Synthesizable Projection

• Synthesizable projection is a paradigm that maps theoretical operators and constructs into experimentally realizable solutions under practical constraints.
• It is applied across adaptive control, operator algebras, quantum logic, and materials design to enforce boundedness and optimize resource-efficient synthesis.
• The approach bridges theoretical models with practical implementations, ensuring stability, correctness, and efficiency in complex system designs.

Synthesizable projection is a mathematical and algorithmic paradigm across control theory, operator algebras, quantum logic synthesis, algebraic geometry, network optimization, session types, and computational chemistry/materials, wherein an operator, map, or constructive procedure “projects” an element, parameter, or function into a domain of allowable, constructible, or experimentally realizable solutions. The term encompasses both analytic projection operators—used, for example, in constrained parameter adaptation and optimization—and algorithmic or syntactic projection methods for constructing entities (e.g., molecules, crystal structures, quantum circuits) that adhere to strict “synthesizability” constraints of practical realization. Synthesizable projection thus delineates the interface between unconstrained generation or evolution in theoretical models and the subset of outcomes that are actually achievable under physical, engineering, or empirical limitations.

1. Projection Operators and Bounded Parameter Adaptation

Synthesizable projection originated in the context of adaptive control as a mechanism that constrains online parameter updates to a prescribed convex set, thereby guaranteeing boundedness and practical stability of adaptive estimates (Lavretsky et al., 2011). Given a convex functional $f(\theta)$ defining the permitted region $\Omega_1$, the projection operator modifies an unconstrained update $y$ according to

$$\text{Proj}(\theta, y, f) = \begin{cases} y - \frac{\nabla f(\theta) (\nabla f(\theta))^T}{\|\nabla f(\theta)\|^2} y f(\theta), & \text{if } f(\theta) > 0,\, y^T\nabla f(\theta) > 0 \ y, & \text{otherwise} \end{cases}$$

This construction ensures that parameter trajectories remain in $\Omega_1$ for all time if initialized in the core set, and that the projection does not “push” estimates away from the admissible region: $$(\theta - \theta^*)^T (\text{Proj}(\theta, y, f) - y) \leq 0$$ Furthermore, matrix-weighted variants (the “$\Gamma$–projection”) extend this framework to adaptation laws where updates are pre- or post-multiplied by symmetric positive-definite matrices.

In practical synthesis, the choice of $f(\theta)$ (e.g., a quadratic bounding function) determines the shape and size of the admissible parameter set. The projection operator becomes an essential tool for the robust on-line synthesis of adaptive laws in continuous-time control systems, as it enforces “synthesizability” by bounding parameter drift under modeling uncertainty.

2. Synthesizable Projection in Operator Algebras and Quantum Logic

Within C*-algebras, synthesizable projection acquires a structural and constructive character via the “projection calculus” (Bice, 2012). Here, projections (self-adjoint idempotents) are synthesized from pairs of given projections $Q, R$ and a continuous function $f$ (obeying $f(0)=0, f(1)=1$ on the spectrum of $QR$), leading to a new projection $P$ such that

$QPQ = f(QRQ)$

The explicit formula for $P$ involves partial isometries and continuous functional calculus. This calculus admits streamlined homotopy constructions (showing any two close projections are connected via a “synthesized” path), strong lifting properties (ensuring projections with specified spectral data can be lifted along C*-algebra homomorphisms), exact excision of pure states, and synthesis of matrix subalgebras for transitivity theorems.

Order-theoretic results further reveal procedures for constructing least upper and greatest lower bounds (joins and meets) of families of projections, all within a synthesizable calculus. These tools unify spectral, algebraic, and order-lattice properties, enabling explicit construction (or “projection”) of entities that obey specified structural or spatial relations—crucial in the assembly of operator-algebraic systems with predetermined invariants or logical properties.

3. Quantum Logic Synthesis via Projection Operations

In quantum circuit synthesis, especially for ternary or multi-valued systems, synthesizable projection refers to decomposing a logic function into a sum of minterms, each realized by explicit “projection operations” (e.g., $L_i$ and $J_i$ acting as ternary-valued selectors) (Mandal et al., 2012). These projection operations, when implemented via Generalized Ternary Gates (GTG) and multi-qutrit extensions, yield circuits that minimize ancillary qutrit requirements and gate complexity.

Through a set of combinatorial simplification rules (e.g., $L_i(a) \cdot 0 = 0$, $L'_i(a) = L_{i+1}(a) + L_{i+2}(a)$), circuit construction is further optimized for depth and resource cost. Notably, synthesizable projection allows direct, ancilla-free realization of the $n$-qutrit sum function, with a predictable, minimized gate count, and superior synthesis metrics compared to previous approaches.

This methodology exemplifies synthesizable projection as a translation from algebraic or logical specification to resource-optimal, physically implementable constructions respecting quantum information constraints.

4. Synthesizable Projection in Algebraic Geometry and Optimization

The notion extends into symbolic computation and polynomial systems, most notably in the context of cylindrical algebraic decomposition (CAD). Here, synthesizable projection refers to operators that reduce a multivariate polynomial system to an explicit sequence (triangular form) amenable to further symbolic or numerical synthesis (Yao et al., 2014).

Brown’s projection operator and Lu Yang’s successive resultant (ResP) operator, though distinct in formulation, are shown to be mathematically equivalent, encapsulating leading coefficients, discriminants, and pairwise resultants. This equivalence ensures that synthesizability—in the sense of producing critical factors for real root isolation, optimization, or decomposition—can be efficiently achieved via either operator, allowing algorithmic choices to be based on computational practicality or downstream requirements.

Similar closed-form projection formulas underpin optimization on nonconvex sets such as intersections of cones and spheres (Bauschke et al., 2017), where synthesizable projection yields explicit constructs usable in iterative numerical algorithms (e.g., copositivity testing in matrix theory).

5. Application to Complex Networks, Graphs, and Communications

In the synthesis of network topologies and distributed protocols, synthesizable projection appears as projective descriptions or automata-based projection operators. For example, in graph design, synthesizable projection refers to constructing topologies by iterative application of constraints on multi-level “projections” (neighborhood unfoldings) of a graph (Melent'ev, 2021). By mapping desired global properties (degree, diameter, girth, etc.) to constraints on the projections, the methodology provides a deterministic synthesis process yielding networks (if they exist) with guaranteed properties, or else certifies infeasibility.

In multiparty session type systems, synthesizable projection is formalized as an automata-theoretic, subset-construction-based operator that maps global protocols into correct-by-construction state machines for each participant, ensuring both soundness and completeness under practical implementability constraints (Li et al., 2023). These advances enable the formal synthesis and verification of distributed communication systems directly from abstract specifications.

In quantum communication, synthesizable projection arises as the programmable reconfiguration of entanglement/single-photon distributions via optical switching and spectral slicing, allowing on-demand adaptation of the network’s connectivity and resource allocation to user or application demands (Laudenbach et al., 2019).

6. Synthesizable Projection in Chemical and Materials Discovery

Recent developments have shifted synthesizable projection into the domain of computational chemistry and materials design, where the focus is on projecting molecular or structural candidates into the subset of chemical/material space that is experimentally realizable.

In drug discovery, synthesizable projection has materialized as transformer-based frameworks (e.g., ReaSyn) that generate explicit synthetic pathways (via “chain-of-reaction” or “postfix” notations) mapping given molecules to synthesizable analogs through sequences of building block/reaction steps, with dense intermediate supervision and reinforcement learning to optimize both synthetic accessibility and desired properties (Luo et al., 7 Jun 2024, Lee et al., 19 Sep 2025). The projection thus ensures each designed molecule is accompanied by a plausible synthesis plan, aligning generative design with laboratory constraints.

Data-driven metrics, such as round-trip scores that integrate retrosynthetic planning and reaction prediction, quantitatively rate the synthesizability of generated molecules, providing actionable feedback to both candidate selection and model training (Liu et al., 13 Nov 2024).

In inorganic materials discovery, synthesizable projection comprises symmetry-guided structure generation, Wyckoff encode–based machine learning for subspace selection, and crystal-likeness scoring, culminating in ab initio calculations restricted to ML-identified synthesizable domains (Xin et al., 14 May 2025). This procedure bridges the gap between “theoretically stable” but unrealizable predictions and the experimentally accessible region of materials space.

7. Theoretical, Algorithmic, and Practical Implications

The unifying principle of synthesizable projection across these domains is the mapping of mathematical or formal objects—be they vectors, functions, circuit representations, network protocols, or molecular graphs—onto the subset of objects or parameter values consistent with external (physical, structural, resource, or implementability) constraints.

Key implications include:

• Control and stability: Guarantees invariant satisfaction of parameter bounds and safe operation in adaptive systems.
• Constructive synthesis: Enables algorithmic design of elements in operator algebras, logic circuits, and network graphs with precisely controlled properties.
• Resource optimization: Reduces ancillary resource requirements (e.g., gate count, circuit depth, FPGA area) in quantum and hardware architectures.
• Bridging theory and practice: Directs generative/optimization outputs into domains that can be verified or realized experimentally, with explicit synthesis recipes or experimentally confirmed prototypes.
• Algorithmic efficiency: Facilitates tractable filtering of vast configuration spaces (as in ML-based subspace selection for material structures) and provides efficient alternatives to brute-force or unconstrained search.
• Verification and correctness: In logic synthesis and protocol projection, ensures that only correct and feasible implementations are generated, with formal guarantees of soundness and completeness.

Synthesizable projection thus serves as a fundamental construct in rigorous scientific computing, unifying mathematical generality with practical realizability through parameterized, algorithmic, and data-driven synthesis paradigms.