Lattices, T-Duality, and Double Field Theory on K3 × T²: A Lean 4 Companion Formalization

{"Revision":[0],"2:":[1],"external":[2],"review":[3],"applied;":[4],"dualScale_eq_iff":[5],"proves":[6],"the":[7,10,41,47,70,80,88,92,100,111,133,155,158,168,215,231,253,255,305,370,373,376,379],"minimizer":[8],"of":[9,49,79,91,103,115,132,157,217,252,282,378],"dual-scale":[11],"bound":[12],"is":[13,207,228,236,330],"unique.":[14],"Closed-string":[15],"compactification":[16],"on":[17,46,64,148,343],"$K3×":[18,96,149,152],"T^2$":[19,97,150],"carries,":[20],"besides":[21],"its":[22,120,264],"continuous":[23],"geometric":[24],"moduli,":[25],"a":[26,56,126,194,268],"discrete":[27],"duality":[28],"structure:":[29,75],"T-duality":[30],"identifies":[31],"backgrounds":[32],"related":[33],"by":[34,40,167,304],"$R\\\\leftrightarrow":[35],"\\\\alpha'/R$":[36],"and,":[37],"more":[38],"generally,":[39],"arithmetic":[42,90,156],"group":[43],"$O(d,d;\\\\mathbb{Z})$":[44,101],"acting":[45],"lattice":[48],"momentum":[50],"and":[51,77,82,95,105,108,119,123,144,151,154,175,191,201,225,263,275,278,287,372],"winding":[52],"charges.":[53],"We":[54,188,247],"report":[55],"Lean":[57,169,306,332],"4":[58,170,307,338],"library,":[59],"DualScaleStream2":[60],"(19":[61],"modules,":[62],"built":[63],"Mathlib,":[65],"toolchain":[66],"v4.33.1),":[67],"that":[68,85,259],"kernel-checks":[69],"algebraic":[71],"skeleton":[72],"underlying":[73],"this":[74,137,198,237,344],"positive-definiteness":[76],"unimodularity":[78],"$E_8$":[81],"hyperbolic":[83],"lattices":[84],"build":[86],"$H^2(K3,\\\\mathbb{Z})$;":[87],"signature":[89],"K3,":[93],"Mukai,":[94],"charge":[98,222],"lattices;":[99],"generators":[102],"Giveon–Porrati–Rabinovici":[104],"their":[106],"charge-norm":[107],"spectrum":[109],"invariance;":[110],"Hull–Zwiebach":[112],"generalized":[113],"metric":[114],"Double":[116],"Field":[117],"Theory":[118],"$B$-field":[121],"shifts":[122],"section":[124],"condition;":[125],"$d$-dimensional":[127],"trace-bound":[128],"generalization,":[129],"$\\\\operatorname{tr}G+\\\\operatorname{tr}G^{-1}\\\\ge":[130],"2d$,":[131],"one-dimensional":[134],"statement":[135,206],"behind":[136],"research":[138],"programme's":[139],"\\"dual-scale\\"":[140],"proposal;":[141],"flux":[142],"integrality":[143],"D-brane":[145],"tadpole":[146],"counting":[147],"K3$;":[153],"Eguchi–Ooguri–Tachikawa":[159],"Mathieu-moonshine":[160],"coefficients.":[161],"All":[162],"100":[163],"theorems":[164,351],"are":[165,299,326],"accepted":[166],"kernel":[171],"with":[172,220,355],"zero":[173],"sorry":[174],"an":[176],"axiom":[177],"footprint":[178],"limited":[179],"to":[180],"Lean's":[181],"three":[182],"standard":[183,356],"axioms":[184,309,354,357],"(propext,":[185],"Classical.choice,":[186,311],"Quot.sound).":[187],"state":[189],"throughout,":[190],"repeat":[192],"in":[193,368],"dedicated":[195],"section,":[196],"what":[197],"formalization":[199,371],"does":[200,202],"not":[203],"establish:":[204],"every":[205,291],"about":[208],"finite-dimensional":[209],"matrices":[210,219],"over":[211],"$\\\\mathbb{Z}$":[212],"or":[213,235,318],"$\\\\mathbb{R}$;":[214],"identification":[216],"these":[218,261],"string-theory":[221],"lattices,":[223],"dualities,":[224],"background":[226],"fields":[227],"quoted":[229],"from":[230],"literature":[232],"(Tier":[233,241],"L)":[234],"project's":[238],"own":[239],"reading":[240],"C),":[242],"never":[243],"re-derived":[244],"as":[245,250],"physics.":[246],"also":[248],"report,":[249],"part":[251],"methods,":[254],"tiered":[256],"multi-agent":[257],"pipeline":[258],"produced":[260],"proofs":[262],"measured":[265],"failure":[266],"modes:":[267],"cheap":[269],"local":[270],"prover":[271],"closes":[272],"concrete":[273],"goals":[274],"nothing":[276],"symbolic,":[277],"two":[279],"agent":[280],"self-reports":[281],"successful":[283],"compilation":[284],"were":[285],"false":[286],"caught":[288],"only":[289],"because":[290],"claim":[292],"was":[293,366],"independently":[294],"recompiled.":[295],"Epistemic":[296],"tiers.":[297],"Claims":[298],"labelled":[300],"Tier":[301,314,319,323],"A":[302,324],"(checked":[303],"kernel;":[308],"propext,":[310],"Quot.sound":[312],"only),":[313],"L":[315],"(literature,":[316],"cited)":[317],"C":[320],"(conjecture).":[321],"Only":[322],"statements":[325],"machine-checked;":[327],"physical":[328],"interpretation":[329],"not.":[331],"artifact:":[333],"SocrateAI-Scientific-Agora-LeanMaster,":[334],"tag":[335],"v3.17.0":[336],"(Lean":[337],"v4.33.1,":[339],"Mathlib).":[340],"Release":[341],"gates":[342],"tag:":[345],"all":[346],"ten":[347],"libraries":[348],"build;":[349],"621":[350],"pass":[352],"#print":[353],"only;":[358],"no":[359],"sorry/admit.":[360],"AI-assisted":[361],"tooling":[362],"(Anthropic":[363],"Claude":[364],"models)":[365],"used":[367],"preparing":[369],"manuscript,":[374],"under":[375],"direction":[377],"author.":[380]}

Authors

Institutions

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22837843
Primary Topic
Black Holes and Theoretical Physics
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
preprint

Lattices, T-Duality, and Double Field Theory on K3 × T²: A Lean 4 Companion Formalization

Xavier Callens, SocrateAI Scientific Agora Collaboration
Zenodo (CERN European Organization for Nuclear Research)
Black Holes and Theoretical Physics
preprint

Lattices, T-Duality, and Double Field Theory on K3 × T²: A Lean 4 Companion Formalization

Xavier Callens, SocrateAI Scientific Agora Collaboration
preprint en

Abstract

Revision 2: external review applied; dualScale_eq_iff proves the minimizer of the dual-scale bound is unique. Closed-string compactification on $K3× T^2$ carries, besides its continuous geometric moduli, a discrete duality structure: T-duality identifies backgrounds related by $R\leftrightarrow \alpha'/R$ and, more generally, by the arithmetic group $O(d,d;\mathbb{Z})$ acting on the lattice of momentum and winding charges. We report a Lean 4 library, DualScaleStream2 (19 modules, built on Mathlib, toolchain v4.33.1), that kernel-checks the algebraic skeleton underlying this structure: positive-definiteness and unimodularity of the $E_8$ and hyperbolic lattices that build $H^2(K3,\mathbb{Z})$; the signature arithmetic of the K3, Mukai, and $K3× T^2$ charge lattices; the $O(d,d;\mathbb{Z})$ generators of Giveon–Porrati–Rabinovici and their charge-norm and spectrum invariance; the Hull–Zwiebach generalized metric of Double Field Theory and its $B$-field shifts and section condition; a $d$-dimensional trace-bound generalization, $\operatorname{tr}G+\operatorname{tr}G^{-1}\ge 2d$, of the one-dimensional statement behind this research programme's "dual-scale" proposal; flux integrality and D-brane tadpole counting on $K3× T^2$ and $K3× K3$; and the arithmetic of the Eguchi–Ooguri–Tachikawa Mathieu-moonshine coefficients. All 100 theorems are accepted by the Lean 4 kernel with zero sorry and an axiom footprint limited to Lean's three standard axioms (propext, Classical.choice, Quot.sound). We state throughout, and repeat in a dedicated section, what this formalization does and does not establish: every statement is about finite-dimensional matrices over $\mathbb{Z}$ or $\mathbb{R}$; the identification of these matrices with string-theory charge lattices, dualities, and background fields is quoted from the literature (Tier L) or is this project's own reading (Tier C), never re-derived as physics. We also report, as part of the methods, the tiered multi-agent pipeline that produced these proofs and its measured failure modes: a cheap local prover closes concrete goals and nothing symbolic, and two agent self-reports of successful compilation were false and caught only because every claim was independently recompiled. Epistemic tiers. Claims are labelled Tier A (checked by the Lean 4 kernel; axioms propext, Classical.choice, Quot.sound only), Tier L (literature, cited) or Tier C (conjecture). Only Tier A statements are machine-checked; physical interpretation is not. Lean artifact: SocrateAI-Scientific-Agora-LeanMaster, tag v3.17.0 (Lean 4 v4.33.1, Mathlib). Release gates on this tag: all ten libraries build; 621 theorems pass #print axioms with standard axioms only; no sorry/admit. AI-assisted tooling (Anthropic Claude models) was used in preparing the formalization and the manuscript, under the direction of the author.

Zenodo (CERN European Organization for Nuclear Research)
Laboratoire de Mathématiques d'Orsay (FR)
Black Holes and Theoretical Physics
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.