A protein begins as a linear sequence of amino acids but may realize an enormous number of possible three-dimensional conformations. Under physiological conditions, many proteins nevertheless reach highly specific folded structures associated with biological function.
The difficulty is not merely to generate possible conformations, but to distinguish the comparatively small set of structurally viable configurations from the vastly larger space of geometrically or physically incompatible possibilities.
For a protein sequence of finite length, let ๐0 denote a finite, discretized candidate conformation space at a chosen structural resolution:
The discretization resolution introduces an unavoidable trade-off: finer sampling of backbone torsion angles, such as ฯ and ฯ, and side-chain states reduces geometric approximation error, but rapidly enlarges the finite candidate-state space.
Each candidate conformation must simultaneously satisfy local and global structural constraints arising from chain connectivity, molecular geometry, steric exclusion, residue compatibility, long-range contacts, and the surrounding environment.
CJM therefore does not initially ask which single candidate is the native fold. Instead, it asks which region of the candidate space remains structurally admissible under the full constraint system.
where
denotes the surviving structurally admissible candidate set. The central change of viewpoint is therefore from direct fold prediction to conformation-space reduction.
Consider a finite structural representation in which residues or short chain segments are assigned candidate local states, together with variables describing permitted contacts and spatial relations. The finite encoding is summarized by six principal constraint families:
Global three-dimensional consistency also introduces Euclidean embedding constraints: locally compatible distances and contacts must admit a mutually consistent realization in three-dimensional Euclidean space. Their Boolean encoding can become one of the dominant sources of clause growth as the number of nonlocal relations and candidate spatial states increases.
Each constraint family expands across the relevant residues, structural states, contacts, and relations of the finite instance. At finite structural resolution and tolerance, these constraints produce a Boolean formula ฮฆ, which is expressed as SAT and normalized through the common CJM 3SAT gate.
The SAT-to-3SAT representation is not itself proposed as a new protein-folding method. Its role in this sketch is one of normalization: heterogeneous molecular constraints are translated into the common 3SAT interface used throughout the CJM Problem Sketch series. The distinct CJM hypothesis begins only after this common representation has been formed.
Conformation-space reduction is not unique to CJM. Existing computational methods already eliminate incompatible or energetically unfavorable candidate states under explicitly defined molecular and energetic criteria. Dead-end elimination (DEE), for example, can provably remove side-chain states that cannot participate in a global minimum-energy conformation under the specified model.
The proposed distinction is therefore not pruning versus no pruning, but the mechanism of discrimination. In earlier CJM applications, the principal output may be interpreted as a binary admissibility judgment. Protein folding asks whether that same judgment can be used over a structured candidate space.
Instead of reproducing candidate elimination through conventional energetic bounds, sequential search, or branch-based pruning, CJM asks whether the normalized 3SAT structure could be subjected to a global physical admissibility judgment that rejects incompatible regions while preserving an admissible subset.
The resulting set ๐adm need not contain only one structure. The CJM question is therefore narrower than complete protein-structure prediction:
If such a mechanism exists, CJM would be used not merely to test one candidate, but to reduce the space in which viable candidates can exist. This distinction remains hypothetical until the physical CJM mechanism is explicitly realized and experimentally demonstrated.
Let the surviving fraction of the finite candidate space be represented schematically by
As structural constraints, resolution, or admissibility tolerance vary, RC need not decrease smoothly. A candidate Changbal Region would be a regime in which the admissible conformation space contracts sharply and reproducibly, indicating a nonlinear reorganization from a broad candidate ensemble toward a much smaller structurally viable region.
Such a reduction would not by itself establish the native fold. It would instead identify a structural transition in the size or organization of the candidate space presented to subsequent physical or biological selection.
The question posed here is closely related to Levinthal's paradox and the later energy-landscape or folding-funnel view of protein folding. The funnel picture explains how a free-energy landscape biased toward native-like structures can guide folding without requiring an exhaustive random search through every possible conformation. CJM does not replace this kinetic and thermodynamic account. It asks a different structural question: whether the same large candidate space can be represented as a finite global constraint system and examined at the level of admissibility before, or alongside, kinetic selection.
A second distinction is equally important. Conformation-space reduction itself is not unique to CJM; methods such as dead-end elimination already prune candidate states under explicit energy criteria. The proposed distinction is therefore not filtering itself, but how the filtering judgment is obtained. CJM asks whether the common 3SAT representation can be subjected to a global physical admissibility judgment rather than reproduced through conventional bound-based pruning or sequential exploration. Whether such a distinction can be physically realized remains an open question.
This sketch does not derive the native structure of a protein, replace molecular dynamics or energy-landscape theory, establish that CJM can uniquely identify a biologically active fold, or claim that protein folding itself is a Changbal Jump.
It also does not claim that conformational discretization, constraint encoding, SAT-based representation, or candidate-space pruning are themselves unique to CJM. Nor does it yet demonstrate that a physical CJM can perform global admissibility discrimination without reproducing an equivalent conventional search or pruning process.
Its purpose is narrower: to represent a finite protein-conformation problem as a structural constraint system, normalize that system through SAT and 3SAT, and formulate the subsequent CJM questionโwhether a physically realized global admissibility judgment could reduce a large candidate space to a smaller admissible subset through a mechanism distinct from conventional sequential or bound-based elimination.