Adaptive Dynamic Fixation Sequences: Engineering Growth Through Stochastic Selection and Deterministic Scheduling

This paper introduces Adaptive Dynamic Fixation Sequences, the most general multiplicative framework where the system dynamically selects at each step which component remains passive and which value drives growth. This maximum flexibility creates a fundamental dichotomy between two paradigms with qualitatively distinct mathematical behaviors.

Stochastic ADFS involves random selection at each step. The paper proves almost-sure divergence for seeds exceeding unity via submartingale arguments, establishes martingale properties (constant expectation despite divergence), characterizes variance growth (linear in time, random walk behavior), and demonstrates almost-sure path uniqueness—different realizations diverge with probability one. No universal closed form exists; each trajectory requires explicit simulation.

Structured ADFS employs deterministic schedules. A complete taxonomy identifies seven canonical subtypes spanning growth rates from exponential through double-exponential to tetrational, each characterized by its fixation pattern and multiplier rule. The Growth Universality Theorem proves that for any computable growth function, an appropriate structured schedule exists achieving that rate—positioning ADFS as a computationally universal framework. The paper explicitly constructs schedules for polynomial, exponential, and tower-exponential targets, providing algorithms for optimal schedule synthesis.

Hierarchy theorems establish that all prior multiplicative classes are strict subsets: CPS emerges from constant fixation, PAMPS from parity-based periodic fixation, RMGS approximates via rapid switching. Applications to supply chain optimization demonstrate three-phase structured scheduling (inventory buildup, diversification, cost optimization) outperforming static allocation policies on real marketplace seller data. The framework applies equally to computational resource allocation, portfolio rebalancing, and multi-armed bandit problems.

ADFS represents the culmination: maximum expressiveness while maintaining tractability through schedule structure versus stochastic chaos.

Position in Series: Paper 5 of 5 — The universal adaptive framework containing all prior classes.