# Omega Naught Class-I frequency-22 corrected verifier package

This package contains the exact frequency-22 certificate payloads and corrected
Python and Julia verifiers for the fixed radial Class-I icosahedral grid
`X_22`.  The mathematical solution payloads and their two canonical solution
digests are unchanged.  The corrections affect minimum-orbit reporting,
decimal rendering, Julia matrix transposition, runtime transcripts, and release
metadata.

## Authority boundary

- `1 <= f <= 9`: theorem of Paper VI v1.3 and frozen
  `CLASS_I_CUBATURE/1.0` data.
- `10 <= f <= 21`: reported post-publication computation.  This package does
  not contain the weight files, moment specification, verifier, transcript, or
  per-file hashes required to call that range publicly certificate-backed.
- `f = 22`: exact positive degree-69 construction and exact degree-70
  obstruction on the fixed node set `X_22`.
- `f >= 23`: open conjecture.

Do not collapse these four status classes into one theorem.

## Corrected frequency-22 facts

- `X_22` has 4,842 nodes in 52 full-icosahedral `H3` orbits.
- The degree-69 moment matrix has rank 49 and a three-dimensional affine
  solution space.
- Every selected degree-69 orbit mass is exactly positive.
- The exact minimum orbit mass is at `(11,11,0)`, not `(22,0,0)`.
- Its orientation decimal is approximately
  `0.003874805888537519508839054528148877`.
- The unique degree-70/71 invariant solution has exactly one negative orbit,
  `(21,1,0)`.
- That mass is approximately
  `-0.006609660200017031631251308016577647`.
- The exact dual spherical mean equals the exact forced negative mass.
- Reynolds averaging excludes arbitrary nonnegative degree-70 weights on the
  same fixed node set, not only orbit-constant starting weights.

## Why the published port failed

The original Python program chose the displayed minimum after converting exact
`Q(sqrt(5))` values to binary64.  Cancellation made `(22,0,0)` appear smaller.
The corrected Python verifier compares differences with the exact `K5.sign()`
method and uses high-precision `Decimal` only for orientation text.

The original Julia verifier already compared the masses exactly, so its stale
assertion expecting `(22,0,0)` failed.  After correcting that assertion, Julia
reached a second implementation error: `Matrix(transpose(matrix71))` attempted
to apply scalar `transpose` to `K5`.  The corrected code uses
`permutedims(matrix71)`, which exchanges matrix axes without acting on field
elements.

## Successful replays

Python 3:

```bash
python3 Omega_Naught_ClassI_f22_exact_verifier.py
```

Julia 1.11.6:

```bash
julia --startup-file=no OmegaNaughtClassIF22ExactVerifier.jl
```

Both programs print `CERTIFICATE PASS`, the minimum orbit `(11,11,0)`, the
negative orbit `(21,1,0)`, and the unchanged solution digests:

```text
degree_69_solution_sha256=0865572f76c7b8c9816ce937a79361c8926abe785ac3f41b06e4f289665bf340
degree_71_solution_sha256=c1e620fc9cf6d99623423cf82450540a5991679e8ded6632bd959914068949f2
```

The two decimal summaries may differ in the last printed rounding digit.  They
are not certificates; the exact `Q(sqrt(5))` payloads are authoritative.

## Source and transcript identities

| File | Bytes | SHA-256 |
| --- | ---: | --- |
| `Omega_Naught_ClassI_f22_exact_verifier.py` | 16972 | `ae615d262ecfc564aed119e5002c4fa9c4fadf35812002426298caf7c6ab89bf` |
| `OmegaNaughtClassIF22ExactVerifier.jl` | 16723 | `5c45650b3baec9889204ac423f8ec62c9cc0782a03e5c30a8467b9cb01a1efea` |
| `Omega_Naught_ClassI_f22_verification_output.txt` | 918 | `7f61a254d04afdc9e65e54355afe84ede6afba9e5874492e755758392a60bf54` |
| `OmegaNaughtClassIF22ExactVerifier.runtime.txt` | 1468 | `9ec4cbbc3ad02fe9291684830b669522cd4d25fdbff0d97449c3f282279a5336` |

These hashes identify this corrected candidate only.  External archival
fixation, peer review, and independent third-party reproduction remain separate
status axes and are not claimed by this README.

## License

Documents, papers, specifications, certificates, JSON, CSV, and data in this
package are licensed under CC BY 4.0 (`LICENSE`). Python and Julia verifier
source is licensed under the MIT License (`LICENSE-CODE`).

ZIP container bytes are excluded from the canonical payload inventory. The
published ZIP is a transport convenience identified separately by its byte
length and SHA-256. Independent compressors are not required to reproduce
identical ZIP bytes. The canonical object is the manifest-listed file set and
inventory digest.