Explicit recovery-preserving joining theorem

Read JOINING_THEOREM.md for the definitions and the proof. SHA-256 7b3ff30da8055eb1ac88fee78ff0449b3e2102a0f1a9df9ac0b0e29349476070Copy. The checker is verify_join.py. Its finite examples supplement the written argument. They are not the argument.

The note uses the 27 October illustrated draft, version 0.3, and says that drafting run did not retrieve the later v0.6 deposit. This page does not attribute the synthesis to Paper IX version 0.6. It does not rewrite the 27 October statement 306 ≤ ℓ* ≤ 360. It does not improve the preprint interval 307 ≤ ℓ* ≤ 352, and it does not compute that optimum. The 360-bit residue witness remains the recovery device in the note.

Stated identities

With $\varphi=(1+\sqrt5)/2$, $\lambda=\varphi/(\varphi+\pi)$, $\Omega_\gamma(t)=1+\gamma(t-1)$ for $0<\gamma<1$, and the note’s matrix $M=13I+A$, $P=M/25$, $q=Q/S$, $D=25S$.

$E(t,w)=t\mathbf 1+Cw$ uses the 120 unit 600-cell vertices as the rows of $C$. The inverse on that image is the note’s mean and $C^T/30$. On the integer image $E_Z(t,x,z)=t\mathbf 1+C_0 x+C_1 z$, with $2C=C_0+\varphi C_1$,

Full remainder records and modulo-eight records translate to each other on valid images, with the reduced output retained. For every finite horizon, the terminal state and the ordered modulo-eight witnesses recover the embedded source. Ideal $\lambda$ and the rational coefficient $q$ are not the same floor: at $n=2\times 10^{15}$ the certified ideal floor is 679899542689557 and the rational floor is 679899542689556.

What this archive reran

python3 verify_join.py and python3 -O verify_join.py on Python 3.10.21. Both exited 0. Stderr was empty. Both reports are the supplied bytes, SHA-256 d7528924f17109ac50bcac944acd5652a91e516924d9fb425398f576fab56c68Copy. The checker’s pinned graph digest is 40bc6a1d52efab3d75cf343e65ac06b314f9c9c375fc02bf6d45bf6bb21f9529Copy, the same fixture as the 27 October pages. It reports 120 positive leading minors of $M-9I$, the integer inverse certificate $MN=NM=hI$ with $h=4948372000$, all 480 entries of $AC=6\varphi C$, and the Gram identity $C^TC=30I_4$. A hash identifies bytes; it does not certify their mathematical correctness.

The supplied EXECUTION.json records Python 3.13.5 and eight rejected mutations. Those mutations were not rerun here. Three inputs named in that file — the version 0.3 PDF, the spectral-obstruction script, and a 4 November review ZIP — are not in this package and are not hosted. $\texttt{exact\_F\_max\_computed}$ and $\texttt{full\_geometric\_proof\_in\_lean}$ are false in the report.

Finite log replay

A separate ASCII log, treeOS_LOG/OMEGA-SRSD/1.0, is 24,941 characters and carries the same date. Its scope line says the profile is compatible with frozen SRSD and is not the archive or an installed system. That sentence is the log’s. This page does not turn it into an SRSD conformance review of Paper III.

The kernel source is saved verbatim as check.py, SHA-256 f31230bf8a31d45b91b637ba9aa2df42b9ec534c3a54baaca4205d90a077d5b7Copy. The log binds that digest. This archive ran python3 check.py treeOS_LOG_OMEGA_Joining.json and the same command under python3 -O, on Python 3.10.21. Both exited 0, stderr was empty, and the stdout bytes match, SHA-256 6a20db4b8d0c8e35635dea4cd2e63c76fdbdb46bf4e728d319a1e1d5a9ce51cbCopy. The verdict is INSTANCE_CHECKED for profile TREEOS-OMEGA-120/1. The matrix root is SHA-256 a6bfb755c12e57d493aedd43936ab10e96ceba5eea0ee2b01e289f72eed31423Copy.

The replay recomputes the pinned graph, 120 positive leading minors, three geometric examples, six history steps, seven scalar examples, and two corruption rejections. A zero-budget decode is recorded as UNRESOLVED. formal_proof_assistant_executed is false. The written statements F1–F7 and G1–G9 stay in the log. They were not reviewed as proofs, and this verdict is not PROOF_CHECKED. The named parent log and the declared legacy source digest were not in the package. Those absences, and the three inputs named in the supplied execution record, are not required by the hosted positive checkers. G9 still does not prove the interval 307 ≤ ℓ* ≤ 352.

Evidence edition 1.1.0

The current package is evidence edition 1.1.0, 254,716 bytes, SHA-256 80d058b1807c4bb3b9ab7ba53163df3e18a5c0cb70ab2befb5cbbb2824f378e9Copy. The earlier 26,470-byte ZIP stays at its original path. It is not replaced. Typed maps are in CONNECTIONS.json. The current index is relationship.json.

Replay from the extracted package: python3 verify.py --self-test and python3 -O verify.py --self-test. This archive ran both on Python 3.10.21. Both exited 0, stderr was empty, and the stdout bytes match, SHA-256 67794f3070ab6a83c7c446762e7a22e5918c8bb4a4e126419b7ae07f6e8c83c1Copy. Status PASS. The stored reports inside the package name Python 3.12.14. Apart from that version field, the report body matches. The run rejected eleven corrupted cases, recovered 32 records and 18 history steps, and did not issue INSTANCE_CHECKED. formal_proof_assistant_executed is false. The written proofs in PROOF.md were not reviewed.

The compact sharing record is treeOS_LOG version 1.1, 3,276 bytes, SHA-256 15a2931cec9df25a7588417917f0817be75d3e9deb7dc5ba5337d163b3640b18Copy. It has no embedded kernel. The INSTANCE_CHECKED receipt stays with the 24,941-character log and is not attached to this record. The frozen TGA 2.1 graph and atlas release 1.0 are unchanged.

Not claimed

• That the scalar constants force this graph.
• That the source works are one theory.
• A bit optimum, a fast general decoder, or a 352-bit volume recertification.
• Peer review, independent third-party reproduction, or a Lean proof.
• That INSTANCE_CHECKED reviews the written proofs or Paper III.
• A tenth Zenodo deposit. Nine deposits remain nine.

Related pages, kept separate: 600-cell signals, the 360-bit witness, and Paper IX.