The Unified Viability Stack: A Composite Viability Certificate for Dynamic Cybernetic Stability

Seven preceding papers each approached adaptive system failure from a distinct theoretical direction — stress accumulation under epistemic deficit (P1), effective information collapse (P2), source- intrinsic relevance decay (P3), constructibility erosion (P4), dynamic cybernetic stability (P5), model currency failure (P6), and feedback semantic inversion (P7) — and each identified a distinct failure mode. This paper proves that these seven failure modes are not independent: they are projections of a single underlying structure onto different observational planes. We introduce the Unified Viability Stack (UVS), a seven-layer formal architecture in which each layer corresponds to one paper’s failure mode, the layers are strictly hierarchically ordered by logical implication, and a single quantity — the Cybernetic Viability Functional Φ(𝑡) — summarizes the joint state of all seven layers simultaneously. We prove the UVS Representation Theorem: Φ(𝑡) > 0 is necessary for Dynamic Cybernetic Stability, and Φ(𝑡) ≥ Φmin provides a bounded stability margin at time 𝑡. We further prove a Viability Trajectory Bound: the time derivativė Φ(𝑡) is bounded above by a sum of layer-specific degradation rates derivable from Papers 1–7, establishing Φ as a trajectory certificate rather than a snapshot condition. Sustaineḋ Φ(𝑡) < 0 implies Φ(𝑡) → 0— unified failure regardless of which layer initiates the degradation. We further prove that the three phase-transition parameters identified across the series — 𝜏 ∗ (P1), 𝜇∗ (P6), 𝜅destab (P7) — are coordinates of a single Tri-Critical Boundary in the system’s parameter space, and that crossing this boundary in any coordinate is sufficient for Φ(𝑡) → 0. The series thus converges: seven failure modes, one functional, one boundary.