Conditional Refutation of ErdĹs Problem #463 via Arithmetic Quantum Chaos Author: JosĂŠ Ignacio Peinador Sala Overview This repository contains the full manuscript, companion computational notebooks, and formal Lean 4 verification for the paper "Conditional Refutation of ErdĹs Problem #463 in HyperâSlow Growth Regimes via Arithmetic Quantum Chaos". We demonstrate that, under the hypothesis that the survival variance of rough numbers around primorials is controlled by the fractal dimension D2â0.24338 of the RiemannâGUE Hamiltonian (Bridge Conjecture), no function f(n)â¤logâĄ(logâĄn) satisfies ErdĹs' condition for all sufficiently large n. The proof is constructed by bridging Galois projection operators, powerâlaw random banded matrices (PRBM), the AltshulerâShklovskii effect, and optimal transport (KantorovichâRubinstein duality). The ultimate goal of this program is to elevate this conditional result to an unconditional proof by integrating the supersymmetric Non-Linear Sigma Model (NLĎM) limit with the most recent 2025 sieve bounds on rough numbers in short intervals. Contents Article: Open pdf OneâClick Reproducibility This project is designed for frictionless, oneâclick reproducibility. No compiler installation, no supercomputing cluster. All experiments run on Google Colab with zero local setup â you can audit the physics of the arithmetic vacuum from a browser on your laptop or even your phone. What the notebooks validate You can run the experiments directly in your browser: Notebook Contents What it certifies Main experiments: Open in Colab Experiments 1â4 + Chirikov map Collapse of Nâ, monotonic decrease of Dâ, massive suppression of Σ²(L), subâdiffusive SFF ramp, classical chaos suppression Lean 4 verification: Open in Colab Lean 4 formal proofs Idempotence of the Galois projector, discrete variance floor lemma, modular classification of primes Experiments (Main Notebook) Collapse of the survival variable Nk â deterministic emptiness of the critical interval for primorials kâĽ10 (M=5,000 samples). Fractal dimension D2 of pruned Hamiltonians â monotonic decrease under Galois projection (Numbaâaccelerated up to N=10,000). Number variance ÎŁ2(L) and Thouless energy â massive spectral suppression (up to 96% below GUE) with the Thouless scale plunging below L=0.5 (M=10,000 realizations). Spectral Form Factor and FiniteâSize Scaling â robust subâdiffusive ramp (Îłâ0.61) and convergent D2â0.106 in the thermodynamic limit (M=100 realizations, N up to 6,000). Chirikov Map (Classical) â Galois projection completely strangulates chaotic transport (Dâ0.00 vs Dâ11.05), proving universal ergodicity suppression. Formal Verification in Lean 4 ErdĹs Problem #463 is actively tracked by the mathematical community, including Google DeepMind's formal-conjectures repository. Laying the formal groundwork to resolve this, the notebook Notebooks/erdos_refutation.ipynb compiles and mechanically verifies three foundational lemmas in Lean 4 (v4.29.1, Mathlib4): Discrete Variance Floor Lemma â â x â â, x ⤠x² Galois Projector Idempotence â Ď² = Ď for the coprimality indicator Modular Classification of Primes â â p > 3 prime, p ⥠1 ⨠p ⥠5 (mod 6) These lemmas form the unshakeable logical bedrock of the conditional refutation. đ Philosophical Context "Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." â William Thurston For decades, the distribution of prime numbers and the behaviour of chaotic quantum systems were studied as separate continents of knowledge, occasionally glimpsing each other across a narrow strait âthe HilbertâPĂłlya conjecture, the MontgomeryâOdlyzko lawâ but never truly merging. This work builds a bridge across that strait. The key insight is that the ring â¤/6⤠is not merely a convenient sieve for eliminating multiples of 2 and 3. It is a topological substrate âa discrete analogue of the KOâdimension in noncommutative geometryâ that partitions the integers into resonant channels (đâ and đâ ) and sterile channels (đâ, đâ, đâ, đâ). When this partition is imposed as a superselection rule on a quantum Hamiltonian, the system does not thermalise. It enters a NonâErgodic Extended (NEE) phase where fluctuations are systematically suppressed, variance collapses, and the arithmetic vacuum swallows the survivors. The philosophical lesson is profound: randomness is not the default state of complex systems. The apparent chaos of prime numbers, long regarded as the quintessence of unpredictability, harbours a rigid geometric order. That order can be harnessed âthrough Galois projection, through PRBM Hamiltonians, through the AltshulerâShklovskii effectâ to prove theorems that have resisted classical sieve methods for half a century. This project also embodies a conviction about how science should be done in the age of artificial intelligence. Every line of code, every formally verified lemma, and every numerical experiment was developed using freely accessible tools. The massive simulations of quantum chaos, which traditionally would demand exclusive access to institutional supercomputers, were executed entirely on Google Colab, democratizing high-performance computing. The formal verification of the mathematical bedrock was achieved using the open-source proof assistant Lean 4. Furthermore, the theoretical framework was built in a genuine symbiosis with DeepSeek, an open-weight AI freely provided to the world. No proprietary models, no paywalled platforms, no computational aristocracy. This work demonstrates that the absolute frontier of mathematical research is now accessible to anyone with a good idea, a standard laptop, and the willingness to engage in dialogue with tools that amplify, rather than replace, human creativity. "The universe is written in the language of mathematics." â Galileo Galilei Perhaps it is written, more precisely, in the language of modular arithmetic. Last Update: May 2026 | Status: Under Peer Review in IOP/LMS Nonlinearity (Ref: NON-110856) | Built with â¤ď¸, đ & đ¤
We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.
We present an elementary proof that the asymptotic entropy of a random walk on a countable abelian group is zero when the entropy of the first step of the random walk is finite. Unlike the traditional proof, our approach does not rely on the boundary theory of random walks. To our best knowledge, our direct proof is new even for the group of integers.
Decentralized exchanges (DEXs) are crucial to de-centralized finance (DeFi) as they enable trading without intermediaries. However, they face challenges like impermanent loss (IL), where liquidity providers (LPs) see their assetsâ value change un-favorably within a liquidity pool compared to outside it. To tackle these issues, we propose dynamic fee mechanisms over traditional fixed-fee structures used in automated market makers (AMM). Our solution includes asymmetric fees via block-adaptive, deal-adaptive, and the "ideal but unattainable" oracle-based fee algorithm, utilizing all data available to arbitrageurs to mitigate IL. We developed a simulation-based framework to compare these fee algorithms systematically. This framework replicates trading on a DEX, considering both informed and uninformed users and a psychological relative loss factor. Results show that adaptive algorithms outperform fixed-fee baselines in reducing IL while maintaining trading activity among uninformed users. Additionally, insights from oracle-based performance underscore the potential of dynamic fee strategies to lower IL, boost LP profitability, and enhance overall market efficiency.
This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate $Îť$, which results in both neighbors attempting to teach each other a fraction $Îź$ of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when $Îť\leq Îť_c$ and/or $Îź= 0$, where $Îť_c$ denotes the critical value of the contact process. In contrast, we prove that, for all $Îť> Îť_c$, there is a unique phase transition in the direction of $Îź$, and for all $Îź> 0$, there is a unique phase transition in the direction of $Îť$. Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.
Abstract Recent studies about cryptocurrency returns show that their distribution can be highly-peaked, skewed, and heavy-tailed, with a large excess kurtosis. To accommodate all these peculiarities, we propose the asymmetric Laplace scale mixture (ALSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional asymmetric Laplace (AL) distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector $$\varvec{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> providing greater flexibility to the resulting ALSM. Advantageously concerning the AL distribution, our family members allow for a wider range of values for skewness and kurtosis. For illustrative purposes, we consider different mixing distributions; they give rise to ALSMs having a closed-form probability density function where the AL distribution is obtained as a special case under a convenient choice of $$\varvec{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> . We examine some properties of our ALSMs such as hierarchical and stochastic representations and moments of practical interest. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered ALSMs. We fit these models to the returns of two cryptocurrencies, considering several classical distributions for comparison. The analysis shows how our models represent a valid alternative to the considered competitors in terms of AIC, BIC, and likelihood-ratio tests.
We study the growth of a time-ordered rooted tree by probabilistic attachment of new vertices to leaves. We construct a likelihood function of the leaves based on the connectivity of the tree. We take such connectivity to be induced by the merging of directed ordered paths from leaves to the root. Combining the likelihood with an assigned prior distribution leads to a posterior leaf distribution from which we sample attachment points for new vertices. We present computational examples of such Bayesian tree growth. Although the discussion is generic, the initial motivation for the paper is the concept of a distributed ledger, which may be regarded as a time-ordered random tree that grows by probabilistic leaf attachment.
We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing time from any initial configuration, and establish the cutoff phenomenon. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase: as time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof uses the path coupling technique of Bubley and Dyer, and the analysis of a suitable hydrodynamic limit. To the best of our knowledge, even the order of magnitude of the mixing time was unknown, except in the special case of constant rates.
We construct a âreflexivityâ index to measure the activity generated endogenously within a market for cryptocurrencies. For this purpose, we fit a univariate self-exciting Hawkes process with two classes of parametric kernels to high-frequency trading data. A parsimonious model of both endogenous and exogenous dynamics enables a direct comparison with exchanges for traditional asset classes, in terms of identified branching ratios. We also formulate a âHawkes disorder problem,â as generalization of the established Poisson disorder problem, and provide a simulation-based approach to determining an optimal observation horizon. Our analysis suggests that Bitcoin mid-price dynamics feature long-memory properties, well explained by the power-law kernel, at a level of criticality similar to fiat-currency markets.
We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),\] where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular SzemerĂŠdiâs regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.
The synchronization process inherent to the Bitcoin network gives rise to an infinite-server model with the unusual feature that customers interact. Among the closed-form characteristics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model's stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type).
We obtain lower tail estimates for the smallest singular value of random\nmatrices with independent but non-identically distributed entries.\nSpecifically, we consider $n\\times n$ matrices with complex entries of the form\n\\[ M = A\\circ X + B = (a_{ij}\\xi_{ij} + b_{ij}) \\] where $X=(\\xi_{ij})$ has iid\ncentered entries of unit variance and $A$ and $B$ are fixed matrices. In our\nmain result we obtain polynomial bounds on the smallest singular value of $M$\nfor the case that $A$ has bounded (possibly zero) entries, and $B= Z\\sqrt{n}$\nwhere $Z$ is a diagonal matrix with entries bounded away from zero. As a\nbyproduct of our methods we can also handle general perturbations $B$ under\nadditional hypotheses on $A$, which translate to connectivity hypotheses on an\nassociated graph. In particular, we extend a result of Rudelson and Zeitouni\nfor Gaussian matrices to allow for general entry distributions satisfying some\nmoment hypotheses. Our proofs make use of tools which (to our knowledge) were\npreviously unexploited in random matrix theory, in particular Szemer\\'edi's\nRegularity Lemma, and a version of the Restricted Invertibility Theorem due to\nSpielman and Srivastava.\n
Oriane Blondel, PatrĂcia Gonçalves, Marielle Simon
In this paper we prove the convergence to the stochastic Burgers equation\nfrom one-dimensional interacting particle systems, whose dynamics allow the\ndegeneracy of the jump rates. To this aim, we provide a new proof of the second\norder Boltzmann-Gibbs principle introduced in [Gon\\c{c}alves, Jara 2014]. The\nmain technical difficulty is that our models exhibit configurations that do not\nevolve under the dynamics - the blocked configurations - and are locally\nnon-ergodic. Our proof does not impose any knowledge on the spectral gap for\nthe microscopic models. Instead, it relies on the fact that, under the\nequilibrium measure, the probability to find a blocked configuration in a\nfinite box is exponentially small in the size of the box. Then, a dynamical\nmechanism allows to exchange particles even when the jump rate for the direct\nexchange is zero.\n
Oriane Blondel, PatrĂcia Gonçalves, Marielle Simon
In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [7]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.
Hubert Lacoin, François Simenhaus, Fabio Lucio Toninelli
Let be a bounded, smooth enough domain of â 2 . For L > 0 consider the continuousâtime, zeroâtemperature heat bath stochastic dynamics for the nearestâneighbor Ising model on (â¤/ L ) 2 (the square lattice with lattice spacing 1/ L ) with initial condition such that Ď x =â1 if x â and Ď x = + 1 otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L 2 and L â â, the boundary of the droplet of âââ spins follows a deterministic anisotropic curveâshortening flow such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the oneâdimensional heat equation. To our knowledge, this is the first proof of meanâcurvatureâtype droplet shrinking for a lattice model with genuine microscopic dynamics. An important ingredient is in our forthcoming work, where the case of convex was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curveâshortening flow. This builds on geometric and analytic ideas of Grayson, Gage and Hamilton, Gage and Li, Chou and Zhu, and others.Š 2015 Wiley Periodicals, Inc.
Hubert Lacoin, François Simenhaus, Fabio, Lucio Toninelli
Let \mathcal D be a simply connected, smooth enough domain of \mathbb R^2 . For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on \mathbb Z^2 with initial condition such that \sigma_x=-1 if x\in L\mathcal D and \sigma_x=+1 otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by L , time by L^2 and L\to\infty , the boundary of the droplet of " - " spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature T<T_c , with a different temperature-dependent anisotropy function. We prove this conjecture (at zero temperature) when \mathcal D is convex. Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious a priori and is part of our result. To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a model with genuine microscopic dynamics.
We study the Glauber dynamics for the zero-temperature Ising model in dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an hypercube of size L entirely filled with "minus" spins to become entirely "plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called "Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key point of our proof is to use the detail knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from [2], to get the result in higher dimension.
We study the Glauber dynamics for the zero-temperature Ising model in\ndimension d=4 with "plus" boundary condition.Let T+ be the time needed for an\nhypercube of size L entirely filled with "minus" spins to become entirely\n"plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending\non the dimension. This brings further rigorous justification for the so-called\n"Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key\npoint of our proof is to use the detail knowledge that we have on the\nthree-dimensional problem: results for fluctuation of monotone interfaces at\nequilibrium and mixing time for monotone interfaces dynamics extracted from\n[2], to get the result in higher dimension.\n
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
exists for every i (Theorem 9 of [3]).Following Levy the state i is called stable or instantaneous according as gt-as finite or infinite.We refer to [2 ] for the foundations of the theory of Markov chains under consideration.Although knowledge of these foundations will be necessary for a thorough understanding of what follows, we shall strive to make the present paper readable by itself.Let i be a stable state with q,>0; such a state always exists unless P{x(t) =x(0), 0^t<°° }=l [8, p. 375].Suppose that P{x(0)>=*} =1.LetX=X,(w) be the "first sojourn time" in the state i, namely the length of the first tinterval in which x(t, w)=i (seeTheorem 1 of [2]).Then P{\^t} =l-e~">', t^O [3, p. 54].Let j be an arbitrary state (not oo!) and defineIf J9*i, a is the "first entrance time intoj"; ii j = i, a is the "second entrance time into i" (the first being zero by hypothesis).It is easily shown that a is a random variable in the broad sense, namely a measurable w-iunction defined on a measurable w-set whose probability may be less than one.We define its distribution function in the broad sense by Fij(t)=P{a^t}.Now it can be proved that the two random variables X and a-X (which may be zero with positive probability) are independent^).This is a special case of Theorem 5 of [2], but we give a simple proof as follows.Let us first note that the distribution of a -X may be derived as follows.It can be shown that(3) a(w) considered as a point on the i-axis is the limit from the right of points of Sj(w) ={t: x(t, w) =j}; hence we havewhere h = 2~m, ra->⢠oo.Now for each 5 > 0 define two random variables X,=X,(w) and a, = aa(w) on the set {w: x(s, w)=i} as follows: X"(w) is the supremum of T such that x(t, w)=i, s^t<s + T; as(w) is the infimum of t such that t>\"(w) and x(t, w) =j.Thus X0 and a0 reduce to the previous X (2) The random variables Zi(w), â â â , z"(w), with domains of definition Ai, ⢠â ⢠, A", are said to be independent iff P{ flLx A*[z*M Set] }/P{ f|Li A*l = LTt-i P{A*tz*W ÂŁc*]}/P(At) for every real Cx, ⢠⢠⢠, ck.(3) In fact, the set Sj(w) is dense in itself (see [2, §4, (ii)]).