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Paper 09
Series Capstone
v0.1.1
2026-08-11
Finite Certificate Frontiers for the Collatz Map
From the Local Affine Atlas and a descent sieve to an integer-anchored hard branch — the capstone of the series. Draws the first eight papers' finite local-dynamics decomposition (finite parity word ↔ unique residue cylinder → exact affine operator) together into the series' capstone summary, while explicitly identifying the global-quantifier gap that remains uncrossed.
The series' capstone: an accounting of exact finite coverage, and the global gap not yet crossed
The paper explicitly does not prove that 𝔉ₖ(N) has a uniform extinction depth for all N, does not prove a closed asymptotic upper bound for K(N), and does not rule out an integer-anchored infinite hard branch — Section 62, "What does this paper not prove?", lays out this global gap verbatim, and it is the boundary all nine papers ultimately stop at.
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 9 / 9