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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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