MeLoCoToN Tensor-Network Formalism

• MeLoCoToN is a tensor-network formalism that transforms classical logical circuits into exact explicit equations for combinatorial inversion, constraint satisfaction, and optimization.
• It follows a structured pipeline—defining variables, constructing circuits, tensorizing the logical operators, and iteratively contracting—to systematically extract solution bits.
• Applications span semiprime factorization and optimization variants (QUBO, QUDO, T-QUDO), with a focus on analytic exactness despite exponential contraction costs.

MeLoCoToN is a classical tensor-network formalism for combinatorial inversion, constraint satisfaction, and optimization in which a problem is first written as a classical logical circuit and then converted into an explicit tensor-network equation whose contraction exposes the solution. In the broadest formulation, the method is presented as yielding exact explicit equations for combinatorial problems, while later works specialize it to semiprime factorization and to QUBO, QUDO, and T-QUDO families. The 2026 QUDO paper explicitly expands the name as Modulated Logical Combinatorial Tensor Networks, whereas the semiprime factorization paper refers to MeLoCoToN as a formalism without expanding the acronym in the excerpted text (Ali, 9 Feb 2025).

1. Definition, scope, and historiography

The foundational MeLoCoToN paper presents the framework as a general recipe for obtaining an exact explicit equation for “every combinatorial problem” of the targeted kinds: inversion, constraint satisfaction, and optimization. Its central claim is not that the resulting equation is efficiently computable, but that the problem can be rewritten as a tensor-network equation obtained from classical logical operators and then solved by iterative contraction and projection (Ali, 9 Feb 2025).

Subsequent papers narrow and operationalize that claim. The prime-factorization work uses MeLoCoToN as the backbone for an exact tensor-network formulation of semiprime factorization, encoding multiplication as a constraint network and recovering factor bits one by one (Ali et al., 29 Jul 2025). The QUBO/QUDO/T-QUDO paper reformulates MeLoCoToN as a quantum-inspired but fully classical optimization methodology based on superposition, imaginary-time evolution, and projective extraction, with particular emphasis on sparse $k$-neighbor lineal-chain structure (Subiñas et al., 30 Mar 2026). The FTNILO paper positions MeLoCoToN as the discrete combinatorial precursor of a continuous-variable generalization built from Field Tensor Networks, stating that MeLoCoToN is recovered when integrals are restricted to natural numbers and Dirac deltas become Kronecker deltas (Ali, 3 May 2025).

Paper	Role in the approach	Distinct emphasis
(Ali, 9 Feb 2025)	General formulation	Exact explicit equations for combinatorial problems
(Ali et al., 29 Jul 2025)	Specialized application	Exact tensor-network equation for semiprime factorization
(Subiñas et al., 30 Mar 2026)	Optimization refinement	QUBO/QUDO/T-QUDO, $k$-neighbor structure, Waterfall
(Ali, 3 May 2025)	Formal extension	Continuous-variable generalization via FTNILO

A persistent feature across these papers is that MeLoCoToN is treated as a formalism or methodology, not merely as a single algorithmic primitive. The formalism centers on logical consistency, amplitude modulation, and selective contraction rather than on variational training or heuristic search.

2. Circuit construction and tensorization

The canonical MeLoCoToN pipeline in the original formulation has four steps: define the variables and rewrite the functions, create the associated classical logical circuit, tensorize the logical circuit, and iterate and contract the tensor network. The original paper introduces a technical vocabulary for these components, including LSTC (Logical Signal Transformation Circuits), LSVC (Logical Signal Verification Circuits), LSMC (Logical Signal Modulation Circuits), CT (Circuit Tensorization), TLC (Tensor Logical Circuit), IOI (Input-Output Indexing), and Half Partial Trace (Ali, 9 Feb 2025).

At the level of a single logical operator, the tensorization rule is straightforward. If an operator takes inputs and produces outputs determined by functions of those inputs, then the corresponding tensor has nonzero entries only when the output indices equal those function values, optionally multiplied by an amplitude factor. The original paper states this in IOI form, while the factorization paper gives the generic tensorization rule

$$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$

so that the output indices are forced to equal the logical outputs of the gate (Ali et al., 29 Jul 2025).

This circuit-first perspective is essential. MeLoCoToN does not begin from a generic numerical tensor ansatz; it begins from a structured logical computation whose signal flow already encodes admissible states. In inversion problems, the circuit computes the forward map and later postselects the known output. In constraint satisfaction, it annihilates invalid assignments by mapping them to zero amplitude. In optimization, it retains all admissible assignments but weights them according to the objective through an imaginary-time factor such as $e^{-\tau C(\vec{x})}$ (Ali, 9 Feb 2025).

The factorization paper restates this architecture as a three-stage formalism: build a classical logical circuit for the function to invert, tensorize that circuit into a tensor network, and iteratively contract it to expose the solution bits one by one. That description is narrower than the four-step presentation but operationally consistent with it (Ali et al., 29 Jul 2025).

3. Extraction of solutions: projection, marginalization, and imaginary time

The most distinctive element of MeLoCoToN is not merely the existence of a tensor network, but the claim that the solution can be extracted by partial contraction procedures that leave one index free at a time. In the original paper, for binary problems, a Minus Vector

$- = (-1,1)$

is contracted against the final free bit so that the sign of the resulting scalar identifies the bit value. This yields the basic explicit rule

$x_i = H(\Omega_i),$

where $H$ is the Heaviside step function and $\Omega_i$ is the scalar obtained from the corresponding contraction (Ali, 9 Feb 2025).

The original work then nests this rule recursively: earlier extracted variables are fed back as projection vectors when later variables are determined. The result is a hierarchy of contracted quantities in which each $x_n$ depends on previously fixed values through embedded Heaviside evaluations. This is the source of the claim that the method supplies an “exact explicit equation” rather than only a search procedure (Ali, 9 Feb 2025).

Later optimization-oriented papers recast the same mechanism in quantum-inspired language. The QUBO/QUDO/T-QUDO paper initializes a superposition over all assignments,

$$\ket{\psi_0}=\bigotimes_{i=0}^{n-1}\left(\sum_{x_i=0}^{d-1}\ket{x_i}\right),$$

weights each configuration by $k$0, and then marginalizes all variables except one to form

$k$1

The optimal value of $k$2 is then extracted either by argmax over the components of $k$3 or, in a binary encoding, by a Heaviside threshold after contraction with $k$4 (Subiñas et al., 30 Mar 2026).

The factorization paper gives the same logic in a purely combinatorial form. After projecting the output legs of the multiplication tensor network onto the known target bits of $k$5, the remaining tensor has a single surviving nonzero configuration at the correct factor bits. A Half Partial Trace-style procedure then leaves one bit open at a time, and the location of the nonzero element determines the bit value. The theorem-level expression in that paper is

$k$6

for an odd semiprime $k$7 with distinct primes $k$8 (Ali et al., 29 Jul 2025).

4. Problem classes and representative instantiations

The original MeLoCoToN paper organizes its target domain into three master classes. Inversion problems seek $k$9 such that $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$0. Constraint satisfaction problems retain only assignments satisfying a specified relation, so the tensor network acts as an indicator over the feasible set. Optimization problems attach amplitude weights $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$1 so that, after appropriate normalization and in the limit $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$2, only the minimum-cost configuration survives (Ali, 9 Feb 2025).

Within that framework the original paper gives a long catalog of examples, including binary addition and multiplication inversion, linear systems $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$3, Single One Input, $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$4-colouring, the Partition Problem, QUBO, QUDO, T-QUDO, HOBO, HODO, T-HODO, shortest path, TSP, vehicle routing, maximum flow, maximum independent set, minimum vertex cover, dominating set, assignment, cutting stock, knapsack, and integer linear, quadratic, and polynomial programming (Ali, 9 Feb 2025). The point of this catalog is structural rather than empirical: the same tensorization logic is claimed to apply across diverse combinatorial encodings.

The semiprime factorization paper provides a detailed instantiation of the inversion/CSP perspective. Multiplication is realized through controlled modular additions and then generalized so that both factors are unknown. The paper defines the recurrence

$$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$5

and then prunes the resulting logical circuit using semiprime-specific constraints: both factors are odd, $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$6 under the assumption $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$7, the most significant bit of $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$8 can be set to $$A_{x,y,z,\mu,\nu,\eta} = A(x,y,z)\, \delta_{\mu,f(x,y,z)} \delta_{\eta,g(x,y,z)} \delta_{\nu,h(x,y,z)},$$9 because $e^{-\tau C(\vec{x})}$0, and several do-nothing operators can be removed. The projected tensor network $e^{-\tau C(\vec{x})}$1 then satisfies

$e^{-\tau C(\vec{x})}$2

after contraction with Kronecker-delta vectors fixing the target number $e^{-\tau C(\vec{x})}$3 (Ali et al., 29 Jul 2025).

The QUBO/QUDO/T-QUDO paper formalizes the optimization branch for discrete variables $e^{-\tau C(\vec{x})}$4. It treats three objective families: $e^{-\tau C(\vec{x})}$5 for QUBO,

$e^{-\tau C(\vec{x})}$6

for QUDO, and

$e^{-\tau C(\vec{x})}$7

for T-QUDO. In the dense QUDO case, the paper factorizes the Boltzmann weight as

$e^{-\tau C(\vec{x})}$8

with local and pairwise factors encoded as tensors. It then specializes to the lineal-chain $e^{-\tau C(\vec{x})}$9-neighbor cost

$- = (-1,1)$0

where the sparsity pattern materially changes contraction complexity (Subiñas et al., 30 Mar 2026).

5. Computational properties, contraction strategies, and approximation

A recurring theme in the MeLoCoToN literature is the separation between formal exactness and computational efficiency. The original paper states plainly that the equations are not necessarily computable in reasonable time, do not surpass the state of the art in computational complexity, and are mainly valuable as a new mathematical viewpoint. It also states that if a physical system were capable of contracting the corresponding tensor networks in polynomial time, then every NP-Hard problem could be solved in polynomial time; this is presented as a complexity-theoretic implication of efficient contraction, not as an achieved algorithmic result (Ali, 9 Feb 2025).

The semiprime factorization paper makes this limitation explicit in asymptotic form. For exact contraction it studies two schemes: Bottom-up exact contraction with cost

$- = (-1,1)$1

and Left-right exact contraction with cost

$- = (-1,1)$2

which becomes $- = (-1,1)$3 in terms of $- = (-1,1)$4. The paper states that exact contraction remains exponential in the number of bits and is worse than brute force in the exact setting. It therefore tests approximate contraction using tensor train (TT) compression, introducing a bond dimension $- = (-1,1)$5 and a minimum $- = (-1,1)$6 needed to preserve the correct answer, together with the compression factor

$- = (-1,1)$7

The experiments show that $- = (-1,1)$8 still grows exponentially with $- = (-1,1)$9, although the required bond dimension is substantially smaller than the uncompressed effective dimension (Ali et al., 29 Jul 2025).

The QUBO/QUDO/T-QUDO paper sharpens this picture by distinguishing dense from structured cases. Dense QUDO contraction has computational complexity

$x_i = H(\Omega_i),$0

and spatial complexity

$x_i = H(\Omega_i),$1

so the dense exact case is again exponentially costly. For $x_i = H(\Omega_i),$2-neighbor lineal chains, however, the 4th-order tensor-contraction implementation has cost $x_i = H(\Omega_i),$3 with sparsity, versus $x_i = H(\Omega_i),$4 without it, and the matrix-vector implementation has $x_i = H(\Omega_i),$5 with sparsity. The same paper introduces Waterfall, which reduces memory use by storing only maximizing indices rather than full intermediates and is reported as especially favorable for the $x_i = H(\Omega_i),$6 case (Subiñas et al., 30 Mar 2026).

Approximation and hybridization are already present in the original work under the heading Motion Onion. There the proposed strategies include Hamiltonian separation, approximation by elimination of some constraint layers, genetic-algorithm hybridization, and Humbucker, which replaces Plus Vectors with complex Phase Vectors to encourage cancellation of unwanted contributions (Ali, 9 Feb 2025). A plausible implication is that MeLoCoToN is best viewed not as a single contraction recipe but as a family of exact formulations plus structured approximation heuristics.

6. Continuous generalization, interpretation, and misconceptions

The FTNILO paper recasts MeLoCoToN as the discrete member of a broader field-tensor-network program. Its central claim is that FTNILO generalizes MeLoCoToN by replacing discrete signals with continuous signals, summations with integrals, and Kronecker deltas with Dirac deltas. In the discrete MeLoCoToN setting, a tensorized logical operator is written with Kronecker constraints,

$x_i = H(\Omega_i),$7

whereas the FTNILO analogue is

$x_i = H(\Omega_i),$8

The paper then states that MeLoCoToN is recovered when integration is restricted to natural numbers and only integer-valued functions are used (Ali, 3 May 2025).

In FTNILO’s inversion formalism, the target condition is encoded by

$x_i = H(\Omega_i),$9

and under a unique solution $H$0, this becomes $H$1. Variable extraction again proceeds one coordinate at a time by integrating out the remaining coordinates. For optimization, the same imaginary-time weighting

$H$2

is used, and the limit $H$3 is claimed to produce a delta concentrated at the global minimizer (Ali, 3 May 2025). This continuity with the discrete MeLoCoToN logic is one of the clearest indications that the “approach” is intended as a general logic-to-tensor methodology rather than a single problem-specific construction.

Two common misconceptions are directly addressed by the literature. The first is that MeLoCoToN supplies a practical polynomial-time solver for NP-hard or cryptographic problems. The cited papers do not make that claim; rather, they emphasize exact explicit formulation and repeatedly acknowledge exponential contraction cost in generic cases (Ali, 9 Feb 2025). The second is that MeLoCoToN is inherently quantum. The papers instead describe it as classical or quantum-inspired: its use of superposition, imaginary-time evolution, and projective extraction is conceptual and structural, while the actual constructions are classical tensor networks executed on classical hardware (Subiñas et al., 30 Mar 2026).

Taken together, these works define the MeLoCoToN approach as a program of logical tensorization: admissible or optimal assignments are encoded in a structured classical tensor network, and solution variables are recovered by controlled contraction, marginalization, and projection. Its strongest contribution is therefore representational and analytic. Its practical performance depends decisively on contraction structure, sparsity, compression quality, and problem topology rather than on the mere existence of the exact equation itself.