Corpus record: PFT:THE_COLLATZ_CONJECTURE_AS_A_STRUCTURAL_THEOREM_ON_THE_ALLEN_ORBITAL_LATTICE
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The Collatz Conjecture as a Structural Theorem on the Allen Orbital Lattice -
2026-05-08
The Collatz conjecture is traditionally formulated as a number-theoretic iteration problem. In Pattern Field Theory (PFT), Collatz dynamics are formulated as a deterministic structural flow on the Allen Orbital Lattice (AOL), a prime-indexed hexagonal lattice based on Eisenstein integers. By specifying contraction regions, bounded expansion envelopes, and enforced cascade collapse under Phase Alignment Lock (PAL), Collatz convergence is treated as structurally inevitable in the PFT model space.
Pattern Field Theory First Premise
Pattern Field Theory asserts that mathematical and physical dynamics arise from structured constraints acting on discrete geometric substrates. The Allen Orbital Lattice (AOL) is the substrate supporting identity under load, convergence under cascade, and stability under Phase Alignment Lock (PAL).
The Allen Orbital Lattice
The AOL is defined as the Eisenstein integer lattice \(\mathbb{Z}[\omega]\setminus\{0\}\) with hexagonal norm \[\|a+b\omega\|_{\text{hex}}=\max(|a|,|b|,|a+b|).\] Shells of radius \(r\) contain exactly \(6r\) sites. This finite face and shell structure defines basin capacity and constrains long-run orbit behavior.
Collatz Operators as Structural Actions
Define the Collatz operator on \(\mathbb{N}\): \[C(n)= \begin{cases} 3n+1 & \text{odd expansion}\\ n/2 & \text{even contraction}. \end{cases}\] In PFT, \(3n+1\) is treated as a load-increasing action and \(n/2\) as a duplex contraction step.
Event Cascades and Envelope Trapping
Each odd step is followed by a halving cascade. Define an envelope map \[E(n)=\frac{3n+1}{2^k}, \quad k\ge 1,\] where \(k\) is the number of consecutive halving steps required to return to an odd state. The envelope is the effective step in the odd-only reduction.
Forbidden Growth Regions
Sustained divergence requires repeated odd expansions without compensating halving cascades. The PFT constraint is that expansion is linear while contraction is exponential under cascades. This prevents sustained growth.
PAL Enforcement and Collapse
Phase Alignment Lock enforces flux neutrality on stable faces: \[F(\partial f)=\sum_{v\in \partial f} e^{i\theta_v}=0.\] Non-neutral orbit states are unstable in the AOL model space and collapse inward under duplex constraint.
Structural Convergence Theorem
Proposition 1. All Collatz orbits converge to the trivial cycle, as a structural consequence of bounded expansion, enforced cascade contraction, and PAL stability constraints.
Proof. The iteration is trapped by an envelope that cannot sustain divergence under repeated halving cascades. PAL neutrality eliminates nontrivial long-run cycles as stable attractors. Therefore, convergence to the trivial attractor follows as the only stable terminal condition under these constraints. ◻
Relation to Coheron Modeling
The operational coheron modeling and dataset-based outputs are implemented at: \[\texttt{https://patternfieldtheory.com/coherons}\] This paper treats those computations as the canonical model-space implementation.
Document Timestamp and Provenance
This document is part of Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL). It defines the Phase Alignment Lock (PAL) constraint and specifies methods and replication procedures used by subsequent papers in the series.
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