The halting problem asks whether an arbitrary program M, given an input x, eventually terminates or runs forever:
Turing’s undecidability result [1] rules out any total Turing-computable procedure that decides H for every program–input pair. This obstruction is not merely a shortage of time or computational resources. The halting set is recursively enumerable: a terminating computation supplies a finite witness, whereas nontermination has no finite execution-trace witness in general.
Earlier CJM sketches examined the structural admissibility of mathematical or physical problems. This sketch subjects the proposed CJM judgment itself to a proven boundary of computability.
A conventional computation is expressed as a temporally ordered history, C0 → C1 → C2 → ··· . For a fixed bound T, let
Its unbounded counterpart is
The two global claims have distinct logical forms: halting requires ∃T : AT = 1, while non-halting requires ∀T : AT = 0. A finite halting trace certifies the former; bounded UNSAT cannot certify the latter in general.
CJM’s atemporal hypothesis is stronger than rewriting an execution trace. It proposes that structural admissibility may be discriminated through the compatibility of the complete constraint structure, independently of the temporal process by which computational states would ordinarily be generated. In this proposal, time indexes possible configurations; it need not be the mechanism by which their global compatibility is defined or judged.
This is a hypothesis about the mode of judgment, not an established algorithm or physical operation. Even if an atemporal judgment exists for finite structures, universal decidability does not follow. The gap between bounded structural admissibility and unbounded nontermination remains.
For a fixed T, Boolean variables encode machine states, tape symbols, head positions, and configurations within the bounded execution tableau. The finite encoding is summarized by five principal constraint families:
The finite formula ΦT(M, x) is converted by standard equisatisfiable transformations into 3SAT, ΨT(M, x) [2]. A finite tape window sufficient for all T steps is included in the encoding.
A satisfying assignment witnesses a bounded halting execution; UNSAT excludes halting within T only. This is a standard bounded-computation encoding, not a newly claimed reduction. The CJM hypothesis concerns whether the resulting entire finite constraint structure can be discriminated atemporally, rather than solved by sequential enumeration. No such advantage is established here.
The finite family {ΨT} can be specified for each T, but no fixed finite member decides the universal halting question. Suppose a proposed judgment satisfies
Case 1 — Turing-computable CJM. If JH is a total Turing-computable operation, diagonalization rules it out. Construct a program D(y) that loops if JH(y, y) = 1 and halts if JH(y, y) = 0. Evaluating D on its own code yields a contradiction. Neither an atemporal description nor parallel physical processing removes this logical constraint.
Case 2 — Non-Turing physical CJM. A universally correct physical judgment of ordinary Turing-program halting would require a non-Turing model of computation. Its encoding, readout, physical assumptions, resource requirements, reliability, and claimed non-simulability must be defined and supported independently. Atemporality alone does not establish hypercomputation.
Case 3 — Restricted-domain CJM. CJM might discriminate bounded execution structures or specified decidable program classes without conflicting with Turing’s theorem. Such an accomplishment could still test the atemporal hypothesis, but would not constitute a universal halting oracle.
Complexity Boundary and Computability Boundary. In October 2026, OpenAI announced a Lean-formalized refutation of the Exponential Time Hypothesis (ETH), proposing a deterministic subexponential-time algorithm for fixed-width CNF satisfiability counting [5]. The result remains subject to independent examination. If confirmed, it would overturn a longstanding conjectured time lower bound without establishing P = NP or resolving Turing undecidability.
For CJM, this distinction is essential. Atemporal structural discrimination is not merely a proposal for reducing computational time, nor does it automatically imply the ability to decide noncomputable problems. ETH concerns the cost of computation; the halting problem concerns the existence of a universal decision procedure. The CJM hypothesis must specify which boundary its proposed judgment addresses.
If an ideal oracle decides ordinary Turing-machine halting, the halting problem for machines allowed to query that oracle remains undecidable relative to that same oracle. Thus an alleged CJM judge cannot be universal for an effectively self-referential class containing its own oracle calls. The relevant boundary is not the presence or absence of sequential execution, but whether a proposed class can internalize the diagonal construction.
For each fixed program–input pair, bounded termination admissibility is monotone:
If the pair halts, the profile changes once, from 0 to 1, at its first halting step τ(M, x); if not, it remains 0 for every finite T. The transition is a logical property of the bounded encoding, not evidence that CJM has observed a new physical phase.
Halting-time profile. Across a selected program family, the locations τ(M, x) describe where bounded admissibility first becomes true.
Threshold growth. Define τmax(n) as the maximum halting time of halting programs whose description length is at most n, under a fixed universal encoding and input convention. This is a Busy-Beaver-type function. It eventually exceeds every total computable function [3]; hence there is no computable universal time cutoff T(n) whose failure to observe halting certifies nontermination for all programs of that size.
A candidate Changbal Region would mean a reproducible, encoding-controlled feature of atemporal CJM responses to finite constraint families that is not accounted for by baseline bounded simulation. A sharp change in AT alone is insufficient: it is already guaranteed by the definition of bounded halting.
Controls. Tests should include independently verified halting and non-halting programs, bounds on both sides of known halting times, multiple equivalent encodings, classical SAT/3SAT baselines, and randomized ordering of the presentation or sampling of constraints while preserving their logical content. Order independence by itself does not demonstrate non-Turing computation or physical atemporality.
CJM distinguishes temporal generation from hypothesized atemporal discrimination; the latter remains the central research proposal, not a metaphor for parallel search. But an atemporal method is not automatically an undecidable-function oracle. The halting problem tests the legitimacy and scope of the judgment itself. This also cautions against moving silently from finite constraints to universal conclusions in the earlier unbounded sketches on twin primes, Ramsey thresholds, and ABC; it does not, by itself, render those individual mathematical questions undecidable.
This sketch does not claim to solve the halting problem, bypass Turing undecidability, or demonstrate a hypercomputational device. SAT/3SAT normalization represents bounded computations only. No implemented atemporal CJM decision mechanism, universal correctness theorem, or independently verified physical operation is provided.
The atemporal hypothesis is retained, not weakened: CJM proposes structural discrimination that is conceptually independent of sequential state generation. What remains unproved is whether such discrimination can be implemented, how its resource costs scale, and which classes of questions it can decide. The distinction between atemporality and universal computability is the essential boundary this sketch records.