Reproduce a certificate

Exact commands for named frozen outputs. Passing a run is implementation conformance for that named object. Conformance tests do not establish theorem validity. G₆ packaging is not a theorem joining the theories. The tuple exists definitionally because unrelated typed objects can always be placed in a tuple. A canonical hash proves equality of a named byte string. Aggregating separate maps into interfaces=PASS is a historical implementation outcome, not current source-map discipline. Agents: start at /archive/MACHINE.json (schema OMEGA_NAUGHT_MACHINE/1.0) for object-scoped commands, hashes, and claims_not_made.

Concrete uses

Each demonstration names its returned data and verification method. Experimental pages keep their own setup and records.

Uses with demonstrations and verification methods.
Use	Demonstration	Verification
Geometric interchange	Export OMEGA_KERNEL_NODES/1.0 and rebuild the labeled kernel.	Compare labels, frame, and true ε against the hosted node log.
Deterministic arithmetic replay	Forward floor map on a seed in D₁₅ using exact BigInt Q, S.	⌊Qd/S⌋ and remainder identity Qd = S d′ + ρ.
Exact reconstruction	Recover 399 from remainders of 399→135→45→15→5→1→0.	Divisibility, remainder bounds, and reverse identity.
Spherical polynomial integration	Apply a published Class-I rule inside its certified degree.	Exact moment identities on the named frequency; 10–21 and 23–24 recovered certificates are hosted.
AI-assisted inspection	Fetch MACHINE.json, resolve an object, verify hashes, run the named checker.	Scoped PASS/FAIL with source and execution evidence. Independent axes stay independent.

Acquisition and run recipes

Worked AI workflow

Fetch /archive/MACHINE.json (schema OMEGA_NAUGHT_MACHINE/1.0).
Select a named object or claim. Independent axes stay independent.
Retrieve its definitions and source artifacts.
Verify the artifact hashes.
Run only the specified checker for that object.
Compare the result against the stated expectation.
Return a scoped result with source and execution evidence.

Resolve aliases certificate-f22-69, certificate-f22-70, and g6-release-1-1 in source_registry. Cubature 10–21 and 23–24 exact packages are hosted; do not collapse them with Paper VI or f=22.

Software dependencies

Setup transcription OMEGA_NAUGHT_Software_Dependencies.docx (45,009 bytes, SHA-256 3841413f…). It is not an additional source paper. The G₆ and OMEGA_NAUGHT/1.0 Rust runners declare no third-party crates: after a Rust toolchain, Cargo, and platform build tools are installed, cargo run --release --offline --locked needs no crate download. Lean/Mathlib acquisition still needs Git and network for a visitor re-run; this archive's lake build PASS on 2026-09-13 is hosted at evidence/current/. Julia companion uses standard libraries Dates, SHA, and TOML only. Plots is not required for that companion.

Declared runtimes and libraries.
Component	Runtime and libraries	Version status
G₆ Rust runner	Rust and Cargo; no external crates	Edition 2021; lock format 4; compiler unpinned
Rust conformance runner	Rust and Cargo; no external crates	Edition 2021; lock format 4; compiler unpinned
G₆ Julia runner	Julia; standard library SHA	No compatibility bound in its Project.toml
Julia proof companion	Julia; standard libraries Dates, SHA, TOML	Compatibility 1.11; recorded runtime 1.11.6
Lean formalization	Lean, Lake, Git; Mathlib and eight inherited packages	Lean 4.19.0; exact Git revisions in lake-manifest.json; lake build PASS on this archive 2026-09-13
Frequency-22 verifiers	Python standard library; Julia SHA and Printf	Archived PASS on Python 3.12.13 and Julia 1.11.6; Python 3.10+ is sufficient. Current Python revision r2 refuses python3 -O
Typed-map audit	Python and SymPy	sympy>=1.12; no exact environment lock
Geometry checks and drift	Python and NumPy; local geometry module	NumPy version unpinned
Julia geometry plot	Julia; LinearAlgebra, Printf; external Plots	Plots is not required for the proof companion; install it in a separate environment
Scalar-interface certificate	Python standard-library fractions	No third-party Python dependency

Historical G₆ artifact — G6_RELEASE/1.1 canonical record

Implementations: Rust, Julia, and the pinned TypeScript snapshot.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir g6-repro && cd g6-repro
curl -fsSO https://omega-naught.grok.me/releases/G6_RELEASE_1.1/G6_RELEASE_1.1.zip
curl -fsSO https://omega-naught.grok.me/releases/G6_RELEASE_1.1/G6_RELEASE_1.1.zip.sha256
curl -fsSO https://omega-naught.grok.me/releases/G6_RELEASE_1.1/RELEASE.json
sha256sum -c G6_RELEASE_1.1.zip.sha256
python3 - <<'PY'
import hashlib, pathlib
got=hashlib.sha256(pathlib.Path('RELEASE.json').read_bytes()).hexdigest()
assert got=='f63ae1c2d595b4805ba9d36687a828fd87200febe01079b184723f7d5eaf746c'
print('RELEASE.json matches frozen envelope')
PY
unzip -q G6_RELEASE_1.1.zip
cd G6_RELEASE_1.1/g6
# Requires an installed Julia runtime, a Rust toolchain, Cargo, and platform build tools.
# The G6 Rust runner declares no third-party crates.
julia --startup-file=no julia/run.jl
(cd rust && cargo run --release --offline --locked)

Named run commands

cd G6_RELEASE_1.1/g6

julia --startup-file=no julia/run.jl

(cd rust && cargo run --release --offline --locked)

Expected: canonical_sha256=92187a21a7ad8ecef384eda3a654aaa49daf46a31422e168eb5871d218c949fa

Requires an installed Rust toolchain, Cargo, and the platform build tools. The G₆ Rust runner declares no third-party crates; cargo run --release --offline --locked is the conformance run. cargo test is the optional test suite, not the conformance run. The live specification HTML is a provenance copy of a frozen contract; it is not the current semantic architecture.

OMEGA_NAUGHT/1.0 named manifest outputs

Implementations: Python, Julia, Java, and Rust.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-naught-10 && cd omega-naught-10
curl -fsSO https://omega-naught.grok.me/conformance/omega_naught_conformance.py
curl -fsSO https://omega-naught.grok.me/g6/OMEGA_NAUGHT_1_0.txt
python3 omega_naught_conformance.py OMEGA_NAUGHT_1_0.txt

Named run commands

python3 omega_naught_conformance.py OMEGA_NAUGHT_1_0.txt

Separate OMEGA_NAUGHT/1.0 conformance: Python, Julia, Java, and Rust.

Frequency-22 exact certificate

Implementations: Python 3.10+ (archived PASS on 3.12.13) and Julia 1.11.6, run separately.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-f22 && cd omega-f22
curl -fsSO https://omega-naught.grok.me/cubature/Omega_Naught_ClassI_f22_certificate_package.zip
python3 - <<'PY'
import hashlib, pathlib
got=hashlib.sha256(pathlib.Path('Omega_Naught_ClassI_f22_certificate_package.zip').read_bytes()).hexdigest()
assert got=='f63c43e1dbd1ac5b7009e66fe577b27e8a231f90fe6b01eef779147826ae55eb', got
print('zip digest matches')
PY
unzip -q Omega_Naught_ClassI_f22_certificate_package.zip
# Flat archive: files extract into the current directory. Python 3.10+ is sufficient.
sha256sum -c SHA256SUMS.txt
python3 Omega_Naught_ClassI_f22_exact_verifier.py
julia --startup-file=no OmegaNaughtClassIF22ExactVerifier.jl
# Current website revision r2. Not a zip member. Original inventory is unchanged.
curl -fsSLo Omega_Naught_ClassI_f22_exact_verifier.r2.py https://omega-naught.grok.me/cubature/Omega_Naught_ClassI_f22_exact_verifier.r2.py
python3 - <<'PY'
import hashlib, pathlib
got=hashlib.sha256(pathlib.Path('Omega_Naught_ClassI_f22_exact_verifier.r2.py').read_bytes()).hexdigest()
assert got=='d9b2450ef16b138f73f89cbc1ee3a298010d27bcf1c184d5de57246a34376387', got
print('r2 digest matches')
PY
python3 Omega_Naught_ClassI_f22_exact_verifier.r2.py
# python3 -O Omega_Naught_ClassI_f22_exact_verifier.r2.py  # must refuse before certificate computation

Named run commands

sha256sum -c SHA256SUMS.txt

python3 Omega_Naught_ClassI_f22_exact_verifier.py

python3 Omega_Naught_ClassI_f22_exact_verifier.r2.py

julia --startup-file=no OmegaNaughtClassIF22ExactVerifier.jl

Current Python verifier is website revision r2: it refuses python3 -O, python3 -OO, and PYTHONOPTIMIZE before certificate computation. The original 16,972-byte verifier (SHA-256 ae615d26…) remains the zip and SHA256SUMS inventory member. Revision r2 does not change the certificate, the theorem, or the canonical solution digests. Published Julia source executed on official Julia 1.11.6; both the Python and Julia verifiers print CERTIFICATE PASS and the unchanged canonical solution digests. Implementation conformance: PASS. The corrected certificate package was executed with Python 3.12.13 and Julia 1.11.6 on Linux x86-64. Both exact implementations report (11, 11, 0) as the minimum positive degree-69 orbit, (21, 1, 0) as the negative degree-71 orbit, and identical canonical solution digests. Python and Julia pass separately. This is a two-implementation conformance check, not a claim of external independent reproduction.

X₂₃ and X₂₄ exact certificate

Implementations: Python 3.11+ (replay_recovered.py); historical verifier uses assertions.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-f23-f24 && cd omega-f23-f24
curl -fsSO https://omega-naught.grok.me/cubature/ClassI_Cubature_f23_f24_exact_certificate_v1.1.json
curl -fsSO https://omega-naught.grok.me/cubature/verify_ClassI_f23_f24_exact_v1.1.py
curl -fsSO https://omega-naught.grok.me/cubature/ClassI_Cubature_f23_f24_verification_ledger_v1.1.txt
curl -fsSO https://omega-naught.grok.me/cubature/replay_recovered.py
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('ClassI_Cubature_f23_f24_exact_certificate_v1.1.json','e886510ec03afe014ad143d6d0b45677fc2bb58a7df495374b5780cf79d5b5af'),
 ('verify_ClassI_f23_f24_exact_v1.1.py','41d83727473850ebfbe0bbea0f57f38a64c08d9849170a880b23163769509ce1'),
 ('ClassI_Cubature_f23_f24_verification_ledger_v1.1.txt','4d56dc2c65466fc89e6c275fdc38012dbca2c5b74456cf045e7ceedb30a38e6e'),
 ('replay_recovered.py','a3bf98e765f18d32f4ac10867c4e914269c61a763d0f7e0b2d1a236b5f41355a'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('f23/f24 originals and launcher match recorded hashes')
PY
# Python 3.11 or newer. Do not use python3 -O; the launcher must refuse.
python3 replay_recovered.py
# python3 -O replay_recovered.py  # must refuse before certificate computation
# Printed audit extract and LaTeX-emitted JSON 3ecbf665… are distinct files; do not substitute them.

Named run commands

python3 replay_recovered.py

python3 -O replay_recovered.py

The three original f=23,24 files are hosted under /cubature/ with their recorded hashes: ClassI_Cubature_f23_f24_exact_certificate_v1.1.json (1,407,835 bytes, SHA-256 e886510e…), verify_ClassI_f23_f24_exact_v1.1.py (6,818 bytes, SHA-256 41d83727…), ClassI_Cubature_f23_f24_verification_ledger_v1.1.txt (10,905 bytes, SHA-256 4d56dc2c…). The guarded launcher replay_recovered.py (1,621 bytes, SHA-256 a3bf98e7…) hash-pins those three files and refuses python3 -O. This-archive Python 3.11.2 replay PASS; that is implementation conformance of recovered source, not independent third-party reproduction and not peer review. The printed finite-audit extract remains a distinct artifact. The LaTeX-emitted JSON (SHA-256 3ecbf665…) has identical mathematics and different bytes; do not substitute it for the original. Does not enlarge Paper VI. No maximality. Neither argument proves failure or impossibility at any higher degree.

Recovered f=2 through f=21 exact certificates

Implementations: Python 3.11+; historical verifier unmodified; r2 revision.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-f2-f21 && cd omega-f2-f21
curl -fsSO https://omega-naught.grok.me/cubature/classI_exact_certificates_f2_f21.json
curl -fsSO https://omega-naught.grok.me/cubature/verify_classI_f2_f21.py
curl -fsSO https://omega-naught.grok.me/cubature/verify_classI_f2_f21.r2.py
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('classI_exact_certificates_f2_f21.json','7b3945ad58b76341bbc22e7812a359f64b61a68d413b2668012dc155803edb5e'),
 ('verify_classI_f2_f21.py','7c3cd105dae6cf36fd140a8b6525ed722fb02b88bd2ba152c465847befa944fa'),
 ('verify_classI_f2_f21.r2.py','4af07ac2ba14540dfb230920f108cddd0b21e8e80dd2e61dadb79acaad64d7df'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('f2–21 certificate and verifiers match recorded hashes')
PY
# Historical verifier uses assertions; do not run it with python3 -O.
python3 verify_classI_f2_f21.py classI_exact_certificates_f2_f21.json
python3 verify_classI_f2_f21.r2.py classI_exact_certificates_f2_f21.json --ledger f2_f21_ledger.json
# Homogeneous harmonic generators. Distinct from the f=23,24 raw even vertex power sums.
# Does not enlarge Paper VI. No maximality.

Named run commands

python3 verify_classI_f2_f21.py classI_exact_certificates_f2_f21.json

python3 verify_classI_f2_f21.r2.py classI_exact_certificates_f2_f21.json --ledger f2_f21_ledger.json

Recovered exact certificate package for 10 ≤ f ≤ 21 is hosted under /cubature/. Combined file classI_exact_certificates_f2_f21.json (1,815,267 bytes, SHA-256 7b3945ad…). Historical verifier and r2 revision are hosted. Per-frequency records at /cubature/frequencies/f10.json through f21.json. Python 3.11.2 r2 PASS on this archive: 383 identities, 404 positive masses. Post-publication computation; does not enlarge Paper VI. Maximality is not claimed. Not peer review. Not independent third-party reproduction.

15-bit successor search

Implementations: Python 3.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-successors && cd omega-successors
curl -fsSO https://omega-naught.grok.me/dynamics/Omega_Naught_Successor_Search_Verifier.py
curl -fsSO https://omega-naught.grok.me/dynamics/successor_search_certificate.json
python3 Omega_Naught_Successor_Search_Verifier.py

Named run commands

python3 Omega_Naught_Successor_Search_Verifier.py

Omega_Naught_Successor_Search_Verifier.py

Finite exact facts against the frozen 60-decimal enclosure of λ. Not an accepted successor.

OMEGA floor-map fibers (independent replay)

Implementations: Python 3.11+ standard library. Writes no files. Checks remain active under -O and -OO..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-fibers && cd omega-fibers
curl -fsSO https://omega-naught.grok.me/fibers/verify_handoff.py
python3 - <<'PY'
import hashlib, pathlib
got=hashlib.sha256(pathlib.Path('verify_handoff.py').read_bytes()).hexdigest()
assert got=='9f7179c5be49c0c4f797824011226dbd4d9980c7cc71aff1d80bd1a616e121c8', got
print('checker digest matches Appendix C')
PY
python3 verify_handoff.py
python3 -O verify_handoff.py
python3 -OO verify_handoff.py
# Stdlib only. Writes no files. Does not authenticate reported 0.2 cover ledgers.

Named run commands

python3 verify_handoff.py

python3 -O verify_handoff.py

python3 -OO verify_handoff.py

Expected: RESULT PASS

Mathematical replay of the 19 September 2026 handoff. Not a source paper. Reported 0.1/0.2 fiber certificate and COVER.json files were not supplied and are not authenticated. Seed blocks, not cosets. Not a Navier–Stokes or geometry theorem.

paper 8 companion ZIP (this-archive Python replay)

Implementations: Exact checker is standard-library Python. Numerical tests used NumPy 2.2.6 here, not the pinned 2.3.5..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-paper-8 && cd omega-paper-8
curl -fsSO https://omega-naught.grok.me/papers/8/OMEGA_Paper_VIII_Source_and_Code.zip
python3 - <<'PY'
import hashlib, pathlib
got=hashlib.sha256(pathlib.Path('OMEGA_Paper_VIII_Source_and_Code.zip').read_bytes()).hexdigest()
assert got=='cbdf7a29d67942eff0bf839f30fb8071125c5c213a60e8d648f116f07a2c2f75', got
print('companion ZIP matches this archive / Zenodo record 22848213 (corrected release)')
PY
unzip -q OMEGA_Paper_VIII_Source_and_Code.zip
cd OMEGA_Paper_VIII/companion
python3 verify_external_bundle.py sample_certificate
python3 omega_reference_checker.py --out ../../fresh_certificate
# Exact checker is stdlib, Python 3.10+. 5e47b7d5… is the original submitted ZIP recorded in the review, not this release.
# Numerical tests need NumPy/SymPy; pinned versions are numpy==2.3.5 sympy==1.14.0.

Named run commands

python3 verify_external_bundle.py sample_certificate

python3 omega_reference_checker.py --out fresh_certificate

Expected: CERTIFICATE_SHA256=32bcc2fefc44d0fe19575683b64fd67ddcd32327aeff32ab14cc988a507f4a79

Hosted ZIP SHA-256 cbdf7a29… is the corrected release on Zenodo. The review recorded 5e47b7d5… as the original submitted archive; that is a version difference, not a corrupted upload. Internal manifests pass (103 top-level, 74 companion). Checker and certificate hashes match the manuscript. This-archive run is not independent third-party reproduction. Not peer review. Not a Navier–Stokes blowup theorem. Does not enlarge Paper VI.

paper 8 Proof Audit 01 (localized forced flow)

Implementations: Python 3 with SymPy. This-archive 3.10.21 / 1.14.0. Supplied run 3.12.14 / 1.14.0..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-paper-8-audit-01 && cd omega-paper-8-audit-01
curl -fsSO https://omega-naught.grok.me/papers/8/audit-01/independent_checks.py
curl -fsSO https://omega-naught.grok.me/papers/8/audit-01/independent_results.json
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('independent_checks.py','380c32bd6bf54acc3fa055d2200e33e209b2e49dd5aa68cdf76b16f6c49c40da'),
 ('independent_results.json','01b5ece2e3234e6231b9f9ecee08e17fea89f3ba052c6149c74f26b221f10f1a'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('audit-01 script and supplied results match recorded hashes')
PY
python3 independent_checks.py --output this_archive_independent_results.json
# Requires SymPy. This-archive used 1.14.0 on Python 3.10.21; supplied run was Python 3.12.14.
# OMEGA_Localized_Flow_Complete_Proof.tex and original independent_checks.log were not supplied.

Named run commands

python3 independent_checks.py --output this_archive_independent_results.json

python3 -O independent_checks.py --output /tmp/paper8-audit01-opt.json

Expected: 29 check groups PASS; 56 degree-five monomials; degree-six obstruction −1/105

Additional work on Theorems 7.3–7.6 and 8.6, not a relabeling of the adversarial review. Not peer review. Not an additional source paper. Original TEX and original independent_checks.log were not supplied and were not reconstructed. Finite checks are not a proof-assistant certificate. Not a Navier–Stokes blowup theorem. Does not enlarge Paper VI.

OMEGA object exploration 02

Implementations: Python 3 with SymPy. This-archive 3.10.21 / 1.14.0 / mpmath 1.3.0..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-exploration-02 && cd omega-exploration-02
curl -fsSO https://omega-naught.grok.me/geometry/exploration-02/OMEGA_Object_Exploration_02.py
curl -fsSO https://omega-naught.grok.me/geometry/exploration-02/OMEGA_Object_Exploration_02.md
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('OMEGA_Object_Exploration_02.py','4c1b3cfa46262e21d19ece458d4dcb8fccd617d15ca4579a2f013108396c13a6'),
 ('OMEGA_Object_Exploration_02.md','12f3106a7980ce56e328c3ae29392f306c1c321e4c7c92decbce38fa00560513'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('exploration 02 script and note match recorded hashes')
PY
python3 OMEGA_Object_Exploration_02.py --output this_archive_exploration_results.json
python3 -O OMEGA_Object_Exploration_02.py --output this_archive_exploration_results.optimized.json
# Requires SymPy (and mpmath for decimal illustrations). This-archive: Python 3.10.21 / SymPy 1.14.0.
# Canonical seed remains one arcsecond. The 36-degree member is a related family instance.

Named run commands

python3 OMEGA_Object_Exploration_02.py --output this_archive_exploration_results.json

python3 -O OMEGA_Object_Exploration_02.py --output this_archive_exploration_results.optimized.json

Expected: 21 check groups PASS

Additional consequences of the six marks, not a source paper. Canonical seed remains one arcsecond. The 36-degree member is a related family instance. Not a camera-noise model. Not peer review. Does not enlarge Paper VII or paper 8. Frozen G₆ unchanged. Named author hashes 6c35365d… / e50f3443… were not received.

Affine dynamics, sampling, and quadrature

Implementations: Python 3 stdlib and Java 17+. This-archive 3.10.21 / OpenJDK 17.0.20. Julia and Rust sources not hosted..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-sampling-extension && cd omega-sampling-extension
curl -fsSO https://omega-naught.grok.me/geometry/sampling-extension/verify.py
curl -fsSO https://omega-naught.grok.me/geometry/sampling-extension/Verify.java
curl -fsSO https://omega-naught.grok.me/geometry/sampling-extension/expected.txt
curl -fsSO https://omega-naught.grok.me/geometry/sampling-extension/PROOFS.md
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('verify.py','381d091c92928a42a970f79c08f366a3681f74352b4780b4a16b93d862e60ce0'),
 ('Verify.java','4b00c0ee5edd80376d7179ed93721d53fb5fd3cfd5aeca31404e64acef246517'),
 ('expected.txt','7ef550cb3f03d5032769024b9cbeb47f5c143ef45f709a0f1b785128684fd662'),
 ('PROOFS.md','99d6b58e29940ea9e06d8585cccf0162a4f99a864c6748d416f393ec3cf2e636'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('sampling-extension files match recorded hashes')
PY
python3 verify.py | cmp -s - expected.txt
python3 -O verify.py | cmp -s - expected.txt
javac Verify.java && java Verify | cmp -s - expected.txt
# Julia and Rust sources were not hosted. run_all.py is not a this-archive command.

Named run commands

python3 verify.py

python3 -O verify.py

javac Verify.java && java Verify

java -da Verify

Expected: TOTAL 21 PASS; stdout byte-identical to expected.txt

Not a source paper. Canonical seed remains one arcsecond. Author report lists eight matching runtimes; author rust stdout is hosted. This archive reran Python and Java only. verify.jl and the Rust vendor tree were not received and were not reconstructed.

Boundary reconstruction and invariant cubic

Implementations: Python 3 with NumPy. This-archive 3.10.21 / NumPy 2.2.6..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-boundary-reconstruction && cd omega-boundary-reconstruction
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/verify_boundary_reconstruction.py
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/this_archive_verify.stdout.txt
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/Boundary_Reconstruction_Transport_Formal_Proof.pdf
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('verify_boundary_reconstruction.py','ce3e5ea29ca8de60e27e45164903065f340e54826935f88045b0ed715b5e3efe'),
 ('this_archive_verify.stdout.txt','60ad47fa29f0c7ada869555a393e497411f9c8b08344b2850b71c84a840890b1'),
 ('Boundary_Reconstruction_Transport_Formal_Proof.pdf','a8c9e39e29f283272b3c7dc2933c7dad1a02bc2651ce0341508317de3d295571'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('boundary-reconstruction files match recorded hashes')
PY
python3 verify_boundary_reconstruction.py | cmp -s - this_archive_verify.stdout.txt
python3 -O verify_boundary_reconstruction.py | cmp -s - this_archive_verify.stdout.txt
# Requires NumPy. This-archive used 2.2.6 on Python 3.10.21, captured 2026-09-22T12:35:49Z.
# Bind the 897-byte stdout to checker SHA-256 ce3e5ea2…; the superseded vacuous checker produced the same transcript.

Named run commands

python3 verify_boundary_reconstruction.py

python3 -O verify_boundary_reconstruction.py

Expected: TOTAL 30 PASS; stdout 60ad47fa… bound to checker ce3e5ea2… (2026-09-22.r2)

Not a source paper. Reconstruction restricted to H_A ⊕ ℝJ. Not a 13-point degree-six cubature. Alignment required for J(n_i)=0. The PDF and remarks are the written proofs. Checker r2 computes the J² spherical mean exactly; remaining groups are numerical. The 897-byte transcript is also produced by the superseded vacuous checker; bind the run to SHA-256 ce3e5ea2…. Frozen G₆ unchanged.

Explicit ten-sample reconstruction

Implementations: Python 3.10+ standard library. This-archive 3.10.21. Supplied transcripts name Python 3.12.14..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-ten-sample && cd omega-ten-sample
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/ten-sample/Ten_Sample_Reconstruction.md
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/ten-sample/verify_ten_sample.py
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/ten-sample/verification.txt
curl -fsSO https://omega-naught.grok.me/geometry/boundary-reconstruction/ten-sample/verification_optimized.txt
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('Ten_Sample_Reconstruction.md','967bc8afe442d6b0763f5d563b993a368161243d0f23b7274be46936d81639fe'),
 ('verify_ten_sample.py','cc850bda469cb33cc9d37c307dceede7e1b4e5ede002957c3ff7497c7a388c0f'),
 ('verification.txt','a6e04971faf30e665f00fc1b60a16488dddce315a83d715291b5298031268be0'),
 ('verification_optimized.txt','a6e04971faf30e665f00fc1b60a16488dddce315a83d715291b5298031268be0'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('ten-sample checker and supplied transcripts match recorded hashes')
PY
python3 verify_ten_sample.py | cmp -s - verification.txt
python3 -O verify_ten_sample.py | cmp -s - verification_optimized.txt
# Standard library only. This archive: Python 3.10.21 at 2026-09-23T23:53:35Z.
# Supplied transcripts were recorded under Python 3.12.14 and are byte-identical.
# Ten_Sample_Reconstruction.md is the written note, 9973 bytes.

Named run commands

python3 verify_ten_sample.py

python3 -O verify_ten_sample.py

Expected: TOTAL 13 PASS; stdout a6e04971… bound to checker cc850bda…

Not a source paper. Ten samples are minimal on this field and have no surplus consistency equation. The thirteen-sample consistency equations are not an error-correction guarantee. Two rational transport checks do not replace the written argument. The written note is Ten_Sample_Reconstruction.md, 9973 bytes, SHA-256 967bc8af….

OMEGA Synthesis 1.0 focused claim check

Implementations: Python 3.9+ standard library. Checks remain active under -O and -OO..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-synthesis-check && cd omega-synthesis-check
curl -fsSO https://omega-naught.grok.me/synthesis/omega_synthesis_claim_check.py
python3 - <<'PY'
import hashlib, pathlib
got=hashlib.sha256(pathlib.Path('omega_synthesis_claim_check.py').read_bytes()).hexdigest()
assert got=='9397c5300b9eaf125958bf21c6a7064f369d14e388313d458cd21b1d825fb41d', got
print('checker digest matches')
PY
python3 omega_synthesis_claim_check.py
python3 -O omega_synthesis_claim_check.py
python3 -OO omega_synthesis_claim_check.py
# OMEGA_Synthesis(3).pdf was not supplied to this archive; do not invent it.

Named run commands

python3 omega_synthesis_claim_check.py

python3 -O omega_synthesis_claim_check.py

python3 -OO omega_synthesis_claim_check.py

Expected: ALL FOCUSED CLAIM CHECKS PASS

Review of a synthesis document, not a source paper. OMEGA_Synthesis(3).pdf was not supplied here; claimed SHA-256 01f0d19c… is unverified. Not Lean, not cubature, not a decoder-integration audit, not independent third-party reproduction.

OMEGA–Navier–Stokes geometric transport 3.0

Implementations: Python 3.11+ with SymPy 1.14.0. Checks remain active under -O and -OO..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-ns-extension && cd omega-ns-extension
curl -fsSO https://omega-naught.grok.me/extension/OMEGA_NS_Geometric_Extension_v3.pdf
curl -fsSO https://omega-naught.grok.me/extension/verify_omega_ns.py
curl -fsSO https://omega-naught.grok.me/extension/audit_omega_sources.py
curl -fsSO https://omega-naught.grok.me/extension/test_omega_auditor.py
mkdir sources
curl -fsS https://omega-naught.grok.me/extension/sources/LEAN_Proof_Complete_Final%281%29.docx -o 'sources/LEAN_Proof_Complete_Final(1).docx'
curl -fsS https://omega-naught.grok.me/extension/sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1%281%29.zip -o 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('verify_omega_ns.py','f054d343f78c96f18d1a4360453dd1761d3fc8de7757edfb56a746996921e531'),
 ('audit_omega_sources.py','7bb072407b4c2c3ea58c0a8a0ad392d5870fc1d7d339ff20b67679b9ad96f08e'),
 ('test_omega_auditor.py','9cae8025b0b6637ca5b188b30d84235b3c73a122445f0a7fdd9fcf8df6d5f509'),
 ('sources/LEAN_Proof_Complete_Final(1).docx','275da4ea14130d21aa1b797b643ec78192e195780bfd8352bac699a75b73542e'),
 ('sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip','ef8f1eb75b83f1e5763d280374eb8b73d9ab00b8dcb76b1824aee771869cd27a'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
pdf=hashlib.sha256(pathlib.Path('OMEGA_NS_Geometric_Extension_v3.pdf').read_bytes()).hexdigest()
print('scripts and source archives match recorded hashes')
print('received PDF sha256', pdf)
print('revision-notes claimed 42802e00…; this archive reports the actual digest')
PY
python3 verify_omega_ns.py
python3 -O verify_omega_ns.py
python3 audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'
python3 test_omega_auditor.py audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'
# Requires Python 3.11+ and SymPy. Symbolic identities and source audit only.
# Do not treat a pdflatex rebuild as required. No fresh Lean or Julia run.

Named run commands

python3 verify_omega_ns.py

python3 -O verify_omega_ns.py

python3 audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'

python3 test_omega_auditor.py audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'

Expected: TOTAL: 62 exact checks; original package accepted; 38 subprocess checks / 36 mutation rejections

Mathematical extension, not a source paper. Not a blowup result, not Lean, not an enlargement of Paper VI cubature certificates. Received PDF SHA-256 98a3f747… does not match the revision-notes claim 42802e00…. SRSD remains frozen. Version 1.0 remains under /extension/v1/.

Omega-Alpha July 2026 synthesis

Implementations: Python 3.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-alpha && cd omega-alpha
curl -fsSO https://omega-naught.grok.me/omega-alpha/omega_alpha_reference.py
python3 omega_alpha_reference.py

Named run commands

python3 omega_alpha_reference.py

omega_alpha_reference.py

The Python reference checks declared finite identities after the integer protocol constants are fixed. Decimal evaluation of φ, π, and λ is a reproducibility check, not symbolic exactness. Passing tests do not establish theorem validity for Papers I–VI, peer review, or independent third-party reproduction.

Paper VII kernel generator

Implementations: Python 3; verify_geometry.py requires NumPy. Optional Julia visualization requires Plots in a separate environment, not the proof companion..

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-kernel && cd omega-kernel
mkdir -p generated regenerated
curl -fsSLo omega_reference_geometry.py https://omega-naught.grok.me/geometry/kernel/omega_reference_geometry.py
curl -fsSLo verify_geometry.py https://omega-naught.grok.me/geometry/kernel/verify_geometry.py
curl -fsSLo omega_kernel_oriented_r3.jl https://omega-naught.grok.me/geometry/kernel/omega_kernel_oriented_r3.jl
curl -fsSLo generated/OMEGA_local_kernel_v1_6_0.json https://omega-naught.grok.me/geometry/kernel/OMEGA_local_kernel_v1_6_0.json
curl -fsSLo generated/OMEGA_local_kernel_v1_6_0.obj https://omega-naught.grok.me/geometry/kernel/OMEGA_local_kernel_v1_6_0.obj
python3 - <<'PY'
import hashlib, pathlib
checks=[
 ('generated/OMEGA_local_kernel_v1_6_0.json','b280522bcd2cf63927e8dfd6438f78fa3a2e7af0cbfd484ee9cd55994b9c3753'),
 ('generated/OMEGA_local_kernel_v1_6_0.obj','2654f85dab5881e9a0659985729450a89d7cc8df3b07035c22a4c4372b10a079'),
]
for name, expect in checks:
    got=hashlib.sha256(pathlib.Path(name).read_bytes()).hexdigest()
    assert got==expect, (name, got)
print('published JSON and OBJ match declared hashes')
PY
python3 -m pip install numpy
python3 verify_geometry.py
# Independent generation writes regenerated copies; do not overwrite generated/ published exports.
# Generator defaults write into the current working directory; those CWD files are not the published pair.
python3 omega_reference_geometry.py --json regenerated/OMEGA_local_kernel_v1_6_0.json --obj regenerated/OMEGA_local_kernel_v1_6_0.obj
cmp regenerated/OMEGA_local_kernel_v1_6_0.json generated/OMEGA_local_kernel_v1_6_0.json
cmp regenerated/OMEGA_local_kernel_v1_6_0.obj generated/OMEGA_local_kernel_v1_6_0.obj
# Optional visualization. Plots is not a proof-companion dependency.
julia --project=. -e 'using Pkg; Pkg.add("Plots")'
julia --project=. omega_kernel_oriented_r3.jl

Named run commands

python3 -m pip install numpy

python3 verify_geometry.py

python3 omega_reference_geometry.py --json regenerated/OMEGA_local_kernel_v1_6_0.json --obj regenerated/OMEGA_local_kernel_v1_6_0.obj

Expected: PASS all 8 conformance groups, including test_distributed_exports

Place the published JSON and OBJ in generated/ before running the unchanged verifier. Generator defaults write into the current working directory; those CWD files are not the published pair. Independent generation should use --json/--obj into regenerated/. Direct evaluation of the closed landmark formulas, a sampled OBJ polyline, and an optional Julia binary64 visualization. Reconstruction of named coordinates, not a theorem checker for Paper VII.

Kernel Construct node log and scalar certificate

Implementations: Browser Construct mode; Python 3 standard library for the scalar certificate.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-construct && cd omega-construct
curl -fsSO https://omega-naught.grok.me/geometry/construct/verify_scalar_interface_exact.py
curl -fsSO https://omega-naught.grok.me/geometry/construct/omega-kernel-nodes.unit.true.json
python3 verify_scalar_interface_exact.py

Named run commands

python3 verify_scalar_interface_exact.py

Expected: OMEGA scalar interface exact algebra certificate: PASS

Interface and presentation around paper-vii-kernel.ts. The node log is the transfer record of the proved planar kernel, not a unique shell. Replay stops at kernel stage 15. The Python certificate is exact for Ψ and Φ only; it does not prove Paper VII Euclidean uniqueness. Not independent reproduction.

De-branded typed-map audit

Implementations: Python 3 with SymPy ≥ 1.12.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-typed-maps && cd omega-typed-maps
curl -fsSO https://omega-naught.grok.me/network/typed_connection_proof.py
curl -fsSO https://omega-naught.grok.me/network/requirements.txt
curl -fsSO https://omega-naught.grok.me/network/README.txt
python3 -m pip install -r requirements.txt
python3 typed_connection_proof.py

Named run commands

python3 -m pip install -r requirements.txt

python3 typed_connection_proof.py

Expected: FINITE FLOOR CERTIFICATE: PASS for every integer d in [0,32767] (32768 states; q<lambda certified)

Python 3 with SymPy ≥ 1.12. Symbolic identities; exact rational enclosures of √5, π, and λ; exhaustive floor equality on D = {0, …, 32767} with q < λ certified; exhaustive 23-step remainder reconstruction on all 32768 seeds; seven malformed-input classes rejected. No binary64 comparison is used in the floor certificate. Homothety of the six kernel marks is Euclidean scaling of the Paper VII configuration, not a unique shell. Radial template identities in the script are algebraic and are not a physical theorem. Passing the named runtime is implementation conformance of that audit, not independent third-party reproduction and not a joining theorem.

Lean 4 geometric-provenance formalization

Implementations: Python 3.10.21 for the axiom-coverage checker; Lean 4.19.0 / Mathlib v4.19.0 for lake build, which passed on this archive 2026-09-13.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-lean && cd omega-lean
curl -fsSO https://omega-naught.grok.me/formal/SHA256SUMS
python3 - <<'PY'
import hashlib, pathlib, urllib.request
base='https://omega-naught.grok.me/formal/'
for line in pathlib.Path('SHA256SUMS').read_text().splitlines():
    line=line.strip()
    if not line or line.startswith('#'): continue
    digest, name = line.split(None, 1)
    dest=pathlib.Path(name)
    dest.parent.mkdir(parents=True, exist_ok=True)
    urllib.request.urlretrieve(base+name, dest)
    got=hashlib.sha256(dest.read_bytes()).hexdigest()
    if got!=digest: raise SystemExit(f'mismatch {name}')
print('all listed members match SHA256SUMS')
PY
python3 scripts/check_axioms.py evidence/axioms.log
python3 scripts/check_axioms.py evidence/current/axioms.log
# lake build of these sources was run on this archive on 2026-09-13.
# Transcripts: evidence/current/build.log (SHA-256 ddc10580…) and evidence/current/axioms.log (SHA-256 82dae064…).
# Re-running lake build needs Lean 4.19.0 / Mathlib v4.19.0; the toolchain is not hosted.

Named run commands

python3 scripts/check_axioms.py evidence/axioms.log

python3 scripts/check_axioms.py evidence/current/axioms.log

Expected: Checked 147 theorem reports; axioms: Classical.choice, Quot.sound, propext

Hosted .lean sources, lakefile, lean-toolchain, and author evidence logs match the package SHA256SUMS byte-for-byte (28/28). On 2026-09-13 this archive ran lake build and lake env lean Omega/AxiomAudit.lean against those sources: Lean 4.19.0 commit 6caaee842e94, Lake 5.0.0-6caaee8, Mathlib v4.19.0 c44e0c8ee63…. Fresh build.log (1,960 bytes, SHA-256 ddc10580…) ends with Build completed successfully; modules were Built, not Replayed. Fresh axioms.log (11,804 bytes, SHA-256 82dae064…) is byte-identical to the author log: 147 theorem reports; axioms {propext, Classical.choice, Quot.sound}; four theorems depend on no axioms. scripts/check_axioms.py PASS. Hosted sources contain no sorry, admit, or native_decide. The toolchain and .lake cache are not hosted. That kernel-check is machine-checking of the selected maps in this package. It is not an additional source paper, not a joining theorem, not peer review of the eight source works, and not independent reproduction of those papers. Cubature 10–21 and f=23,24 exact packages are hosted separately under /cubature/; they are not produced by this Lean run. Frozen G₆ bytes are unchanged.

Julia mathematical proof companion v0.1

Implementations: Julia 1.11.6, standard library (author-supplied environment; not run on this archive).

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-julia-proofs && cd omega-julia-proofs
curl -fsSO https://omega-naught.grok.me/julia-proofs/downloads/OMEGA_NAUGHT_Julia_Proof_v0.1.reconstructed-from-members.zip
# This ZIP is labeled reconstructed; it is not the author-stated original container.
printf '%s  %s\n' 'b5d4e46ac32ac4ec08f96a6fc609fe8c8c3a291b7226809c2e79c3adac2e26ab' 'OMEGA_NAUGHT_Julia_Proof_v0.1.reconstructed-from-members.zip' | sha256sum -c -
unzip -q OMEGA_NAUGHT_Julia_Proof_v0.1.reconstructed-from-members.zip
cd OMEGA_NAUGHT_Julia_Proof_v0.1
sha256sum -c SHA256SUMS
# Requires Julia 1.11.6. Author O0/O3 logs are archived evidence, not a site run.
julia --startup-file=no --project=. scripts/check_inventory.jl
julia --startup-file=no --project=. run_proofs.jl

Named run commands

julia --startup-file=no --project=. scripts/check_inventory.jl

julia --startup-file=no --project=. run_proofs.jl

julia --startup-file=no --project=. scripts/reproduce.jl

julia --startup-file=no --project=. examples/reconstruction.jl

Expected: O0 and O3 exit 0; six identical certificate files; 147 indexed declarations. Not generated on this archive.

The 68 original members were recovered from OMEGA_NAUGHT_Julia_Proof_v0.1_Complete_Verbatim.docx and match SHA256SUMS byte-for-byte (74,508 source lines). Author-supplied Julia 1.11.6 O0/O3 run records are archived evidence and are not a run performed by this site. The reproduction driver updates run metadata, so archived evidence should be retained separately from any later execution. The original ZIP container was not recovered from the Word file; those same bytes are now hosted as /extension/sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip (1,681,923 bytes, SHA-256 ef8f1eb7…). The reconstructed transport ZIP under /julia-proofs/downloads/ is a later packaging of the recovered members and is not that original container. Cubature 10–21 and f=23,24 exact packages are hosted separately under /cubature/; they are not supplied by this Julia companion. lake build of the Lean package was run on this archive on 2026-09-13; see /formal.

OMEGA drift experiment v0.1

Implementations: Python 3 with NumPy.

From an empty folder

Download, extract if needed, verify hashes, install the named dependency, then run. Commands below use the public origin.

mkdir omega-drift && cd omega-drift
curl -fsSO https://omega-naught.grok.me/geometry/drift/omega_drift_experiment.py
curl -fsSO https://omega-naught.grok.me/geometry/drift/results.json
python3 -m pip install numpy
python3 omega_drift_experiment.py

Named run commands

python3 omega_drift_experiment.py

Numerical experimental control of a hypothesis-layer 4D rotation. Not a theorem of Papers I–VII. The published results.json is the named run (seed 20260831).