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Paper 02
Core Theorem
v0.1.1
2026-08-10
Collatz Local Affine Atlas: Exact Affine Linearization of Finite Parity Words
Finite-Word Affine Closure, Count/Order Decomposition, and a Word-Order Correction. Establishes the series' first core mathematical layer: proves that every finite parity word w of length k corresponds, under the modified Collatz map, to an exact affine operator F_w(x) = (3^u(w) x + b_w) / 2^k.
Exact affine closure of finite parity words: F_w(x) = (3^u x + b_w) / 2^k
The paper explicitly does not prove "∀n, T^j(n)=1" for some j, does not prove every infinite parity sequence eventually contains a descending prefix, and does not derive universal convergence from the average value of u/k — it proves only the exact affine structure of the finite words themselves.
Connections
Relationship to the rest of the series — the source document itself already points to the "next paper," reproduced here as-is.
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