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This work aims to explore the null orbit of the Syracuse map and its influence on the map's divergence behavior.

May 4, 2026Open Access

The Null Orbit of the Syracuse Map

Key Points

This work aims to explore the null orbit of the Syracuse map and its influence on the map's divergence behavior.
Characterised the null orbit through its algebraic, analytic-continuation, and 2-adic properties.
Investigated divergence obstruction and cycle equations within the framework of the Syracuse map.
Provided a logarithmic bound via Hölder–zeta interpolation in a companion study.
The null orbit organizes divergence obstruction through its unique properties.
Established limiting geometric behavior at critical points identified in the map.
Identified three boundary results that inform the non-closure of the framework surrounding the Syracuse map.

Abstract

We identify a single arithmetic object — the null orbit \ (N = \-1{2 2^k: k Z\}\) of the map \ (f (x) = 3x + 1\) — and show that it organises the divergence obstruction of the Syracuse map. The null \ (x₀ = -1/2\) admits three independent characterisations: the algebraic fixed point of \ (f\), the analytic-continuation value of \ (3ⁱ\) evaluated outside the disk of convergence, and (up to the transparent factor 2) the unique \ (2\) -adic fixed point sustaining maximal expansion \ (1\). CYCLE OBSTRUCTION — The Syracuse cycle equation \ ( (2ⁿ - 3T) x = ST^*\) is not algebraically reducible to the pure map's fixed-point equation. Baker's theorem constrains cycles by bounding \ (|2ⁿ - 3T|\) from below, but does not by itself exclude them. The null orbit plays no role here. DIVERGENCE OBSTRUCTION — The null orbit organises the limiting geometry through three mechanisms: the algebraic centre of the dominant mean multiplicative dynamics at \ (q = -1/2 Z^+\), the \ (2\) -adic escape at \ (q = -1 Z^+\), and the regularised accumulation. The open interface is the promotion of static mean contraction together with independent control of the additive correction \ (AT\) to a pointwise orbital drop criterion \ (T 2 > AT\). The companion paper (Envelope Bound, DOI 10. 5281/zenodo. 19937295) supplies a non-circular logarithmic bound \ (AT = O (T) \) via Hölder–\ (\) interpolation, so that any eventual positive co-length margin \ (KT/T > ₂ 3 + \) would imply finite drop for all sufficiently large \ (T\). The underlying difficulty is the Collatz singularity: \ (Z^+₎₃₃\) embeds densely in the odd coset \ (1 + 2Z₂\) but has Haar measure zero, so the known \ (2\) -adic ergodicity does not transfer automatically. Three boundary results identify the non-closure interfaces of the present framework: Baker bounds do not control numerator divisibility; \ (SU (2) \) -density does not imply deterministic orbital equidistribution; and the self-referential remainder cannot be controlled by assuming the surplus it is meant to prove. The framework should be read as a localisation of the Collatz frontier, not as a closure of it. EPISTEMIC MARKERS — Each statement carries a status marker: I = proved, D = definition, N = negative result, O = open interface, B = boundary result. Serie I · v0. 7 · DOI all versions: 10. 5281/zenodo. 19874522

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