Multi-Chain Multi-Tentacle Optimisation: Synthesising MCCO and Spacetime Digital Exhaust for Collapse-Centred NP-Hard Search

This paper synthesises two previously distinct frameworks of the NSD/MOEH programme: the Multi-Chain Collapse-Centred Optimisation (MCCO) framework, which deploys N chains of CPTP evolution as parallel optimisation agents on indeterminate landscapes; and the Spacetime Digital Exhaust (SDE) framework, in which a single chain is augmented with Nₜ peripheral factor observations at each time step. The synthesis is multiplicative rather than additive, producing a four-axis Cartesian object indexed by chain c, factor n, inner state k, and time t, with Nₜ tentacles per (chain, step) cell. The central design choice is the shared factor pool: at each time t, the Nₜ peripheral factors are shared across all N chains, while each chain has its own tentacle weight function w^ (c). This sharing produces implicit cross-chain information flow: an observation that updates one chain's posterior also informs the others through correlated weighting. The shared-pool architecture is the structural feature distinguishing the multi-chain multi-tentacle (MCMT) framework from a trivial parallelisation of single-chain SDE. The paper's contributions are theoretical. The MCMT architecture is defined with the shared factor pool and chain-specific tentacle weighting; the tensor evolution law is derived showing exact decomposition into per-chain CPTP dynamics and cross-chain factor accumulation; three structural propositions are proven with full proofs (CPTP preservation, information-theoretic gain over independent SDE, consistency with both parent frameworks as boundary cases) ; the theoretical performance analysis derives complexity bounds and information-theoretic orderings (Propositions on complexity and entropy ordering) ; the cross-box applicability is established for five NP-Hard problem classes from the seven-box landscape taxonomy (Multidimensional Knapsack, Graph Colouring, Vehicle Routing, Quadratic Assignment, Sequential Decision) ; bridges to the FOG framework and DRF pipeline are sketched. This is a programme-internal Working Paper Version 1. All performance claims are theoretical (asymptotic complexity bounds and information-theoretic orderings) rather than empirical. Empirical validation on operational systems—including the extension of MCCO v3's 3. 56× speedup benchmark and cross-box measurement across the five NP-Hard problem classes—is the subject of a separate companion paper that will be developed when the MCMT implementation infrastructure is in place. The paper bridges to FOG-4 (10. 5281/zenodo. 20152165) as the discrete-time, finite-tentacle instantiation of the continuous-time path-functional framework; consumes the diagnostic output of NP Landscape Classification; and provides the optimisation engine for the forthcoming DE-5 pipeline capstone. 24 pages, bilingual English/Traditional Chinese. Builds on: MCCO Standalone (10. 5281/zenodo. 20104942), Spacetime DE (10. 5281/zenodo. 19750775), NP Landscape Classification (10. 5281/zenodo. 19653758), FOG-4 v2 (10. 5281/zenodo. 20152165).