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Paper 04
Core Theorem
v0.1
2026-08-10
Bidirectional Residue-Class Translation: 2ᵏ Cylinders and 3ᵘ Progressions
From the Collatz Local Affine Atlas to exact inverse fibers, the odd skeleton, and bidirectional-tree reconstruction. Proves that every admissible parity word w of length k corresponds to a unique source residue cylinder Ω_w, and that T^k(r_w + 2^k a) = m_w + 3^u a holds exactly.
Exact bidirectional residue-class translation: the affine relationship between source cylinders and T^k
Proves an exact bidirectional translation between finite words and their residue classes; the NEXT paper turns to the closer-to-global question of which charts must eventually descend at sufficiently large scale — this paper does not yet answer that.
Connections
Relationship to the rest of the series — the source document itself already points to the "next paper," reproduced here as-is.
Series progressCore series 4 / 9