Collapse as a Unifying Language for NP-Hard Optimisation: Seven Problem Classes, One Operation

We propose that all effective solvers for NP-hard optimisation problems perform the same fundamental operation: collapse — an irreversible, observation-driven reduction of uncertainty. We demonstrate this unifying language across seven problem classes (MKP, TSP, Graph Coloring, SAT, VRP, QAP, Sequential Decision), identifying each class's collapse chain equivalent. Experimental validation includes MKP +5.7% via BPTT, QAP +6.1% via Sinkhorn Jacobian, Graph Coloring -55% via Gumbel-Softmax, and VRP matching specialist tools. CDCL's constraint propagation is identified as SAT's native collapse mechanism, resolving the hard wall as a tool limitation rather than a conceptual one. Part of the NSD research programme. Version 1 (April 2026) remains available as a standalone record. Version 2 is a substantially expanded version of the original working paper. The core thesis is unchanged: all effective solvers for NP-hard optimisation perform the same fundamental operation—collapse (observe, reduce uncertainty, propagate). Version 2 adds the following over Version 1: Formal definitions of the collapse operator and collapse chain, with proofs of monotone uncertainty reduction and collapse failure modes A dedicated Background section reviewing the seven-box landscape classification framework Each of the seven problem boxes expanded with full problem structure analysis, mathematical derivation of the collapse operator, and independent experimental tables TSP boundary analysis: dense-perspective experiments confirming that TSP's landscape is smooth at observable scales, validating the hard boundary of the framework The Information Monotonicity property of collapse chains (global uncertainty is non-increasing despite local reversals) Expanded Related Work covering No Free Lunch, Algorithm Selection, Constraint Programming, Survey Propagation, and Belief Propagation A dedicated Experiments section with full protocol documentation Four-step protocol illustrated with a new scheduling problem example References expanded from 9 to 19 v3 update: Added Section "Unified MCCO Engine: Cross-Box Validation" — elevates the unification from linguistic (a shared vocabulary) to operational (a shared engine). A single MCCO engine with interchangeable collapse operator plug-ins (LP relaxation for Box 1, Gumbel-Softmax for Box 2, angular cutting for Box 4, Sinkhorn projection for Box 5) matches domain-specific specialist algorithms to within 0.7% across four problem boxes, validated with 30 independent runs and Wilcoxon signed-rank tests per configuration (14 configurations total). The specialist is statistically better at some scales (p < 0.05), but the practical gap never exceeds 0.7% — one engine, four plug-ins, four boxes. Protocol upgraded from four steps to five steps: added Step 2 "Reconnoitre and reshape the landscape" integrating the NP Landscape reconnaissance-transformation loop and DRF pipeline as prerequisites before collapse operator deployment. Complete three-phase pipeline: Phase 1 (NP Landscape) → Phase 2 (DRF) → Phase 3 (MCCO with plug-in collapse operator). Part of the NSD / MOEH research programme.