When structure
determines
computation.
Two domains. One algebraic engine.

Meltalice develops mathematical engines for systems whose hierarchical structure
makes classical computation intractable.
The core result — in preparation for publication — reduces exponential complexity
to linear by exploiting an algebraic property of the system’s architecture.

rheox.meltalice.com
Polymer rheology — G′, G″, η* in real time
→

quant.meltalice.com
Quantitative finance — exact Greeks, sub-linear cost
→

Tree(A) Φ →

Mathematical approach

Algebraic structure
as computational engine.

The systems we work on — branched polymer melts, exotic derivative payoffs —
share a common feature: their complexity arises from a hierarchical tree structure
whose depth makes direct computation exponentially expensive.
Our approach exploits a specific algebraic property of these structures
to reduce computation to linear cost without approximation.
The mathematical framework is in preparation for publication.

Hierarchical tree algebras

Complex systems with branching structure — polymer topologies, derivative payoff trees — are naturally represented as elements of decorated tree algebras. This algebraic language makes the structure of the problem visible and exploitable.

Algebraic factorisation

A structural theorem — whose proof draws on the theory of operadic morphisms and co-algebraic renormalisation — establishes that the system’s response factors through a canonical projection. The computational consequence is a reduction from exponential to linear complexity. No approximation is involved.

Quantitative error bounds

We are extending the framework with quantitative polynomial estimates — providing explicit, non-asymptotic bounds on the error of each local computation. This is the foundation for exact Greeks in quantitative finance.

The core result

Linear complexity.
Exact output.
No heuristic.

Classical computation on hierarchical branched systems scales exponentially
with the depth of the structure.
The algebraic property we exploit collapses this cost to O(n)
— linear in the number of architectural elements —
regardless of branching depth or topological complexity.

This is not a numerical approximation, a neural network, or a regression model.
It is a consequence of the algebraic structure of the system itself.

Validated industrially on polymer rheology (RHEOX engine).
Extension to quantitative finance under active development.

Formal mathematical treatment in preparation — arXiv, 2026