EXPLICIT LSMC RESTRUCTURING OF TRADE SETTLEMENT OPTIMIZATION
A Detailed Mathematical Derivation and Comparative Analysis
Financial Infrastructure · Tensor Networks · Combinatorial Optimization · Quantum-Ready Computation
THE PROBLEM
Trade settlement presents an unusually concrete combinatorial problem.
Given a large network of transactions, balances, credit limits, priorities and dependencies, which transactions should settle?
With N transactions, the apparent decision universe contains 2N possible settlement combinations. At realistic institutional scale, direct enumeration is impossible.
But the exponential universe is misleading.
Settlement decisions are not independent. Each transaction modifies balances, affects subsequent feasibility and interacts with a relatively structured financial network. The real computational object is therefore not the collection of all possible settlement subsets, but the system of relationships generating them.
THE IDEA
The paper restructures settlement as a Logical Signal Modulation Circuit — LSMC.
Instead of representing every possible settlement configuration explicitly, financial operations such as balance updates, credit-limit checks, transaction acceptance and constraint propagation are expressed as local logical operators.
Those operators are then tensorized and connected into a network.
The resulting tensor network does not enumerate the exponentially large outcome space. It represents the computational process that generates and constrains that space.
FROM COMBINATORIAL EXPLOSION TO STRUCTURE
This produces an important change of perspective.
Rather than asking:
Which of 2N possible transaction combinations should settle?
the problem becomes:
What is the structure of the network through which settlement feasibility propagates?
Sparsity, limited transaction interactions and the topology of the financial network can then become computational assets rather than incidental characteristics of the problem.
The paper develops the transformation from the original constrained optimization problem through logical circuit construction and operator tensorization to solution extraction, together with explicit analysis of treewidth, contraction complexity and feasibility.
CLASSICAL NOW, QUANTUM READY
The restructuring also illustrates an important principle in Prime Radiant's quantum research.
A problem should not necessarily be handed directly to a quantum optimizer in its original exponential representation.
The LSMC first uses classical structure to identify and compress the feasible manifold into much smaller bond spaces. Quantum optimization can then operate inside the structured representation, rather than searching an overwhelmingly irrelevant ambient space.
The quantum architecture therefore emerges from the mathematical restructuring of the problem rather than being attached to it afterward.
WHY IT MATTERS
Settlement is deliberately unglamorous.
It is also a large, real and economically important infrastructure problem whose apparent exponential complexity emerges from many individually simple interactions.
That makes it a useful test of a broader proposition:
sometimes the solution to combinatorial complexity is not to search the combinations more efficiently, but to stop representing the problem as combinations.
By replacing the universe of outcomes with the structure that generates those outcomes, an apparently exponential problem can acquire a radically different computational geometry.
RESEARCH STATUS
Independent research · 2026
Working research paper