Plate-Level Period-3 Seeds and the Activation Law of the Newton Z₃
Abstract
Paper 36 in the "Geometry of the Critical Line" programme We study the origin of the Newton-side Z₃ transient previously resolved structurally as a mesoscopic period-3 core in the transient block T (RN6), but left without a generative mechanism. Rather than seeking a new SCT bridge object, we investigate the holomorphic dynamics of the Cover Function itself, z → C_α(z) = z − exp(−α/z), as a third arc of the programme. We show numerically that the iteration z → C_α(z) possesses repelling period-3 cycles across the tested α-range, and that these cycles lie on the Julia-set skeleton rather than in dynamically featureless regions. This establishes genuine plate-level period-3 material. We then test the direct-parent hypothesis that the Newton Z₃ band is a simple readout of this plate-level structure. That hypothesis fails: period-3 cycles persist outside the Newton Z₃ window, and their lowest multipliers vary monotonically rather than peaking in the live band. A more refined picture survives. Near α ≈ 0.39, the lowest-multiplier period-3 family undergoes a sharp change in corridor residence, from a region external to the Newton relay corridor into a corridor-resident configuration. This family-switch event supplies the enabling condition. Within the active band, Newton-side stagnation geometry acts as an amplitude modulator, while routing-floor reweighting suppresses the signal at shutdown. We formulate this as a three-factor activation law: plate-level period-3 seeds, corridor entry by family switch, and projection-layer gating by Newton transport geometry. The main conclusion is that the Newton Z₃ is neither a purely Newton-internal accident nor a direct cross-arc bridge image. It is best understood as a projection-selective activation of persistent plate-level period-3 material in the holomorphic dynamics of C_α.
Fields of Study & Subjects
Nonlinear Dynamics and Pattern Formation
Electrical and Electromagnetic Research
Earthquake Detection and Analysis
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 20 March 2026
- Language
- EN
- Document type
- Preprint