This paper presents a comprehensive cross-era analysis of the algorithmic evolution of Large Language Models (LLMs) through four developmental epochs: Before Transformer (pre-2017), Transformer (post-2017), Instruction-tuned \& Open-source LLMs, and Multimodal Agents (2024-2025). A novel innovation pathway framework is introduced that traces causal relationships between architectural breakthroughs and emergent capabilities, addressing critical research gaps in three dimensions: (1) Cross-paradigm synthesis connecting statistical foundations to modern multimodal systems, (2) Causal innovation mapping demonstrating how architectural choices propagate through model generations, and (3) Cross-domain capability analysis quantifying transfer between representation learning, knowledge acquisition, behavioral alignment, and multimodal integration. This analysis reveals that LLM progression represents fundamental paradigm shifts rather than incremental improvements, with transformer architectures, human feedback mechanisms, and open-source ecosystems collectively enabling the transition from specialized NLP tools to general reasoning systems. We provide empirical evidence through case studies of capability emergence, quantify innovation impacts using performance metrics, and examine safety implications through recent jailbreak analysis and refusal mechanism studies. The contributions include: (a) a unified lifecycle synthesis with original analytical framework, (b) innovation trajectory mapping with causal pathway analysis, and (c) validated evolutionary principles for forecasting next-generation AI capabilities.