What question did this study set out to answer?

This research aims to determine if the CPSE certification for canonical exits applies to all integers in the tunnel class.

May 9, 2026Open Access

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Shift-Robust Compensation and the Canonical-to-Tunnel Extension in Collatz Dynamics

KMKyung-Up Moon

Key Points

This research aims to determine if the CPSE certification for canonical exits applies to all integers in the tunnel class.
Proved the Tunnel Shift Formula T^k(n) = x_can + 2c·3^k.
Checked c ∈ {0,...,2^13-1} for complete residue cover at modulus 2^14.
Verified across 400 high-energy words with max delay of 54 steps.
Demonstrated shift-robust compensation with zero failures.
Achieved exhaustive verification of tunnel class extension.

Abstract

Papers A and B of this series established an energy budget framework and exhaustive symbolic closure for high-energy Collatz corridors. This paper addresses the critical gap: does the CPSE certification of canonical exits extend to the entire tunnel class of actual integers? We prove the Tunnel Shift Formula Tᵏ (n) = xcan + 2c·3ᵏ and show that checking c ∈ 0,. . . , 2¹3-1 constitutes a complete residue cover at modulus 2¹4. Exhaustive verification across 400 high-energy words confirms shift-robust compensation with zero failures and max delay 54 steps. Paper C of a four-part series. Paper A: Moon, K. -U. (2026). Zenodo. https: //doi. org/10. 5281/zenodo. 20068553Paper B: Moon, K. -U. (2026). Zenodo. https: //doi. org/10. 5281/zenodo. 20068640

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Shift-Robust Compensation and the Canonical-to-Tunnel Extension in Collatz Dynamics

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Abstract

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