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 N_t 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 N_t tentacles per (chain, step) cell. The central design choice is the shared factor pool: at each time t, the N_t 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).