Generalization v0.1.1 2026-08-10

Generalized (mx+r) Systems and Residue-Class Operation Translation

From the Collatz special case to commutative scalar affine dynamics, a phase boundary, and a generalized local atlas. Strips out Collatz's specific coefficients (3, 1) and studies the parity-preserving generalized system defined by positive odd integer parameters m and r, showing that Collatz sits inside a larger class: Residue-Class Operation Translation (RCOT).

RCOT: Collatz is a special case of a larger class of generalized (mx+r) systems Generalizes the first six papers' structure from Collatz's specific coefficients (3, 1) to general odd parameters m and r; the scope is which algebraic properties survive generalization, not a new convergence proof.

Connections

Relationship to the rest of the series — the source document itself already points to the "next paper," reproduced here as-is.

PreviousPaper 06 · Valuation Language and the Accelerated Collatz Map

NextPaper 08 · Algebraic Domains of Validity and Structural Breakage Theorems

Series progressCore series 7 / 9

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