# OMEGA explicit joining theorem

Read JOINING_THEOREM.md for the precise scope and complete proof.
Run with Python 3.10+ (standard library only):

    python verify_join.py --output results_normal.json
    python -O verify_join.py --output results_optimized.json

The theorem connects marked scale, the affine scalar map, uniform and four
geometric modes of the 600-cell real network, an explicit integer embedding
derived from the rational and golden parts of its coordinates, the rational floor map, full
remainder records, modulo-eight records, and finite witnessed histories.
The original graph, coefficient, and domains are unchanged.

The key new synthesis is a collection of proved commuting relations and
inverse record translations, not a claim that shared constants force a unique
model or that every earlier source work has become the same theory. Several
constituent identities already occur in the supplied earlier Paper IX draft.

Normal and optimized reports match byte-for-byte. Eight deliberately corrupted
executions were rejected. General proofs are in the Markdown, not inferred
from finite-state test success. No end-to-end Lean proof or maximum-fiber
calculation is claimed. The current v0.6 deposit could not be fetched; the
accessible illustrated draft was version 0.3, and is identified as such.

Files: proof, standalone checker, normal/optimized JSON and logs, corruption
results, execution/provenance report, and SHA-256 manifest. Original source
papers are not bundled; the input hashes appear in EXECUTION.json.