[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86410-en":3,"doc-seo-86410-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86410,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","The Extremum Stack as Optimal Memory for Rate-Independent Sequence Models","This paper studies a functional on discrete sequences that is rate-independent, meaning it is invariant under monotone time reparametrisations and depends only on the ordered local extrema of the input rather than their timing. The central construct is the Preisach extremum stack Πn, formed by alternating maxima/minima that survive Preisach hysteresis wiping-out. The work provides complete information-theoretic and algorithmic characterisations, proving representation, Kolmogorov and Shannon minimality, and sharp online update-complexity guarantees including worst-case maintenance costs.","arXiv :2606 .05245v2 [ cs .DS] 13 Jul 2026  \nThe Extremum Stack as Optimal Memory for Rate-Independent Sequence Models: Information-Theoretic Foundations and Online  \nComplexity  \nPiotr Frydrycha  \na Faculty of Mechatronics, Warsaw University of Technology, Warsaw, Poland  \nAbstract  \nA functional on discrete sequences is rate-independent if it is invariant under monotone time reparametrisations: it responds only to the order of local extrema of its input, not to their timing. The central algorithmic object of this class is the Preisach extremum stack Πn: the nested sequence of alternating local maxima and minima of u0:n ∈ GnL+1 that survive the classical wiping-out rule of the Preisach hysteresis operator. We give a complete information-theoretic and algorithmic characterisation of this object.  \nFirst, a characterisation theorem: a computable functional F is rateindependent if and only if F [u](n) = f(Πn) for a computable f; the stack is a complete invariant of rate-independence.  \nSecond, Kolmogorov minimality: K(Πn) − O(1) ≤ KR (u0:n) ≤ K(Πn ) + O(1), where KR (u0:n) is the length of the shortest program answering every query in the class R of computable rate-independent functionals, and the O(1) overhead is independent of both the sequence length n and the stack depth k (it depends only on the grid resolution L) . Any compression of a hysteresis-driven stream that preserves the full class R must retain at least K(Πn) − O(1) bits.  \nThird, Shannon minimality: under any probability measure on u0:n , I(u0:n;Πn) ≤ I(u0:n;S) for every random variable S from which all R-queries are computable, with equality characterising S as informationally equivalent to Πn.  \nFourth, worst-case update complexity: (i) any compact exact R-minimal  \nEmail address: [piotr.frydrych@pw.edu.pl](piotr.frydrych@pw.edu.pl) (Piotr Frydrych)  \nrepresentation incurs Θ(k) output changes per step in the worst case, in a model-independent output-change metric; (ii) the monotone ordering of the Preisach wiping property enables binary search, reducing boundary detection to O(log k), though physical deletion remains Θ(d); (iii) a finger-tree implementation achieves O(log k) worst-case time per step for both search and deletion, while maintaining exact R-minimality with no approximation error.  \nTogether these results identify the extremum stack as the canonical minimal representation for rate-independent computation — in the worst-case (Kolmogorov), average-case (Shannon), and online algorithmic senses—and settle its maintenance complexity.  \nKeywords: Preisach attention, extremum stack, rate-independence, sequence modelling, sufficient statistic, Kolmogorov complexity, Shannon mutual information, worst-case complexity, KV-cache compression, finger tree  \n1. Introduction  \nThe Preisach Attention Layer (PAL) [1] is a recently proposed sequence modelling architecture that replaces softmax attention with a stack of binary relays from classical hysteresis theory. PAL is Turing-complete at depth O(1), and its function class is provably incomparable with the transformer’s: the separating property is rate-independence — PAL responds only to the ordered sequence of local extrema of its input, not to absolute token positions or temporal spacing. By discarding timing and retaining only the alternation structure, PAL reduces per-token inference from O (n2 ) to O (nlog n) and replaces the O (n · dmodel) KV-cache with an O (k · dmodel) extremum stack, where k ≤ n is the current stack depth.  \nThe central algorithmic object of PAL is this extremum stack Πn: the nested sequence of alternating local maxima and minima of the input u0:n ∈ GnL+1 that survive the classical wiping-out rule [2]—a new extremum erases all prior extrema of smaller magnitude. The stack is maintained online by a simple algorithm (Algorithm 1) running in O(1) amortised time per step. Rate-independence—and the extremum stack—also appears in the modelling of ferromagnetic materials [3 , 2], elastoplastic s","cbCaivv9lszADi5B","https://ap.wps.com/l/cbCaivv9lszADi5B","pdf",704187,3,1,31,"English","en",105,"# Introduction\n## Preisach Attention Layer (PAL) and motivation\n## The extremum stack Πn as central object\n## Three guiding questions (Q1–Q3)\n# Characterisation, minimality, and online complexity","[{\"question\":\"What does “rate-independent” mean for the functionals studied in the paper?\",\"answer\":\"A functional is rate-independent if it is unchanged under monotone time reparametrisations, so it responds only to the order of local extrema and not to their timing.\"},{\"question\":\"What is the Preisach extremum stack Πn and how is it constructed?\",\"answer\":\"Πn is the nested sequence of alternating local maxima and minima of the input that survive the classical Preisach hysteresis wiping-out rule, where a new extremum erases prior extrema of smaller magnitude.\"},{\"question\":\"How do the paper’s Kolmogorov and Shannon minimality results differ?\",\"answer\":\"Kolmogorov minimality gives worst-case shortest-program length bounds for representing the class of rate-independent functionals, while Shannon minimality gives an average-case mutual-information minimisation under probability measures.\"}]",1784211571,78,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"the-extremum-stack-as-optimal-memory-for-rate-independent-sequence-models","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/the-extremum-stack-as-optimal-memory-for-rate-independent-sequence-models/86410/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does “rate-independent” mean for the functionals studied in the paper?","Question",{"text":75,"@type":76},"A functional is rate-independent if it is unchanged under monotone time reparametrisations, so it responds only to the order of local extrema and not to their timing.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the Preisach extremum stack Πn and how is it constructed?",{"text":80,"@type":76},"Πn is the nested sequence of alternating local maxima and minima of the input that survive the classical Preisach hysteresis wiping-out rule, where a new extremum erases prior extrema of smaller magnitude.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the paper’s Kolmogorov and Shannon minimality results differ?",{"text":84,"@type":76},"Kolmogorov minimality gives worst-case shortest-program length bounds for representing the class of rate-independent functionals, while Shannon minimality gives an average-case mutual-information minimisation under probability 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