Liquid-State Mathematics: Decoupling Combinatorial Complexity via Continuous Relational Fields and Concurrent Topological Flow
{"Traditional":[0],"combinatorial":[1,43,131],"optimization":[2,69],"relies":[3],"on":[4],"discrete":[5,50],"sequential":[6],"search":[7],"trees,":[8],"treating":[9],"problem":[10,116,139],"size":[11],"(\\\\bm{n})":[12],"as":[13,45],"a":[14,33,135,149,162],"multiplier":[15],"of":[16,110,151,167],"temporal":[17],"sequence":[18],"and":[19,102],"encountering":[20],"exponential":[21],"scaling":[22],"walls":[23],"(e.g.,":[24],"\\\\bm{O(n!)}":[25],"or":[26],"\\\\bm{O(2^n)}).":[27],"This":[28,93],"collection":[29],"introduces":[30],"Liquid-State":[31],"Mathematics,":[32],"native":[34],"computational":[35,105],"architecture":[36],"that":[37,125],"bypasses":[38],"explicit":[39],"enumeration":[40,160],"by":[41,148],"recasting":[42],"complexity":[44],"spatial":[46],"geometry.":[47],"By":[48],"translating":[49],"variables":[51],"into":[52],"continuous":[53,112],"topographical":[54],"gradients":[55],"at":[56],"an":[57,103],"ultra-fine":[58],"arbitrary":[59],"resolution":[60],"(\\"Dark":[61],"Scale\\"),":[62],"the":[63,83,89,96],"system":[64],"abandons":[65],"path-by-path":[66],"search.":[67],"Instead,":[68],"is":[70],"achieved":[71],"via":[72],"concurrent":[73],"topological":[74],"wavefront":[75],"propagation,":[76],"utilizing":[77],"floating":[78],"transient-state":[79],"memory":[80],"to":[81,85,121,142],"force":[82],"optimum":[84],"arrive":[86],"first":[87],"within":[88],"lowest":[90],"energy":[91],"basin.":[92],"submission":[94],"includes":[95],"theoretical":[97],"protocol,":[98],"virtual":[99],"execution":[100,109],"rules,":[101],"empirical":[104],"test":[106],"report.":[107],"Experimental":[108],"this":[111],"relational":[113],"field":[114,156],"across":[115],"sizes":[117],"ranging":[118],"from":[119,158],"\\\\bm{n=12}":[120],"\\\\bm{n=50,000}":[122],"nodes":[123],"demonstrates":[124],"convergence":[126,145,157],"depth":[127,146],"does":[128],"not":[129],"track":[130],"state-space":[132],"growth.":[133],"Across":[134],"100x":[136],"increase":[137],"in":[138],"scale":[140],"(500":[141],"50,000":[143],"nodes),":[144],"increased":[147],"factor":[150],"only":[152],"7.12x,":[153],"successfully":[154],"decoupling":[155],"classical":[159],"with":[161],"sublinear":[163],"power":[164],"law":[165],"fit":[166],"\\\\bm{D(n)":[168],"\\\\approx":[169],"67.13":[170],"n^{0.423}}.":[171]}
Authors
- Matt Webb (ORCID: https://orcid.org/0009-0006-6784-1877)
Publication Details
- Journal
- Zenodo (CERN European Organization for Nuclear Research)
- Published
- 2026-09-11
- DOI
- https://doi.org/10.5281/zenodo.22714637
- Primary Topic
- Slime Mold and Myxomycetes Research
- Type
- preprint