We prove that for every open $U \subseteq \mathbb{R}^n$ the spherical projection of every gradient trajectory $\theta(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb{R}$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We extend our conclusions from the Euclidean metric to a general class of ambient metrics including Riemannian metrics and we establish uniform local upper bounds for the lengths of the spherical projections of gradient flow trajectories. We finally apply our findings to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).
The halting problem asks whether a program eventually halts, with unrestricted resources. Engineering systems instead certify bounded halting: given a program, an input, and a step budget T, decide whether the program halts within T steps. This paper studies two of its uses. First, it defines an exact finite-budget problem, placing the simulated program and the checker on the same unit-cost timescale, so the checker can be embedded in an adversarial program. A diagonal argument with explicit additive overhead shows that no fixed deterministic checker decides every instance within any fixed fraction of T below one, for large T. In the nondeterministic case, T bounds the branch depth allowed for a stable observable decision. Second, bounded-halting instances give adversarial tests for finitary universality, since a universal model must support arbitrary finite computations as T grows. The tests check resource growth with T, the absence of a fixed deliberation bound, and later access to intermediate data. They separate a fixed model with extendable external resources, a model family that grows with T, and a larger system built around a fixed component. For machine-learning systems, this identifies the resource growth a universality claim assumes and whether it fails for deployed models.
Our goal is to construct mathematical operations that combine indeterminism measured from quantum randomness with computational determinism so that non-mechanistic behavior is preserved in the computation. Formally, some results about operations applied to computably enumerable (c.e.) and bi-immune sets are proven here, where the objective is for the operations to preserve bi-immunity. While developing rearrangement operations on the natural numbers, we discovered that the bi-immune rearrangements generate an uncountable subgroup of the infinite symmetric group (Sym$(\mathbb{N})$) on the natural numbers $\mathbb{N}$. This new uncountable subgroup is called the bi-immune symmetric group. We show that the bi-immune symmetric group contains the finitary symmetric group on the natural numbers, and consequently is highly transitive. Furthermore, the bi-immune symmetric group is dense in Sym$(\mathbb{N})$ with respect to the pointwise convergence topology. The complete structure of the bi-immune symmetric group and its subgroups generated by one or more bi-immune rearrangements is unknown.
The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzm\"uller and Scott's Notes on G\"odel's and Scott's variants of the ontological argument, reaches beyond the modal object language: its quantifiers range over domains containing the nominals and satisfaction operators of hybrid logic. A proof using one proves a theorem of the embedding, not necessarily of the modal logic. Whether a result is one of the modal logic can be settled by replaying it in an explicit calculus, or, more cheaply and for all results at once, by analysing what the embedding's proofs substitute. Every statement the Notes prove has a proof in the embedding substituting nothing outside the object language, 294 of them explicit. The Notes' own proofs substitute none either; what it flags are terms a prover introduced. A soundness theorem over general models closed under the object language, and a sentence of that language the embedding proves but which fails in such a model, fix what the certificate says. The Notes' three open questions are settled too, without nominals, but by the empty instance of the conjunction axiom, too cheaply to convince. Generalised to G\"odel's "any number of summands", the axiom covers the empty conjunction and the singleton. This article restricts it to at least two different conjuncts, a non-degenerate reading of G\"odel's footnote, and settles them again without nominals. In effect only the empty conjunction is excluded; the singleton comes back with the other axioms. In one variant the restriction refutes the possible existence of a God-like being; the answers return, with no side condition, once the Notes' non-emptiness clause is confined to G\"odel's essence. Every theorem here is verified in Isabelle/HOL and independently in Lean 4, the separating model in its propositional part; the countermodels are Nitpick's, checked by the build.
A Markov logic network (MLN) $\mathbb{M}$ determines a probability distribution $\mathbb{P}_n^\mathbb{M}$ on the set $\mathbf{W}_n$ of structures, or ``possible worlds'', with domain $\{1, \ldots, n\}$. We study the properties of such distributions as $n$ tends to infinity. We show that with mild assumptions on an MLN $\mathbb{M}$ with one soft constraint with an arbitrary positive weight the distribution $\mathbb{P}_n^\mathbb{M}$ will behave quite differently from the uniform distribution $\mathbb{P}_n^{uni}$ on $\mathbf{W}_n$ for all sufficiently large $n$. For a language with only one relation symbol $R$ which has arity 1 we give an almost complete characterization of the possible asymptotic behaviours of $\mathbb{P}_n^\mathbb{M}$ as $n \to \infty$, where $\mathbb{M}$ may be any MLN for this language. The asymptotic behaviour depends on the soft constraints and weights of the MLN. This characterization is used to show that if the language under consideration contains at least one relation symbol of arity 1 then the following holds: (a) There is an MLN $\mathbb{M}$ such that for every lifted Bayesian network (LBN) $\mathbb{G}$ there are infinitely many $n$ such that $\mathbb{M}$ and $\mathbb{G}$ determine different distributions on $\mathbf{W}_n$. (b) There is an LBN $\mathbb{G}$ such that for every MLN $\mathbb{M}$ there are infinitely many $n$ such that $\mathbb{G}$ and $\mathbb{M}$ determine different distributions on $\mathbf{W}_n$. We also show that, in the limit, the weight dimension and the domain size dimension may behave completely differently.