# Encoded-ball exact projection and fresh full-factor SVD verification

The new integer projection backend is exact across the declared encoded2^-48 vector-noise ball. Fresh legacy-GEMM and integer-backend runs on the known cyclic record produce identical projected matrices and SVD factors; both certify exactly209 sketch singular values above the fixed2^-24 cutoff. The maximum validated singular-value error is<2.555e-13. These results narrow the floating bridge without proving uniform SVD/phase convergence.

| Result | Status / scope |
|---|---|
| New integer backend, all encoded-ball projections | EXACT; numerator bound2504646576090828<2^53, safe int64 accumulation |
| Legacy GEMM, a priori whole-ball projection | CONDITIONAL arithmetic model; error<1.580822e-9, insufficient for inherited phase envelope |
| Fresh actual legacy projection | EXACT equality to integer N/2^55 on this one cyclic record |
| Both fresh full-factor SVD outputs | CERTIFIED order209; error<2.555e-13; all321 cutoff intervals checked |
| Fresh original legacy proposer | Emitted209 integer phases byte-identical to paired legacy run |
| Fresh legacy vs archived cyclic | Basis/shift/eigenvalues/angles/emission exactly identical; no new phase bound issued |
| Fresh source recovery and raw replay | Four original hashes regenerated; three raw controls replayed; both eligible Grams exact |
| External nonconstructor replay | PENDING; checklist and blank receipt supplied |

No dense Hankel matrix is stored or allocated. This certificate concerns an8191×321 trajectory SKETCH. The retained sketch count209, native count1893, historical1146 diagnostics and nominal747-dimensional dynamical observer kernel are distinct. The latter remains unobservable; no hidden-state reconstruction is claimed.

## Executions and provenance

The freeze predates the paired runs. The actual source rank is not forced. Two paired estimator/SVD runs plus one separate unchanged original legacy proposer ran (three fresh estimator executions total). The paired paths agree on this record and are not independent statistical replications. Code/factor hashes, deltas, all321 singular values, complete left/right factors and emitted proposals are retained. No source phases/rates/jitter/coefficients are loaded by the paired proposer. Uniform projection uses a new backend and must not be advertised as proof that every legacy BLAS projection is exact.

The separate same-team recovery restored four complete PR195 records (75,160,616 scalar entries), then checked three raw controls (56,370,462 scalar differences) and exactly reproduced both eligible PR204 sketch Gram sets. Dense-unit retains raw-ball HOLD (squared integer norm62 versus1) despite an earlier phase PASS. The separate archived replay preserves two phase certificates and four negative controls. A supplementary raw projection replay checks another18,790,154 differences and all2,629,311 integer sketch entries. These counts describe computations on synthetic reconstructed records, not new apparatus readings.

Three new corrupted full-factor fixtures are refused (zeroU, zeroVt, changed projection delta/hash); originals unchanged. Known source data and post-execution structural/provenance drivers are disclosed. Fresh-environment and separately implemented same-team checks do not satisfy nonconstructor independence. The external checklist/schema is a handoff, not an attestation. An initial wrong-directory raw-replay launch failed before executing the script; its log is preserved beside the corrected successful replay.

## Portable witness replay

Extract CORE and every WITNESS ZIP into one EMPTY folder (reject absolute, traversal, duplicate or symlink members). With Python3.12, install requirements.txt and run:

```bash
OPENBLAS_NUM_THREADS=1 python replay_entry.py
```

Byte-identical paired NPZ outputs are losslessly deduplicated with explicit hash-bound aliases. The wrapper reconstructs those duplicate files from retained canonical bytes, then verifies package and array hashes before importing the exact certificate, then recomputes both complete dyadic Grams/cross-Grams and residual/cutoff enclosures. It does not run a proposer or reconstruct raw records. The complete retained factors independently support each actual SVD certificate; no opaque SVD accuracy promise is assumed.

For new raw reconstruction, follow handoff/INDEPENDENT_REPLAY_HANDOFF.md using immutable parent captures. Stage sibling floating-timing-work, uniform-two-work and this directory named rounding-replay-work; then run raw_verify.py. The original run.py refuses an existing results directory. Do not overwrite retained evidence: use a separate work copy with only generated results removed. Negative-control scripts under controls/ have disclosed scratch-relative staging assumptions and are retained source/receipts, not automatically executed by portable replay.

## Open returns

Uniform output-quality/convergence of the floating SVD and downstream phase estimator; externally attributable nonconstructor replay; SI/source/preparation/clock/noise/metrology and native metric qualification. Original instrument-contract204/205 HOLD, modulo-pi ambiguity, full-Hankel admission and review HOLD remain unchanged. See THEOREMS.md for exact hypotheses and supported witness-format limitations.

Physical promotion0. Development — no registry entry. O03 tolerances are protocol controls, not theory constants. Public Site unchanged this turn.