Cascade Information Theory: Unified Framework for Multi-Witness Inference and Environmental Transformations

Cascade Information Theory provides a unified framework for analysing how information about an initial latent state propagates through a sequence of stochastic transformations and is partially recovered through multiple observational channels (witnesses). This paper introduces three formal contributions: (1) The Echo — E := I(S₀; W₁:ₙ) — a global information invariant measuring the total mutual information between the initial latent state and all witnesses jointly. The Echo quantifies the theoretical upper bound on recoverability from any reconstruction algorithm, generalising the Information Efficiency Metric (IEM) introduced in the Trace Forensics series to the full multi-witness, multi-layer setting. (2) The Synergy Decomposition — Δ := I(S₀; W₁:ₙ) − Σ I(S₀; Wᵢ) — formalising the multi-witness information gain observed empirically in earlier work and providing the theoretical explanation for why witness diversity improves reconstruction beyond simple additivity. (3) Fisher Information Contraction — Jᵢ ≤ (∇Tᵢ)ᵀ Jᵢ₋₁ (∇Tᵢ) — the information-geometric mechanism underlying cascade degradation, instability, and the computational phase transition in which systems remain theoretically identifiable yet become numerically non-reconstructable. The framework connects information theory, information geometry, and multi-view inference, and provides a principled foundation for reasoning about reconstructability under sequential transformations. A complete worked example in the linear-Gaussian setting grounds all three contributions analytically. Explicit connections are drawn to the Trace Forensics Research Series (Papers 1–10, Davidson 2026), which provides empirical validation of the theoretical constructs introduced here. Applications are discussed across physical systems, quantum measurement, biological neural processing, and distributed sensor networks.