/exact-LS assumption about the archived floating algorithm.

## 4. Actual eigensystem, argument output and source targets

For actual saved Va,Λa and inverse proposal Ha, verify rhoa=||VaHa−I||2<1. With epsilon_a≥||TaVa−VaΛa||2,

Delta_lambda=epsilon_a||Ha||2/(1−rhoa)

bounds a matrix perturbation whose exact eigenvalues are the actual saved Λa. Passing to the polar basis multiplies this perturbation byκa=sqrt((1+ηa)/(1−ηa)). Orthogonal alignment Q adds no norm factor.

The baseline source reduction D0 and its validated inverse bound h0, norm d0 and intertwining residual LJ0 come from the pinned PR197 receipt, with all relevant baseline array hashes checked. The exact source J is normal and orthogonal; its rotation approximation was charged in that receipt. Therefore the actual emitted eigenvalues are compared with J through radius

R=h0·LJ0+κ0 h0 d0 (b+κa Delta_lambda),
κ0=sqrt((1+η0)/(1−η0)).

Require R<1,2R<the complete209-target minimum chord, and2(asin(R)+argument_error)<that same chord. Each emitted2^-60 integer must equal ties-to-even quantization of its actual saved binary64 argument; validated complex arguments bound argument_error. All209 source labels must have a unique distinct actual eigenvalue in their disks. The resulting circular phase bound is radians per TWO-STEP sample. Original frequency remains ambiguous moduloπ; source-labeled lifts are conditional.

## 5. Controls and interpretation

The controls were archived and their prior results known before this protocol was frozen. No new proposer/estimator run, blinding or independent execution is claimed. New work consists of exact source projections, basis alignments, residual proofs and replays under a newly frozen contract.

Baseline and the PR201 cyclic-unit record are inside the raw ball. PR200 dense-unit changes all62 coordinates by one quantum per vector, so its norm sqrt62·2^-48 is outside this ball. Its prior actual phase certificate passes, but this new acquisition/quality/phase CONTRACT must HOLD on raw eligibility. A phase certificate alone cannot admit raw data.

For every eligible record, zeroing Ca must violate the exact sketch-fit envelope. Zeroing Ha produces Va0−I with Frobenius residual sqrt209≥1 and must violate the inverse gate. These are concrete counterfactual proposals evaluated through the same exact equations; original arrays and frozen protocol remain unchanged. They are not claims that all bad proposals or adversarial records have been enumerated.

This return narrows the gap between an ideal theorem and executable, record-wise floating-output admission. It does not prove uniform floating convergence over the noise ball, full-Hankel acceptance, physical observation/source/clock/metric mapping, SI frequencies, hidden-state reconstruction or independent review. O03 budgets remain protocol controls; original204/205 HOLD and the nominal747-dimensional dynamical observer kernel remain unchanged.