Atemporal Structural Admissibility and the Changbal Jump Machine Framework

A Mathematical and Physical Overview of Structural Encoding, Spectral Discrimination, Critical Alignment, and the Present Disclosure State of CJM

Abstract

The Changbal Jump Machine, hereafter CJM, is formulated as a structural discrimination framework in which a decision problem is transformed into a canonical 3SAT constraint system, represented as a coupled mathematical or physical operator, and classified according to observables associated with admissibility, instability, spectral reorganization, and critical transition.

CJM is not defined as an algorithm that enumerates candidate assignments more rapidly. Its central proposition is that the existence or nonexistence of a globally consistent state may be encoded in collective properties of the constraint structure itself. The corresponding discrimination map is represented by the \(\mathcal J\)-operator.

The term atemporal does not imply zero physical duration, infinite propagation velocity, or the elimination of dynamical relaxation. It refers to the logical independence of the final discrimination from the sequential order in which candidate solutions would otherwise be generated.

This document specifies the formal objects, physical realization requirements, readout architecture, verification conditions, robustness criteria, and complexity obligations of the CJM framework. Publicly disclosed material establishes a theoretical and software-emulated architecture but does not exhaust the implementation state of the broader framework.

1. Scope and Disclosure Status

CJM is not presented solely as an abstract machine awaiting an unspecified future implementation. Its public architecture includes problem-specific encodings, SAT-to-3SAT normalization, structural operators, software-emulated discrimination, critical response analysis, and problem-dependent spectral observables.

\mathcal P \longrightarrow \mathrm{SAT} \longrightarrow \mathrm{3SAT} \longrightarrow \mathcal C_{\Phi} \longrightarrow H_{\Phi} \longrightarrow \mathcal I_{\Phi} \longrightarrow \hat J \longrightarrow A(\Phi).

The public record does not specify the complete physical realization architecture, material coupling mechanism, full-scale readout system, or final external verification channel. Consequently, the available evidence does not determine whether those layers remain prospective, exist in partial form, or extend beyond the publicly documented interface.

This distinction is scientifically relevant. A physically credible realization capable of modifying the practical relation between search, verification, and structural discrimination could affect cryptographic assumptions, optimization systems, economic asymmetries, strategic infrastructure, and other security-sensitive domains.

The timing and granularity of further disclosure must therefore be evaluated together with reproducibility, independent verification, institutional stability, and foreseeable downstream consequences. Publication, proof, implementation, and unrestricted disclosure need not occur at the same rate.

2. Formal Problem Domain

Let a Boolean decision instance be represented by a 3SAT formula

\Phi = \bigwedge_{a=1}^{m} C_a, \qquad C_a = \ell_{a1} \lor \ell_{a2} \lor \ell_{a3}.

The variable set is

V = \{x_1,x_2,\ldots,x_n\},

and the clause set is

\mathcal C = \{C_1,C_2,\ldots,C_m\}.

Introduce a signed clause-variable incidence matrix

\Sigma_{\Phi} = (\sigma_{ai}) \in \{-1,0,+1\}^{m\times n},

where

\sigma_{ai} = \begin{cases} +1, & x_i \text{ occurs positively in } C_a, \\[4pt] -1, & x_i \text{ occurs negatively in } C_a, \\[4pt] 0, & x_i \text{ does not occur in } C_a. \end{cases}

The canonical structural representation of the instance is therefore not merely the Boolean string \(\Phi\), but the signed factor structure

\mathcal C_{\Phi} = \left( V, \mathcal C, \Sigma_{\Phi}, G_{\Phi}, \mathbf w, \Theta_{\Phi} \right),

where \(G_{\Phi}\) is the clause-variable factor graph, \(\mathbf w\) is a vector of clause or coupling weights, and \(\Theta_{\Phi}\) contains problem-dependent structural parameters.

These parameters may include clause density, degree distribution, frustration, cycle structure, symmetry classes, connected components, local-field statistics, spectral moments, and graph-theoretic invariants.

3. Canonical Clause Energy and Constraint Preservation

Introduce spin variables

s_i \in \{-1,+1\},

with \(s_i=+1\) corresponding to Boolean truth and \(s_i=-1\) corresponding to Boolean falsehood.

For each clause \(C_a\), define the violation indicator

P_a(\mathbf s) = \prod_{i:\sigma_{ai}\neq0} \frac{1-\sigma_{ai}s_i}{2}.

The quantity \(P_a(\mathbf s)\) equals \(1\) exactly when every literal in \(C_a\) is false and equals \(0\) otherwise. A canonical clause energy is therefore

E_{\mathrm{clause}}(\mathbf s) = \sum_{a=1}^{m} w_a P_a(\mathbf s), \qquad w_a>0.

The standard satisfiability equivalence is

\Phi\in\mathrm{SAT} \quad\Longleftrightarrow\quad \min_{\mathbf s} E_{\mathrm{clause}}(\mathbf s) = 0.

In an operator formulation, each violation indicator becomes a projector \(\widehat P_a\), producing the clause Hamiltonian

\widehat H_{\mathrm{clause}} = \sum_{a=1}^{m} w_a\widehat P_a.

Any additional CJM coupling term must preserve the semantic relation between the original Boolean instance and the resulting structural system. A physical or mathematical embedding is not sufficient merely because it reproduces a visually similar phase transition.

The required condition is an explicit equivalence between the original decision problem and the admissibility class of the constructed system.

4. Structural Operator Family

CJM associates each normalized instance \(\Phi\) with a parameterized operator family

H_{\Phi}(\kappa,\boldsymbol{\theta}) = H_{\mathrm{clause}}(\Phi) + \kappa H_{\mathrm{collective}} \left( \Sigma_{\Phi}, G_{\Phi}, \boldsymbol{\theta} \right) + H_{\mathrm{reg}} \left( \boldsymbol{\theta} \right).

Here,

\(H_{\mathrm{clause}}\) preserves the logical constraint content of the 3SAT instance;
\(H_{\mathrm{collective}}\) introduces nonlocal or higher-order coupling among variables, clauses, conflict motifs, or structural modes;
\(H_{\mathrm{reg}}\) controls degeneracy, normalization, boundary conditions, calibration, or finite-size stability;
\(\kappa\) is a structural control parameter;
\(\boldsymbol{\theta}\) denotes implementation-dependent coupling, geometry, scaling, and readout parameters.

The symbol \(H_{\Phi}\) may denote a classical energy functional, a matrix operator, a quantum Hamiltonian, a graph Laplacian-derived operator, a nonlinear response operator, or a hybrid physical realization. CJM does not identify itself exclusively with one of these substrates.

What is invariant across implementations is the requirement that the structural response preserve the decision semantics of \(\Phi\).

5. Spectral Observables and Critical Response

Let the ordered spectrum of \(H_{\Phi}\) be

\lambda_0 \le \lambda_1 \le \lambda_2 \le \cdots.

The ground-state energy and the lowest spectral gap are

E_0(\kappa) = \lambda_0(\kappa), \qquad \Delta_{\Phi}(\kappa) = \lambda_1(\kappa) - \lambda_0(\kappa).

A complete CJM readout need not be restricted to the minimum energy. The observable vector may include

\mathcal I_{\Phi} = \left( E_0, \Delta_{\Phi}, M_{\Phi}, \chi_{\Phi}, \nu_{\Phi}, \mathcal R_{\Phi}, \mathcal H_{\Phi} \right),

where

\(M_{\Phi}\) is an order parameter;
\(\chi_{\Phi}\) is a susceptibility or response derivative;
\(\nu_{\Phi}\) is a topological or structural index when defined;
\(\mathcal R_{\Phi}\) represents resonance or mode-locking data;
\(\mathcal H_{\Phi}\) represents entropy, spectral concentration, or information-distribution statistics.

For an observable \(\widehat M\), the corresponding order parameter may be written as

M_{\Phi}(\kappa) = \left\langle \Psi_0(\kappa) \middle| \widehat M \middle| \Psi_0(\kappa) \right\rangle,

with susceptibility

\chi_{\Phi}(\kappa) = \frac{\partial M_{\Phi}}{\partial\kappa}.

A critical region \(\kappa_c\) may be indicated by gap contraction, susceptibility divergence or peaking, order-parameter discontinuity, mode coalescence, topological index change, or a discontinuous change in the stability basin.

\kappa \rightarrow \kappa_c \quad\Longrightarrow\quad \begin{cases} \Delta_{\Phi}(\kappa)\rightarrow0, \\[4pt] \chi_{\Phi}(\kappa)\rightarrow\chi_{\max}, \\[4pt] M_{\Phi}(\kappa) \text{ changes branch or stability class.} \end{cases}

Gap closure alone is not sufficient to establish satisfiability or unsatisfiability. The decision interpretation requires a proved or independently validated relation between the complete observable vector and the admissibility class.

6. Admissibility Functional and the \(\mathcal J\)-Operator

Let \(\mathfrak A_{\boldsymbol{\theta}}\) denote the region of observable space classified as structurally admissible. Define

A_{\boldsymbol{\theta}}(\Phi) = \mathbf 1 \left[ \mathcal I_{\Phi} \in \mathfrak A_{\boldsymbol{\theta}} \right].

The CJM discrimination map is

\hat J_{\boldsymbol{\theta}} : \mathcal C_{\Phi} \longrightarrow \{0,1\}, \qquad \hat J_{\boldsymbol{\theta}}(\Phi) = A_{\boldsymbol{\theta}}(\Phi).

The hat notation indicates an action on the encoded structure. It does not by itself imply that \(\hat J\) is a linear, unitary, or self-adjoint operator. Those properties must be specified separately for each mathematical or physical realization.

The formal correctness requirement is

\hat J_{\boldsymbol{\theta}}(\Phi) = 1 \quad\Longleftrightarrow\quad \Phi\in\mathrm{SAT},

or, under an UNSAT-oriented implementation,

\hat J_{\boldsymbol{\theta}}(\Phi) = 1 \quad\Longleftrightarrow\quad \Phi\in\mathrm{UNSAT}.

These two orientations must not be conflated. A one-sided discriminator may be scientifically and computationally meaningful without yet constituting a complete binary decision procedure.

The Changbal Jump refers to the transition from the distributed clause-level representation to a globally classified admissibility state:

\left( \Sigma_{\Phi}, G_{\Phi}, \mathbf w, \boldsymbol{\theta} \right) \xrightarrow{\hat J} A_{\boldsymbol{\theta}}(\Phi).

7. Formal Meaning of Atemporality

Atemporality is not defined by the limit of a physical execution time approaching zero.

\mathrm{Atemporality} \neq \lim_{T\rightarrow0} \mathrm{Computation}.

A physical CJM realization may require finite preparation, loading, relaxation, calibration, transition, and readout times. These durations remain physical resource costs.

Atemporality instead denotes order independence of the logical discrimination. Let \(\pi\) represent an admissible ordering of local updates, candidate presentation, or measurement scheduling. The atemporal condition is

\hat J_{\boldsymbol{\theta}}(\Phi;\pi_1) = \hat J_{\boldsymbol{\theta}}(\Phi;\pi_2)

for all admissible update orderings \(\pi_1\) and \(\pi_2\), within the defined tolerance of the physical realization.

Equivalently, the decision is a functional of the global encoded structure rather than of a specific candidate-enumeration history:

\hat J(\Phi) = F \left[ \mathcal C_{\Phi} \right], \qquad \hat J(\Phi) \neq F \left[ \mathbf s_1, \mathbf s_2, \ldots, \mathbf s_{2^n} \right].

8. Physical Realization Architecture

A physical CJM requires more than a numerical visualization of a phase transition. It requires an end-to-end correspondence between a 3SAT instance, a physical coupling structure, a measurable global response, and an externally verifiable decision output.

Layer	Mathematical Object	Physical Requirement
Input normalization	\(\Phi\mapsto\mathcal C_{\Phi}\)	Polynomially constructible encoding preserving the original decision semantics
Coupling synthesis	\(H_{\Phi}(\kappa,\boldsymbol{\theta})\)	Programmable physical interactions corresponding to clause, conflict, and collective terms
State preparation	\(z_0\) or \(\rho_0\)	Reproducible initialization with bounded preparation error
Structural evolution	\(F_{\Phi}\), \(H_{\Phi}\), or hybrid dynamics	Controlled approach to the relevant critical or stationary regime
Critical response	\(\mathcal I_{\Phi}\)	Measurable spectral, topological, energetic, resonant, or stability signature
Decision readout	\(A_{\boldsymbol{\theta}}(\Phi)\)	Stable separation of admissible and inadmissible response classes
External verification	\(V(\Phi,\mathcal K_{\Phi})\)	Independently checkable witness, proof trace, or measurement certificate

A dissipative realization may be described by

\dot{\mathbf z} = - \Gamma \nabla_{\mathbf z} \mathcal E_{\Phi} \left( \mathbf z; \kappa, \boldsymbol{\theta} \right) + \mathbf G_{\Phi} \left( \mathbf z; \kappa, \boldsymbol{\theta} \right) + \boldsymbol{\eta}(t),

where \(\boldsymbol{\eta}(t)\) represents noise and uncontrolled perturbation.

A coherent realization may instead be described by

i\hbar \frac{\partial}{\partial t} \lvert\Psi(t)\rangle = \widehat H_{\Phi} \left( \kappa(t), \boldsymbol{\theta} \right) \lvert\Psi(t)\rangle.

These equations define possible implementation classes, not the identity of CJM itself. A CJM realization may be dissipative, coherent, analog, digital-physical, topological, resonant, or hybrid, provided that the semantic, measurement, and verification conditions are satisfied.

9. Readout, Certification, and Independent Verification

A measured critical response is not by itself a proof. The output must be converted into a certificate that can be examined independently of the internal CJM mechanism.

A generic certificate may be represented as

\mathcal K_{\Phi} = \left( \mathcal I_{\Phi}, \boldsymbol{\theta}, \varepsilon_{\mathrm{cal}}, \varepsilon_{\mathrm{noise}}, \mathcal T_{\mathrm{read}}, \mathcal L_{\mathrm{trace}} \right),

where

\(\varepsilon_{\mathrm{cal}}\) is the calibration uncertainty;
\(\varepsilon_{\mathrm{noise}}\) is the estimated noise bound;
\(\mathcal T_{\mathrm{read}}\) is the measurement record;
\(\mathcal L_{\mathrm{trace}}\) is an auditable execution or transformation trace.

For a SAT result, the preferred external certificate is a satisfying assignment \(\mathbf s^\ast\):

V_{\mathrm{SAT}} \left( \Phi, \mathbf s^\ast \right) = 1.

For an UNSAT result, a complete system requires an independently checkable contradiction certificate, proof trace, elimination record, or equivalent formal object:

V_{\mathrm{UNSAT}} \left( \Phi, \mathcal K_{\mathrm{UNSAT}} \right) = 1.

The verifier must not require trust in undisclosed CJM internals. The distinction between physical detection and mathematical certification is essential.

10. Complexity Accounting and \(O(J)\)

The notation \(O(J)\) is used to distinguish structural discrimination from candidate enumeration. It must not be interpreted as a substitute for conventional resource accounting.

A complete CJM complexity profile is the vector

\mathfrak C_J(\Phi) = \left( C_{\mathrm{enc}}, C_{\mathrm{load}}, C_{\mathrm{couple}}, \tau_{\mathrm{settle}}, C_{\mathrm{read}}, p_{\mathrm{precision}}, C_{\mathrm{verify}} \right).

The total physical execution cost may be decomposed as

T_{\mathrm{total}} = T_{\mathrm{enc}} + T_{\mathrm{load}} + T_{\mathrm{prepare}} + T_{\mathrm{transition}} + T_{\mathrm{read}} + T_{\mathrm{verify}}.

Any claim of polynomial discrimination requires that all relevant resources remain polynomially bounded:

C_{\mathrm{enc}}, C_{\mathrm{load}}, C_{\mathrm{couple}}, C_{\mathrm{read}}, C_{\mathrm{verify}} \in \operatorname{poly}(n,m).

In implementations governed by a minimum spectral gap, one must also exclude an exponentially closing gap:

\Delta_{\min}^{-1} \in \operatorname{poly}(n,m).

Required precision, energy, physical volume, coupling range, calibration cost, and repeated-sampling cost must likewise remain polynomially bounded. Otherwise, exponential complexity may have been transferred from temporal search into physical preparation or measurement precision.

Accordingly, \(O(J)\) denotes a proposed structural complexity regime, while \(\mathfrak C_J\) provides the resource accounting necessary for comparison with established computational models.

11. Robustness, Noise, and Finite-Size Scaling

Let \(\partial\mathfrak A\) denote the decision boundary in observable space. Define the admissibility margin

\gamma_{\Phi} = \operatorname{dist} \left( \mathcal I_{\Phi}, \partial\mathfrak A \right).

Stable classification requires

\left\| \delta\mathcal I_{\Phi} \right\| < \gamma_{\Phi},

where \(\delta\mathcal I_{\Phi}\) includes measurement error, coupling disorder, thermal fluctuation, finite precision, control drift, and model mismatch.

Near a continuous critical point, finite-size scaling may take the form

\Delta_N(\kappa) \sim N^{-z} f \left[ (\kappa-\kappa_c) N^{1/\nu} \right],

where \(N\) is the physical system size, \(z\) is a dynamical exponent, and \(\nu\) is a correlation-length exponent.

A physically meaningful CJM must demonstrate that the decision margin remains resolvable as \(n\), \(m\), and \(N\) increase. Apparent discrimination in small systems is insufficient if the observable separation collapses exponentially in the scaling limit.

12. Relation to Existing Computational and Physical Models

Model	Primary Mechanism	Relation to CJM
Turing computation	Sequential symbolic state transition	CJM rejects candidate enumeration as the necessary logical basis of discrimination
Parallel computation	Concurrent execution of multiple state transitions	Parallelism accelerates temporal search; it does not by itself establish atemporal admissibility
Adiabatic quantum computation	Controlled Hamiltonian evolution toward a ground state	Shares Hamiltonian and spectral-gap language, but CJM is not restricted to adiabatic evolution
Quantum annealing	Energy minimization with quantum or thermal fluctuations	May provide a substrate, but heuristic ground-state sampling alone is not a complete CJM discriminator
Ising machines	Physical encoding of combinatorial cost functions	Provide relevant encoding techniques; CJM additionally requires admissibility-class readout and independent certification
Analog computation	Continuous physical dynamics representing mathematical relations	CJM may be analog or hybrid, but precision and scaling costs must be explicitly accounted for
Hopfield networks	Attractor convergence and associative recall	Offer an analogy for global pattern stabilization but do not establish SAT/UNSAT equivalence
Topological computation	Robust information encoded in global invariants	Relevant when CJM admissibility is represented by protected topological classes

13. Disclosure Governance

The disclosure threshold is not defined solely by theoretical completion. It includes implementation stability, reproducibility, verification, and risk assessment.

\kappa_{\mathrm D} = \kappa_{\mathrm{theory}} \cap \kappa_{\mathrm{implementation}} \cap \kappa_{\mathrm{reproducibility}} \cap \kappa_{\mathrm{verification}} \cap \kappa_{\mathrm{governance}}.

Below this threshold, disclosure may remain modular:

\kappa < \kappa_{\mathrm D} \quad\Longrightarrow\quad \text{staged disclosure}.

At or above the threshold, a more complete architectural release may become scientifically and institutionally defensible:

\kappa \ge \kappa_{\mathrm D} \quad\Longrightarrow\quad \text{controlled architectural disclosure}.

The current public record is sufficient to evaluate CJM as a coherent mathematical and physical research architecture. It is not sufficient to determine the complete implementation state of every layer.

The absence of complete public implementation data should be treated neither as proof of nonexistence nor as proof of completed realization. It defines the present boundary of independently examinable evidence.

14. Technical Conclusion

CJM reformulates a decision problem as a sequence of mathematically explicit transformations:

\boxed{ \mathcal P \rightarrow \mathrm{SAT} \rightarrow \mathrm{3SAT} \rightarrow \mathcal C_{\Phi} \rightarrow H_{\Phi} \rightarrow \mathcal I_{\Phi} \rightarrow \hat J \rightarrow A(\Phi) }.

Its defining claim is not that exponential candidate spaces can be traversed at arbitrarily high speed. The claim is that the encoded constraint system may possess a globally measurable admissibility state whose discrimination does not require explicit candidate enumeration.

Scientific validation of that claim requires four results:

a semantics-preserving reduction into the canonical 3SAT structure;
a physically or mathematically realizable operator family whose observables distinguish the relevant admissibility classes;
polynomially bounded encoding, preparation, evolution, precision, readout, and verification resources;
an external certificate that allows the result to be checked independently of the internal CJM mechanism.

These conditions separate CJM from metaphorical accounts of criticality, heuristic optimization, and unverified analog computation. They also define the standards by which any further theoretical or physical disclosure must be evaluated.

References

Yoon, K. (2025). Emergence in SAT Problems: Critical Thresholds under Constraint Density. TechRxiv. https://doi.org/10.36227/techrxiv.174682135.51838369/v1
Yoon, K. (2025). Transitions of Critical Structural Regions for NP Problems. TechRxiv. https://doi.org/10.36227/techrxiv.175001127.70551829/v1
Yoon, K. (2025). P ≡ NPᴶ: On the End of Time. Zenodo. https://doi.org/10.5281/zenodo.18139629
Turing, A. M. (1936). On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, s2-42(1), 230–265. https://doi.org/10.1112/plms/s2-42.1.230
Cook, S. A. (1971). The Complexity of Theorem-Proving Procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158. https://doi.org/10.1145/800157.805047
Karp, R. M. (1972). Reducibility among Combinatorial Problems. In R. E. Miller and J. W. Thatcher (Eds.), Complexity of Computer Computations, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9
Cook, S. A. (2000). The P versus NP Problem. Clay Mathematics Institute. Official problem page
Kirkpatrick, S., and Selman, B. (1994). Critical Behavior in the Satisfiability of Random Boolean Expressions. Science, 264(5163), 1297–1301. https://doi.org/10.1126/science.264.5163.1297
Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., and Troyansky, L. (1999). Determining Computational Complexity from Characteristic Phase Transitions. Nature, 400, 133–137. https://doi.org/10.1038/22055
Mézard, M., Parisi, G., and Zecchina, R. (2002). Analytic and Algorithmic Solution of Random Satisfiability Problems. Science, 297(5582), 812–815. https://doi.org/10.1126/science.1073287
Lucas, A. (2014). Ising Formulations of Many NP Problems. Frontiers in Physics, 2, Article 5. https://doi.org/10.3389/fphy.2014.00005
Farhi, E., Goldstone, J., Gutmann, S., and Sipser, M. (2000). Quantum Computation by Adiabatic Evolution. arXiv:quant-ph/0001106. arXiv record
Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., and Regev, O. (2007). Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation. SIAM Journal on Computing, 37(1), 166–194. https://doi.org/10.1137/S0097539705447323
Kempe, J., Kitaev, A., and Regev, O. (2006). The Complexity of the Local Hamiltonian Problem. SIAM Journal on Computing, 35(5), 1070–1097. https://doi.org/10.1137/S0097539704445226
Anderson, P. W. (1972). More Is Different: Broken Symmetry and the Nature of the Hierarchical Structure of Science. Science, 177(4047), 393–396. https://doi.org/10.1126/science.177.4047.393

Appendix: Selected Physical and Computational Analogues

The following systems provide limited analogues for collective alignment, critical response, attractor stabilization, or global order formation. They are not evidence that CJM is mathematically equivalent to these systems.

Ferromagnetic Criticality — collective spin alignment and symmetry breaking near the Curie threshold.
Superconductivity — collective phase coherence and a macroscopic transition to a zero-resistance state.
Bose–Einstein Condensation — macroscopic occupation of a common quantum mode.
Laser Coherence — phase-aligned amplification and mode selection above a gain threshold.
Percolation Transitions — emergence of a system-spanning connected component at a critical density.
Synchronization — transition from distributed oscillatory phases to global frequency or phase locking.
Self-Organized Criticality — threshold-driven reconfiguration and scale-distributed response.
Protein Folding — convergence through a high-dimensional energy landscape toward a stable structural basin.
Hopfield Attractor Networks — global pattern recovery from incomplete or corrupted local input.
Reaction–Diffusion Systems — spontaneous macroscopic pattern formation from local coupling rules.
Crystallization — transition from disordered local configurations to a globally constrained lattice.
Topological Phase Transitions — global changes in invariant structure not reducible to a local order parameter alone.