Fine-Phase Defect Rigidity in Record-Critical Accelerated Collatz Cycles

Program position. This paper is the first public module of the periodic / positive-nontrivial-cycle branch of a modular research program for the accelerated Collatz map. This branch is logically separate from the aperiodic Entry-Exit branch. In the aperiodic branch, Future-Minimum Critical Corridors and Beatty Clock Rigidity in Accelerated Collatz Dynamics (Entry Program I) supplies the physical front end: future-minimum critical corridors, short-gap Beatty rigidity, first-passage geometry, and a capacity barrier. A separate centered-residue rigidity theorem supplies the downstream arithmetic layer, referred to in the author's research architecture as the Delta-Core Exit. The deterministic bridge from the Entry-side physical structure to repeated subcritical centered-residue visitation / integer realizability remains open. Positive periodic cycles require a different treatment. They do not simply fall into the aperiodic centered-residue Exit mechanism, so the periodic obstruction is developed as a separate cycle-rigidity branch. The present paper isolates an explicit record-critical sector of hypothetical primitive positive accelerated-Collatz cycles and proves a conditional rigidity chain under hypotheses (H1)-(H5). Within that sector, endpoint-state information determines the physical fine correction. Sufficiently long repeated physical fine factors force recurrence. This yields logarithmically syndetic noncanonical defects, which are strengthened by an entropy argument to positive-density fine-phase defects. An affine comparison then places the parent checkpoint states on the full capacity scale. The theorem chain established here is: explicit record-critical hypotheses (H1)-(H5)→ endpoint-state fine-correction determinization→ logarithmic fine-factor rigidity→ defect syndeticity→ positive-density fine-phase defects→ capacity-scale localization. This is a conditional rigidity theorem, not a terminal cycle-exclusion theorem. The paper does not prove that every hypothetical positive nontrivial Collatz cycle satisfies (H1)-(H5), does not prove the global reduction of all positive cycles to the stated record-critical sector, does not prove complete nontrivial-cycle exclusion, and does not prove the Collatz conjecture. The current public program therefore has two distinct branches: Aperiodic branch:Entry Program I→ deterministic centered-residue visitation / integer-realizability bridge [OPEN]→ centered-residue rigidity / Delta-Core Exit. Periodic branch:Cycle Exclusion Program I→ global cycle reduction / terminal exclusion [OPEN]. The deposited LaTeX source reproduces the public manuscript. The accompanying verification package reproduces selected finite examples and numerical constants only; no theorem in this paper depends on computer-assisted proof.