{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T07:38:15Z","timestamp":1774251495897,"version":"3.50.1"},"reference-count":55,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T00:00:00Z","timestamp":1774224000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Comput. Neurosci."],"abstract":"<jats:p>\n                    Computation separates time from space: nondeterministic problems are exponential in time (the \u201cTime Dragon\u201d) but polynomially simulable in space (the \u201cSpace Dragon\u201d), as formalized by Savitch's theorem (NPSPACE\u2286PSPACE). We propose that the brain physically instantiates this theorem through Recursive Condensation, a topological mechanism that converts intractable high-dimensional search into efficient low-dimensional navigation. Drawing on Urysohn's Lemma, we demonstrate that separability is a property of connectivity, not volume; a stable decision boundary exists independent of ambient dimension provided the underlying manifolds are topologically disjoint. To manufacture this disjointness, the cortex employs a parity alternation strategy: it alternates between odd-parity metric expansion (exploratory search) to untangle local geometry, and even-parity topological contraction (closure\/condensation) to lock in validated invariants. This cycle acts as a biological \u201cSavitch Machine,\u201d mediating a Topological Trinity Transformation (TTT),\n                    <jats:italic>Search<\/jats:italic>\n                    \u2192\n                    <jats:italic>Closure<\/jats:italic>\n                    \u2192\n                    <jats:italic>Navigation<\/jats:italic>\n                    , that compiles high-entropy exploration paths into low-energy quotient tokens. Under Memory-Amortized Inference (MAI), the cortex slays the Space Dragon by collapsing vast state spaces into compact metric singularities, and tames the Time Dragon by memoizing these traversals as structural priors. Evolution's \u201ccheat code,\u201d linear cortical growth yielding exponential cognitive gain, emerges as a physical law of topological inference: exponential search in time becomes polynomial reuse in space via recursive metric collapse.\n                  <\/jats:p>","DOI":"10.3389\/fncom.2026.1778902","type":"journal-article","created":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T06:44:00Z","timestamp":1774248240000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["The two dragons of cognition: recursive condensation for predictive processing"],"prefix":"10.3389","volume":"20","author":[{"given":"Xin","family":"Li","sequence":"first","affiliation":[{"name":"Department of Computer Science, University at Albany","place":["Albany, NY, United States"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1965","published-online":{"date-parts":[[2026,3,23]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"e3000487","DOI":"10.1371\/journal.pbio.3000487","article-title":"Alpha oscillations and traveling waves: signatures of predictive coding?","volume":"17","author":"Alamia","year":"2019","journal-title":"PLoS Biol"},{"key":"B2","doi-asserted-by":"publisher","first-page":"695","DOI":"10.1016\/j.neuron.2012.10.038","article-title":"Canonical microcircuits for predictive coding","volume":"76","author":"Bastos","year":"2012","journal-title":"Neuron"},{"key":"B3","doi-asserted-by":"publisher","first-page":"1401","DOI":"10.1523\/JNEUROSCI.3175-15.2016","article-title":"Phase of spontaneous slow oscillations during sleep influences memory-related processing of auditory cues","volume":"36","author":"Batterink","year":"2016","journal-title":"J. Neurosci"},{"key":"B4","volume-title":"Dynamic Programming","author":"Bellman","year":"1957"},{"key":"B5","volume-title":"A Brief History of Intelligence: Evolution, AI, and the Five Breakthroughs that Made Our Brains","author":"Bennett","year":"2023"},{"key":"B6","doi-asserted-by":"publisher","first-page":"3207","DOI":"10.5555\/2567709.2567766","article-title":"Counterfactual reasoning and learning systems: the example of computational advertising","volume":"14","author":"Bottou","year":"2013","journal-title":"J. Mach. Learn. Res"},{"key":"B7","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/0004-3702(91)90053-M","article-title":"Intelligence without representation","volume":"47","author":"Brooks","year":"1991","journal-title":"Artif. Intell"},{"key":"B8","doi-asserted-by":"publisher","first-page":"1877","DOI":"10.5555\/3495724.3495883","article-title":"Language models are few-shot learners","volume":"33","author":"Brown","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780195301069.001.0001","author":"Buzsaki","year":"2006","journal-title":"Rhythms of the Brain"},{"key":"B10","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1002\/hipo.22488","article-title":"Hippocampal sharp wave-ripple: a cognitive biomarker for episodic memory and planning","volume":"25","author":"Buzs\u00e1ki","year":"2015","journal-title":"Hippocampus"},{"key":"B11","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1038\/nn.3304","article-title":"Memory, navigation and theta rhythm in the hippocampal-entorhinal system","volume":"16","author":"Buzs\u00e1ki","year":"2013","journal-title":"Nat. Neurosci"},{"key":"B12","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1146\/annurev-neuro-062111-150444","article-title":"Mechanisms of gamma oscillations","volume":"35","author":"Buzs\u00e1ki","year":"2012","journal-title":"Annu. Rev. Neurosci"},{"key":"B13","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1146\/annurev.neuro.31.060407.125639","article-title":"Spike timing-dependent plasticity: a hebbian learning rule","volume":"31","author":"Caporale","year":"2008","journal-title":"Annu. Rev. Neurosci"},{"key":"B14","first-page":"151","article-title":"\u201cThe complexity of theorem-proving procedures,\u201d","volume-title":"Proceedings of the Third Annual ACM Symposium on Theory of Computing","author":"Cook","year":"1971"},{"key":"B15","doi-asserted-by":"publisher","first-page":"15160","DOI":"10.1073\/pnas.0801819105","article-title":"Spontaneous neural activity during human slow wave sleep","volume":"105","author":"Dang-Vu","year":"2008","journal-title":"Proc. Nat. Acad. Sci"},{"key":"B16","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1038\/nrn2536","article-title":"Perceiving is believing: a Bayesian approach to explaining the positive symptoms of schizophrenia","volume":"10","author":"Fletcher","year":"2009","journal-title":"Nat. Rev. Neurosci"},{"key":"B17","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1038\/nrn2787","article-title":"The free-energy principle: a unified brain theory?","volume":"11","author":"Friston","year":"2010","journal-title":"Nat. Rev. Neurosci"},{"key":"B18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1162\/neco.1992.4.1.1","article-title":"Neural networks and the bias\/variance dilemma","volume":"4","author":"Geman","year":"1992","journal-title":"Neural Comput"},{"key":"B19","article-title":"\u201cAmortized inference in probabilistic reasoning,\u201d","author":"Gershman","year":"2014","journal-title":"Proceedings of the Annual Meeting of the Cognitive Science Society, Volume"},{"key":"B20","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1142\/S0129065709002002","article-title":"Spiking neural networks","volume":"19","author":"Ghosh-Dastidar","year":"2009","journal-title":"Int. J. Neural Syst"},{"key":"B21","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1016\/j.plrev.2025.02.001","article-title":"Dark brain energy: toward an integrative model of spontaneous slow oscillations","volume":"52","author":"Gong","year":"2025","journal-title":"Phys. Life Rev"},{"key":"B22","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.tics.2015.04.006","article-title":"Hierarchical process memory: memory as an integral component of information processing","volume":"19","author":"Hasson","year":"2015","journal-title":"Trends Cogn. Sci"},{"key":"B23","volume-title":"Algebraic Topology","author":"Hatcher","year":"2005"},{"key":"B24","doi-asserted-by":"publisher","first-page":"1158","DOI":"10.1126\/science.7761831","article-title":"The \u201cwake-sleep\u201d algorithm for unsupervised neural networks","volume":"268","author":"Hinton","year":"1995","journal-title":"Science"},{"key":"B25","doi-asserted-by":"publisher","first-page":"837","DOI":"10.1098\/rstb.2005.1623","article-title":"The cortical column: a structure without a function","volume":"360","author":"Horton","year":"2005","journal-title":"Philos. Trans. R. Soc. B: Biol. Sci"},{"key":"B26","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1038\/nature09552","article-title":"Rethinking schizophrenia","volume":"468","author":"Insel","year":"2010","journal-title":"Nature"},{"key":"B27","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1162\/089976606775093882","article-title":"Polychronization: computation with spikes","volume":"18","author":"Izhikevich","year":"2006","journal-title":"Neural Comput"},{"key":"B28","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/978-1-4684-2001-2_9","author":"Karp","year":"1972"},{"key":"B29","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.neuron.2018.10.003","article-title":"Predictive processing: a canonical cortical computation","volume":"100","author":"Keller","year":"2018","journal-title":"Neuron"},{"key":"B30","doi-asserted-by":"publisher","first-page":"e1602026","DOI":"10.1126\/sciadv.1602026","article-title":"Coordinated infraslow neural and cardiac oscillations mark fragility and offline periods in mammalian sleep","volume":"3","author":"Lecci","year":"2017","journal-title":"Sci. Adv"},{"key":"B31","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"},{"key":"B32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1103\/RevModPhys.81.1","article-title":"Colloquium: the physics of maxwell's demon and information","volume":"81","author":"Maruyama","year":"2009","journal-title":"Rev. Mod. Phys"},{"key":"B33","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1109\/JRPROC.1961.287775","article-title":"Steps toward artificial intelligence","volume":"49","author":"Minsky","year":"1961","journal-title":"Proc. IRE"},{"key":"B34","volume-title":"Computation: Finite and Infinite Machines","author":"Minsky","year":"1967"},{"key":"B35","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1152\/jn.1957.20.4.408","article-title":"Modality and topographic properties of single neurons of cat's somatic sensory cortex","volume":"20","author":"Mountcastle","year":"1957","journal-title":"J. Neurophysiol"},{"key":"B36","doi-asserted-by":"publisher","first-page":"663","DOI":"10.1085\/jgp.107.6.663","article-title":"The hippocampus as a cognitive graph","volume":"107","author":"Muller","year":"1996","journal-title":"J. Gen. Physiol"},{"key":"B37","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1109\/MSP.2019.2931595","article-title":"Surrogate gradient learning in spiking neural networks: bringing the power of gradient-based optimization to spiking neural networks","volume":"36","author":"Neftci","year":"2019","journal-title":"IEEE Signal Process. Mag"},{"key":"B38","doi-asserted-by":"publisher","first-page":"566","DOI":"10.1038\/s41586-023-06098-1","article-title":"Geometric constraints on human brain function","volume":"618","author":"Pang","year":"2023","journal-title":"Nature"},{"key":"B39","doi-asserted-by":"publisher","first-page":"676","DOI":"10.1073\/pnas.98.2.676","article-title":"A default mode of brain function","volume":"98","author":"Raichle","year":"2001","journal-title":"Proc. Nat. Acad. Sci"},{"key":"B40","doi-asserted-by":"publisher","first-page":"062719","DOI":"10.1103\/PhysRevE.91.062719","article-title":"Neural field theory of nonlinear wave-wave and wave-neuron processes","volume":"91","author":"Robinson","year":"2015","journal-title":"Phys. Rev. E"},{"key":"B41","first-page":"107","author":"Robinson","year":"2015"},{"key":"B42","author":"Rudin","year":"1976","journal-title":"Principles of Mathematical Analysis"},{"key":"B43","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/S0022-0000(70)80006-X","article-title":"Relationships between nondeterministic and deterministic tape complexities","volume":"4","author":"Savitch","year":"1970","journal-title":"J. Comput. Syst. Sci"},{"key":"B44","first-page":"467","article-title":"The architecture of complexity","volume":"106","author":"Simon","year":"1962","journal-title":"Proc. Am. Philos. Soc"},{"key":"B45","volume-title":"Introduction to the Theory of Computation, 3rd Edn","author":"Sipser","year":"2012"},{"key":"B46","doi-asserted-by":"publisher","first-page":"1679","DOI":"10.1038\/nn.4119","article-title":"Hierarchical nesting of slow oscillations, spindles and ripples in the human hippocampus during sleep","volume":"18","author":"Staresina","year":"2015","journal-title":"Nat. Neurosci"},{"key":"B47","doi-asserted-by":"publisher","first-page":"634","DOI":"10.1016\/j.biopsych.2018.05.015","article-title":"The predictive coding account of psychosis","volume":"84","author":"Sterzer","year":"2018","journal-title":"Biol. Psychiatry"},{"key":"B48","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.1093\/cercor\/10.12.1185","article-title":"Origin of slow cortical oscillations in deafferented cortical slabs","volume":"10","author":"Timofeev","year":"2000","journal-title":"Cereb. Cortex"},{"key":"B49","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1023\/A:1008839312043","article-title":"Simulation of gamma rhythms in networks of interneurons and pyramidal cells","volume":"4","author":"Traub","year":"1997","journal-title":"J. Comput. Neurosci"},{"key":"B50","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1098\/rstb.1952.0012","article-title":"The chemical basis of mokphogenesis","volume":"237","author":"Turing","year":"1952","journal-title":"Phil. Trans. R. Soc. Lond. B Biol. Sci"},{"key":"B51","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1007\/BF01208659","article-title":"\u00dcber die m\u00e4chtigkeit der zusammenh\u00e4ngenden mengen","volume":"94","author":"Urysohn","year":"1925","journal-title":"Math. Ann"},{"key":"B52","volume-title":"Statistical Learning Theory","author":"Vapnik","year":"1998"},{"key":"B53","article-title":"\u201cAttention is all you need,\u201d","author":"Vaswani","year":"2017","journal-title":"Advances in Neural Information Processing Systems 30"},{"key":"B54","doi-asserted-by":"publisher","first-page":"5362","DOI":"10.1109\/TPAMI.2024.3367329","article-title":"A comprehensive survey of continual learning: theory, method and application","volume":"46","author":"Wang","year":"2024","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell"},{"key":"B55","doi-asserted-by":"publisher","first-page":"5635","DOI":"10.1007\/s11263-024-02117-4","article-title":"Generalized out-of-distribution detection: a survey","volume":"132","author":"Yang","year":"2024","journal-title":"Int. J. Comput. Vis"}],"container-title":["Frontiers in Computational Neuroscience"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fncom.2026.1778902\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T06:44:03Z","timestamp":1774248243000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fncom.2026.1778902\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,23]]},"references-count":55,"alternative-id":["10.3389\/fncom.2026.1778902"],"URL":"https:\/\/doi.org\/10.3389\/fncom.2026.1778902","relation":{},"ISSN":["1662-5188"],"issn-type":[{"value":"1662-5188","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,23]]},"article-number":"1778902"}}