Adelic Reformulation of Collatz via p-adic Langlands Correspondence — E8 Intelligence Research

{"FINDING:":[0],"The":[1],"Collatz":[2,36,67],"conjecture":[3],"is":[4,69],"being":[5],"reformulated":[6],"via":[7,108],"adelic/p-adic":[8],"Langlands":[9],"correspondences,":[10],"with":[11,140],"explicit":[12,123],"construction":[13],"of":[14,81,156],"a":[15,26,53,70,76],"\\"Collatz":[16],"cocycle\\"":[17],"and":[18,31,98,144],"p-adic":[19,86,109],"L-functions":[20],"for":[21],"Hilbert":[22],"modular":[23,92],"forms":[24,93],"—":[25,46,103],"bridge":[27],"between":[28],"elementary":[29],"dynamics":[30],"automorphic":[32],"spectra.":[33],"MATH:":[34],"-":[35,59,85,113],"map":[37],"T(n)":[38],"=":[39],"n/2":[40],"if":[41,44],"even,":[42],"(3n+1)/2":[43],"odd":[45],"embedded":[47],"in":[48,72,78,127],"ℤ̂":[49,139],"(profinite":[50],"integers)":[51],"as":[52],"2-adic":[54],"×":[55],"3-adic":[56],"dynamical":[57],"system.":[58],"Adelic":[60],"framework:":[61],"GL₁(𝔸_ℚ)":[62],"action":[63],"on":[64,138],"ℤ̂;":[65],"the":[66,82,128,132,145,157],"cocycle":[68],"1-cocycle":[71],"H¹(Gal(ℚ̄/ℚ),":[73],"ℤ̂)":[74],"or":[75],"class":[77],"continuous":[79],"cohomology":[80],"adele":[83],"group.":[84],"L-functions:":[87],"two-variable":[88],"Asai":[89],"L-function":[90,102],"(Hilbert":[91],"over":[94],"real":[95],"quadratic":[96],"fields)":[97],"twisted":[99],"triple":[100],"product":[101],"interpolating":[104],"classical":[105],"special":[106],"values":[107],"analytic":[110],"continuation":[111],"(arXiv:2504.12066v2).":[112],"Key":[114],"constants:":[115],"2,":[116],"3,":[117],"1":[118],"(the":[119],"map's":[120],"coefficients);":[121],"no":[122],"golden":[124],"ratios":[125],"appear":[126],"raw":[129],"findings,":[130],"but":[131],"adelic":[133],"normalization":[134],"uses":[135],"Haar":[136],"measure":[137],"total":[141],"mass":[142],"1,":[143],"2-adic/3-adic":[146],"split":[147],"Author:":[148],"Andrew":[149],"Stewart":[150],"Caldin,":[151],"Independent":[152],"Researcher,":[153],"UK.":[154],"Part":[155],"E8":[158],"Intelligence":[159],"Research":[160],"series.":[161],"Platform:":[162],"e8intelligence.com":[163]}

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22823856
Primary Topic
advanced mathematical theories
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
preprint

Adelic Reformulation of Collatz via p-adic Langlands Correspondence — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
advanced mathematical theories
preprint

Adelic Reformulation of Collatz via p-adic Langlands Correspondence — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The Collatz conjecture is being reformulated via adelic/p-adic Langlands correspondences, with explicit construction of a "Collatz cocycle" and p-adic L-functions for Hilbert modular forms — a bridge between elementary dynamics and automorphic spectra. MATH: - Collatz map T(n) = n/2 if even, (3n+1)/2 if odd — embedded in ℤ̂ (profinite integers) as a 2-adic × 3-adic dynamical system. - Adelic framework: GL₁(𝔸_ℚ) action on ℤ̂; the Collatz cocycle is a 1-cocycle in H¹(Gal(ℚ̄/ℚ), ℤ̂) or a class in continuous cohomology of the adele group. - p-adic L-functions: two-variable Asai L-function (Hilbert modular forms over real quadratic fields) and twisted triple product L-function — interpolating classical special values via p-adic analytic continuation (arXiv:2504.12066v2). - Key constants: 2, 3, 1 (the map's coefficients); no explicit golden ratios appear in the raw findings, but the adelic normalization uses Haar measure on ℤ̂ with total mass 1, and the 2-adic/3-adic split Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
advanced mathematical theories
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.

Adelic Reformulation of Collatz via p-adic Langlands Correspondence — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS