{ "nodes": [ { "id": "state_eq", "label": "The Geometric State Equation", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "κ = (h ln 2 / (n−1))². Three postulates — information flux, hierarchical topology, geometric fidelity — yield a unique equation with zero free parameters relating entropy rate h, curvature κ, and embedding dimension n.", "size": 5 }, { "id": "n2_univ", "label": "n = 2 Universality", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "The embedding dimension n = 2.00 ± 0.06 across all domains tested — genomes, viruses, proteins, neural systems, AI, language. Strict bifurcating hierarchies require exactly 2 dimensions (Sarkar embedding + Theorem 1).", "size": 4 }, { "id": "viral_r", "label": "Viral Validation (r = 0.996)", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "15 viral families spanning 5-year outbreaks to ancient lineages. Predicted vs measured κ: Pearson r = 0.996, p < 10⁻⁶, explaining 99.3% of variance. Curvature tracks phylogenetic depth (ρ = 0.84), not mutation rate (ρ = 0.12).", "size": 4 }, { "id": "protein_jump", "label": "Protein Curvature Jump (3.1×)", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "14 Pfam families: κ_protein = 3.80 ± 0.60, n = 2.03 ± 0.10. Predicted from h_protein ≈ 2.85 bits: κ ≈ 3.90 (within 2.6%). The 3.1× jump from nucleotide to amino acid curvature follows directly from the larger effective alphabet (20 vs 4).", "size": 3 }, { "id": "neural_kappa", "label": "Neural κ = 0.485", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "39 Neuropixels sessions (mouse visual cortex). κ = 0.485 ± 0.005 via triangle excess on SPD(180) manifold. Volume entropy yields h = 1.04, giving n = 2.03 ± 0.36. The state equation holds for spike trains on a ~2.4s timescale.", "size": 4 }, { "id": "criticality", "label": "Criticality Corollary", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "κ > 0 ⟹ J < 1. The brain's distance from criticality is not tuned by homeostasis — it is set by the entropy rate through the state equation: 1 − J = κ/(h₀ ln 2)². Twenty years of neuroscience, derived in one line.", "size": 4 }, { "id": "icosahedral", "label": "Icosahedral Mind Atlas", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "At κ*, the boundary sphere S² admits an optimal partition into 12 regions with icosahedral symmetry (Cohn-Kumar). The dvādasāyatana of early Buddhist phenomenology predicts exactly 12 sense bases. Match: 10/12, z = 1.91, p < 0.0001.", "size": 3 }, { "id": "alphabet_conv", "label": "Alphabet Convergence", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "DNA (4 bases → ~3 effective via Ts/Tv bias), language (~40 phonemes → ~3 effective via articulatory channelling), proteins (20 AA → ~7 effective via BLOSUM). DNA and language converge on h ≈ log₂3 ≈ 1.58 bits → same κ ≈ 1.2.", "size": 4 }, { "id": "geo_gap", "label": "Geometric Gap (Bio vs AI)", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "Biological κ ≈ 0.43–0.49 vs artificial κ ≈ 0.27–0.34. Gap of 0.16–0.20 (p < 0.0001) persists across all architectures (GPT-2, BERT, ViT, CLIP), modalities, and training objectives. Substrate-independent but biology-specific.", "size": 3 }, { "id": "lean_proofs", "label": "Lean 4 Verification (9 theorems)", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "524 lines, 23 named lemmas, 0 sorry stubs in the logical chain. Existence, uniqueness, monotonicity in h and n, quadratic scaling, growth-rate matching, Lyapunov non-negativity, zero-iff-equilibrium, minimum at κ*. Machine-checked against Mathlib.", "size": 3 }, { "id": "convergence_5seed", "label": "5-Seed Convergence (5,550 genomes)", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "Five independent training runs on 5,550 genomes. Mean pairwise Procrustes r = 0.948 ± 0.016. Inter-domain κ ≈ 1.28–1.34 (scale-dependent). Stable organism coordinates: e.g. H. sapiens r = 0.908, θ = 168.4°.", "size": 3 }, { "id": "ling_kappa", "label": "Linguistic κ = 1.18–1.31", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "1,015 languages, 18 families, 16,496 sound change rules (Index Diachronica). Phonemic transition entropy h = 1.653 bits. Cross-entropy excess (ASJP, 106K pairs): h = 1.568 bits. Both yield κ in the genomic range.", "size": 3 }, { "id": "rg_invariance", "label": "RG Invariance of κ", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "When EEG covariance matrices are progressively block-averaged (reducing effective dimensionality), κ remains stable. Ratio before/after coarse-grain: 0.97 ± 0.03. Curvature is a structural property, not a scale-dependent artifact.", "size": 2 }, { "id": "paper1", "label": "Paper I: Evolution as Active Geometry", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "bioRxiv 2026.03.09.710612v2. The empirical paper: 107K+ taxa, 15 viral families, 15 protein families, 5,550-genome convergence. Inter-domain κ ≈ 1.28–1.34; intra-domain species trees κ = 3–16. Pearson r = 0.996 for viruses.", "size": 4 }, { "id": "paper2", "label": "Paper II: The Geometric State Equation", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "The mathematical foundations paper. Three postulates → state equation. 9+ theorems in Lean 4. Lyapunov stability proof: κ* is a global attractor. Existence, uniqueness, monotonicity.", "size": 4 }, { "id": "paper3", "label": "Paper III: Cognition as Active Geometry", "cluster": "core", "year": 2026, "authors": "Fenn & Fenn", "desc": "The neural paper. 39 Neuropixels sessions, ABIDE fMRI, EEGBCI EEG, 6 AI architectures. κ = 0.485 (neural), geometric gap 0.16–0.20. Criticality as corollary. Consciousness-state dependence (EO > EC, p < 0.001).", "size": 4 }, { "id": "manning1979", "label": "Manning 1979", "cluster": "riemannian", "year": 1979, "authors": "A. Manning", "desc": "Topological entropy for geodesic flows. Ann. Math. 110, 567–573. The central theorem: h_top = (n−1)√κ for geodesic flow on compact n-manifold of constant negative curvature K = −κ. This is the mathematical bridge that converts entropy rate into curvature.", "size": 5 }, { "id": "ratcliffe2006", "label": "Ratcliffe 2006", "cluster": "riemannian", "year": 2006, "authors": "J.G. Ratcliffe", "desc": "Foundations of Hyperbolic Manifolds, 2nd ed. Springer. Volume of geodesic balls in H^n grows as exp(r(n−1)√κ) — exponential growth that matches informational growth of branching hierarchies. Foundational reference.", "size": 3 }, { "id": "gromov1987", "label": "Gromov 1987", "cluster": "riemannian", "year": 1987, "authors": "M. Gromov", "desc": "Hyperbolic groups. Essays in Group Theory, Springer, 75–263. Introduces δ-hyperbolicity: tree metrics have δ = 0 (perfectly hyperbolic). Provides the large-scale geometric framework for understanding why trees embed into hyperbolic space.", "size": 4 }, { "id": "besson1995", "label": "Besson–Courtois–Gallot 1995", "cluster": "riemannian", "year": 1995, "authors": "G. Besson, G. Courtois, S. Gallot", "desc": "Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799. Volume entropy rigidity: among negatively curved manifolds with fixed topology, the locally symmetric metric uniquely minimizes volume entropy. Connects Manning's theorem to geometric rigidity.", "size": 3 }, { "id": "katok1995", "label": "Katok & Hasselblatt 1995", "cluster": "riemannian", "year": 1995, "authors": "A. Katok, B. Hasselblatt", "desc": "Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press. Comprehensive treatment of topological entropy, Lyapunov exponents, and ergodic theory. Foundation for the dynamical systems perspective on the state equation.", "size": 2 }, { "id": "milnor1968", "label": "Milnor 1968", "cluster": "riemannian", "year": 1968, "authors": "J. Milnor", "desc": "A note on curvature and fundamental group. J. Diff. Geom. 2, 1–7. Growth rate of the fundamental group of a negatively curved manifold is exponential — an early result connecting topology to exponential growth that Manning later quantified.", "size": 2 }, { "id": "shannon1948", "label": "Shannon 1948", "cluster": "infotheory", "year": 1948, "authors": "C.E. Shannon", "desc": "A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423. The origin of information theory. Channel capacity of the genetic code: H_raw = log₂(4) = 2 bits per substitution, the upper bound on the biological entropy rate h.", "size": 5 }, { "id": "cover_thomas", "label": "Cover & Thomas 2006", "cluster": "infotheory", "year": 2006, "authors": "T.M. Cover, J.A. Thomas", "desc": "Elements of Information Theory, 2nd ed. Wiley. Defines entropy rate for stationary stochastic processes — the asymptotic per-symbol entropy. The framework's h is precisely this quantity applied to mutation processes and neural spike trains.", "size": 3 }, { "id": "kolmogorov1958", "label": "Kolmogorov 1958", "cluster": "infotheory", "year": 1958, "authors": "A.N. Kolmogorov", "desc": "New metric invariant of transient dynamical systems. Dokl. Akad. Nauk SSSR 119, 861–864. Introduces metric (measure-theoretic) entropy for dynamical systems — the measure-theoretic counterpart of topological entropy that Manning's theorem relates to curvature.", "size": 3 }, { "id": "ratedistortion", "label": "Rate-Distortion Theory", "cluster": "infotheory", "year": 1959, "authors": "Shannon 1959; Berger 1971", "desc": "The state equation can be recast as a rate-distortion problem: the potential U(h,κ,n) = (h ln 2 − (n−1)√κ)² measures the squared gap between information rate and geometric capacity. The equilibrium κ* minimizes this \"distortion.\"", "size": 2 }, { "id": "kimura1968", "label": "Kimura 1968", "cluster": "molevol", "year": 1968, "authors": "M. Kimura", "desc": "Evolutionary rate at the molecular level. Nature 217, 624–626. The neutral theory of molecular evolution. Purifying selection removes 5–8% of substitutions from heritable variation, bounding the effective entropy rate h.", "size": 4 }, { "id": "kimura1980", "label": "Kimura 1980", "cluster": "molevol", "year": 1980, "authors": "M. Kimura", "desc": "A simple method for estimating evolutionary rates of base substitutions. J. Mol. Evol. 16, 111–120. Quantifies transition/transversion bias (~2:1 ratio), reducing effective entropy from 2.0 to ~1.9 bits per substitution.", "size": 3 }, { "id": "bird1980", "label": "Bird 1980", "cluster": "molevol", "year": 1980, "authors": "A.P. Bird", "desc": "DNA methylation and the frequency of CpG in animal DNA. Nucleic Acids Res. 8, 1499–1504. CpG dinucleotide hypermutation (10–50× elevation) introduces context-dependent predictability, further reducing per-site entropy to ~1.7–1.9 bits.", "size": 3 }, { "id": "eigen1971", "label": "Eigen 1971", "cluster": "molevol", "year": 1971, "authors": "M. Eigen", "desc": "Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58, 465–523. The error threshold: above a critical mutation rate, the genetic code degrades via error catastrophe. Provides the physical upper bound on κ — above κ*, fidelity fails.", "size": 4 }, { "id": "lynch2007", "label": "Lynch 2007", "cluster": "molevol", "year": 2007, "authors": "M. Lynch", "desc": "The Origins of Genome Architecture. Sinauer Associates. Comprehensive theory of how mutation, drift, and selection shape genome structure. The non-adaptive forces that set the effective entropy rate h at the population level.", "size": 2 }, { "id": "adami2002", "label": "Adami 2002", "cluster": "molevol", "year": 2002, "authors": "C. Adami", "desc": "What is complexity? BioEssays 24, 1085–1094. Defines physical complexity of genomes as mutual information between genome and environment. Connects Shannon entropy to evolutionary information content — the \"h\" in the state equation from a digital-life perspective.", "size": 2 }, { "id": "koonin2011", "label": "Koonin 2011", "cluster": "molevol", "year": 2011, "authors": "E.V. Koonin", "desc": "The Logic of Chance. FT Press. Universal patterns in genome evolution — mutation biases, selection pressures, and horizontal transfer as perturbations to the tree topology (Postulate 2). Framework for understanding when n > 2.", "size": 2 }, { "id": "sarkar2012", "label": "Sarkar 2012", "cluster": "phylo", "year": 2012, "authors": "R. Sarkar", "desc": "Low distortion Delaunay embedding of trees in hyperbolic plane. Graph Drawing 2011, LNCS 7034, 355–366. Any finite weighted tree on N nodes embeds into H² with multiplicative distortion 1 + O(ε). The rigorous foundation for n = 2 optimality.", "size": 4 }, { "id": "parks2022", "label": "Parks et al. 2022 (GTDB)", "cluster": "phylo", "year": 2022, "authors": "D.H. Parks et al.", "desc": "GTDB: an ongoing census of bacterial and archaeal diversity. Nucleic Acids Res. 50, D199–D207. Source of the domain-level species trees: 107,340 bacterial tips (κ = 16.4) and 5,932 archaeal tips (κ = 12.7).", "size": 3 }, { "id": "li2021", "label": "Li et al. 2021 (Fungi)", "cluster": "phylo", "year": 2021, "authors": "Y. Li et al.", "desc": "A genome-scale phylogeny of the kingdom Fungi. Curr. Biol. 31, 1653–1665. Source of the fungal species tree (1,610 tips, κ = 3.0 ± 0.1). Independent phylogenetic reference for the fungal telescope experiment.", "size": 3 }, { "id": "katoh2013", "label": "Katoh & Standley 2013 (MAFFT)", "cluster": "phylo", "year": 2013, "authors": "K. Katoh, D.M. Standley", "desc": "MAFFT multiple sequence alignment software version 7. Mol. Biol. Evol. 30, 772–780. Alignment tool used for 89,247 viral genomes across 15 families.", "size": 2 }, { "id": "billera2001", "label": "Billera–Holmes–Vogtmann 2001", "cluster": "phylo", "year": 2001, "authors": "L.J. Billera, S.P. Holmes, K. Vogtmann", "desc": "Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27, 733–767. The tree space is a CAT(0) cubical complex — non-positively curved. Geodesics in tree space are unique. This geometric perspective on phylogenetics connects directly to Gromov hyperbolicity.", "size": 3 }, { "id": "gavryushkin2016", "label": "Gavryushkin & Drummond 2016", "cluster": "phylo", "year": 2016, "authors": "A. Gavryushkin, A.J. Drummond", "desc": "The space of ultrametric phylogenetic trees. J. Theor. Biol. 403, 197–208. Ultrametric trees (molecular clocks) form a particularly well-behaved subset of BHV tree space — these are the trees where radial coordinate = time depth, matching the framework's interpretation.", "size": 2 }, { "id": "nickel2017", "label": "Nickel & Kiela 2017", "cluster": "hyperML", "year": 2017, "authors": "M. Nickel, D. Kiela", "desc": "Poincaré embeddings for learning hierarchical representations. NeurIPS 30, 6338–6347. Demonstrated that hierarchical data embeds efficiently into hyperbolic space. The framework extends this by deriving the *specific curvature* rather than treating it as a tunable hyperparameter.", "size": 4 }, { "id": "sala2018", "label": "Sala et al. 2018", "cluster": "hyperML", "year": 2018, "authors": "F. Sala et al.", "desc": "Representation tradeoffs for hyperbolic embeddings. ICML 2018, 4460–4469. Characterizes the precision-dimension tradeoff in hyperbolic embeddings. The framework shows this tradeoff has a unique optimum at the state equation solution.", "size": 3 }, { "id": "ganea2018", "label": "Ganea et al. 2018", "cluster": "hyperML", "year": 2018, "authors": "O. Ganea et al.", "desc": "Hyperbolic neural networks. NeurIPS 31. Extends standard neural network operations (MLPs, attention) to hyperbolic space using the Möbius gyrovector formalism. Foundation for trainable models that respect hyperbolic geometry.", "size": 3 }, { "id": "chami2019", "label": "Chami et al. 2019", "cluster": "hyperML", "year": 2019, "authors": "I. Chami et al.", "desc": "Hyperbolic graph convolutional neural networks. NeurIPS 32. HGCN: message-passing on graphs in hyperbolic space. Outperforms Euclidean GNNs on hierarchical graphs — consistent with the framework's claim that biological hierarchies require hyperbolic embedding.", "size": 2 }, { "id": "kochurov2020", "label": "Kochurov et al. 2020 (Geoopt)", "cluster": "hyperML", "year": 2020, "authors": "M. Kochurov et al.", "desc": "Geoopt: Riemannian optimization in PyTorch. arXiv:2005.02819. The Geoopt library used by BiosphereCodec for Riemannian optimization on the Poincaré ball. Enables gradient descent on manifolds.", "size": 2 }, { "id": "nguyen2024", "label": "Nguyen et al. 2024 (HyenaDNA)", "cluster": "hyperML", "year": 2024, "authors": "E.P. Nguyen et al.", "desc": "HyenaDNA: long-range genomic sequence modeling at single nucleotide resolution. NeurIPS 37. Hyena operators used in BiosphereCodec's 4-layer encoder-decoder for mapping genomic sequences to Poincaré ball coordinates.", "size": 2 }, { "id": "pearce2025", "label": "Pearce et al. 2025 (Evo 2)", "cluster": "hyperML", "year": 2025, "authors": "M. Pearce et al.", "desc": "Finding the tree of life in Evo 2. Goodfire Research. Independent confirmation: sparse autoencoder applied to Evo 2 (7B DNA LM, 9.3T nucleotides, no evolutionary supervision) found phylogenetic geometry as a curved manifold. Geodesic distances track branch lengths.", "size": 3 }, { "id": "tifrea2019", "label": "Tifrea et al. 2019", "cluster": "hyperML", "year": 2019, "authors": "A. Tifrea et al.", "desc": "Poincaré GloVe: Hyperbolic Word Embeddings. ICLR 2019. Word embeddings in hyperbolic space capture hierarchical relationships (hypernymy, entailment) that flat embeddings miss. Consistent with linguistic hierarchies having intrinsic curvature.", "size": 2 }, { "id": "mathieu2019", "label": "Mathieu et al. 2019", "cluster": "hyperML", "year": 2019, "authors": "E. Mathieu et al.", "desc": "Continuous hierarchical representations with Poincaré variational auto-encoders. NeurIPS 32. Generative models in hyperbolic space — latent space curvature emerges from data hierarchy. Connects to the framework's claim that curvature is intrinsic, not imposed.", "size": 2 }, { "id": "beggs2003", "label": "Beggs & Plenz 2003", "cluster": "neuro", "year": 2003, "authors": "J.M. Beggs, D. Plenz", "desc": "Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167–11177. Discovery of neuronal avalanches with power-law size distributions, suggesting the cortex operates near a critical point. The framework explains this as a *corollary* of κ > 0: the brain is subcritical, not critical.", "size": 4 }, { "id": "steinmetz2019", "label": "Steinmetz et al. 2019", "cluster": "neuro", "year": 2019, "authors": "N.A. Steinmetz et al.", "desc": "Distributed coding of choice, action and engagement across the mouse brain. Nature 576, 266–273. Source of the 39 Neuropixels sessions yielding κ = 0.485, h = 1.04, n = 2.03. The primary neural electrophysiology dataset.", "size": 3 }, { "id": "dimartino2014", "label": "Di Martino et al. 2014 (ABIDE)", "cluster": "neuro", "year": 2014, "authors": "A. Di Martino et al.", "desc": "The ABIDE: towards large-scale evaluation of the intrinsic brain architecture in autism. Mol. Psychiatry 19, 659–667. Source of the fMRI dataset: κ = 0.49 ± 0.06. Bridges single-neuron to whole-brain scale validation.", "size": 2 }, { "id": "schalk2004", "label": "Schalk et al. 2004 (BCI2000)", "cluster": "neuro", "year": 2004, "authors": "G. Schalk et al.", "desc": "BCI2000: a general-purpose brain-computer interface system. IEEE Trans. Biomed. Eng. 51, 1034–1043. Source of the EEGBCI dataset (64-channel EEG, 20 subjects). Yields κ = 0.18 ± 0.03, and the consciousness-state experiment (EO vs EC).", "size": 2 }, { "id": "wilting2019", "label": "Wilting & Priesemann 2019", "cluster": "neuro", "year": 2019, "authors": "J. Wilting, V. Priesemann", "desc": "25 years of criticality in neuroscience — established results, open controversies, novel concepts. Curr. Opin. Neurobiol. 58, 105–111. Comprehensive review: cortex is slightly subcritical (m̂ ≈ 0.98). The framework derives this: 1 − J = κ/(h₀ ln 2)².", "size": 3 }, { "id": "munoz2018", "label": "Muñoz 2018", "cluster": "neuro", "year": 2018, "authors": "M.A. Muñoz", "desc": "Colloquium: Criticality and dynamical scaling in living systems. Rev. Mod. Phys. 90, 031001. Near-criticality across biological scales — from gene regulation to neural circuits. The state equation provides a geometric mechanism for the ubiquity of near-criticality.", "size": 3 }, { "id": "gallego2017", "label": "Gallego et al. 2017", "cluster": "neuro", "year": 2017, "authors": "J.A. Gallego et al.", "desc": "Neural manifolds for the control of movement. Neuron 94, 978–984. Low-dimensional neural manifolds underlie motor control. The framework predicts these manifolds should be hyperbolic (negative curvature) when the neural computation is hierarchical.", "size": 2 }, { "id": "cunningham2014", "label": "Cunningham & Yu 2014", "cluster": "neuro", "year": 2014, "authors": "J.P. Cunningham, B.M. Yu", "desc": "Dimensionality reduction for large-scale neural recordings. Nat. Neurosci. 17, 1500–1509. Standard methods (PCA, factor analysis) assume Euclidean geometry. The framework shows curvature is invisible in Euclidean projections — visible only on SPD manifolds.", "size": 2 }, { "id": "cocchi2017", "label": "Cocchi et al. 2017", "cluster": "neuro", "year": 2017, "authors": "L. Cocchi et al.", "desc": "Criticality in the brain: a synthesis of neurobiology, models and cognition. Prog. Neurobiol. 158, 132–152. Brain criticality across scales — from ion channels to whole-brain dynamics. The framework unifies this: all scales share the same κ (RG invariance).", "size": 2 }, { "id": "radford2019", "label": "Radford et al. 2019 (GPT-2)", "cluster": "neuro", "year": 2019, "authors": "A. Radford et al.", "desc": "Language models are unsupervised multitask learners. OpenAI Blog. GPT-2 as the primary AI baseline: κ = 0.34 ± 0.04, h = 0.84. Below biological κ by ~0.15 — the geometric gap.", "size": 2 }, { "id": "shew2013", "label": "Shew & Plenz 2013", "cluster": "neuro", "year": 2013, "authors": "W.L. Shew, D. Plenz", "desc": "The functional benefits of criticality in the cortex. Neuroscientist 19, 88–100. Benefits of near-critical operation: maximized dynamic range, information transmission, and computational capability. The framework says these are consequences of κ > 0, not independent tuning.", "size": 2 }, { "id": "calvo2026", "label": "Calvo et al. 2026", "cluster": "neuro", "year": 2026, "authors": "R. Calvo, C. Martorell, A. Roig, M.A. Muñoz", "desc": "Robust scaling in human brain dynamics despite correlated inputs and limited sampling distortions. PRL 136, 068402. After correcting for autocorrelated inputs and subsampling artifacts, resting-state fMRI reveals the brain is slightly subcritical — not critical. Near-critical dynamics emerge from reverberant network activity, not fine-tuning. From Muñoz's own group.", "size": 4 }, { "id": "priesemann2014", "label": "Priesemann et al. 2014", "cluster": "neuro", "year": 2014, "authors": "V. Priesemann et al.", "desc": "Spike avalanches in vivo suggest a driven, slightly subcritical brain state. Front. Syst. Neurosci. 8, 108. The title is the criticality corollary. Measured in awake rats: the subcritical distance is stable across behavioral states — consistent with geometric origin rather than homeostatic tuning.", "size": 3 }, { "id": "fosque2021", "label": "Fosque et al. 2021", "cluster": "neuro", "year": 2021, "authors": "L.J. Fosque et al.", "desc": "Evidence for quasicritical brain dynamics. PRL 126, 098101. The brain sits in a Griffiths phase — an extended near-critical region. The framework explains why: κ > 0 forces J < 1, but Lyapunov stability makes κ* an attractor, so the system hovers near the geometric optimum.", "size": 2 }, { "id": "touboul2017", "label": "Touboul & Destexhe 2017", "cluster": "neuro", "year": 2017, "authors": "J. Touboul, A. Destexhe", "desc": "Can power-law scaling and neuronal avalanches arise from stochastic dynamics? Phys. Rev. E 95, 012413. Power-law avalanche distributions can arise in non-critical systems. Undermined the original Beggs & Plenz evidence — the field was looking for criticality when the mechanism produces subcriticality.", "size": 2 }, { "id": "stringer2019", "label": "Stringer et al. 2019", "cluster": "neuro", "year": 2019, "authors": "C. Stringer et al.", "desc": "High-dimensional geometry of population responses in visual cortex. Science 364, eaav7893. 10,000+ simultaneous neurons. Population activity lies on a manifold whose dimensionality scales as a power law. The framework predicts n = 2 for hierarchical processing; effective dimensionality is low and highly structured.", "size": 3 }, { "id": "gallego2017", "label": "Gallego et al. 2017", "cluster": "neuro", "year": 2017, "authors": "J.A. Gallego et al.", "desc": "Neural manifolds for the control of movement. Neuron 94, 978–984. Low-dimensional neural manifolds underlie motor control. Manifold structure preserved across days/conditions. If hyperbolic (as predicted for hierarchical computation), intrinsic dimension should be 2.", "size": 2 }, { "id": "jazayeri2021", "label": "Jazayeri & Ostojic 2021", "cluster": "neuro", "year": 2021, "authors": "M. Jazayeri, S. Ostojic", "desc": "Interpreting neural computations by examining intrinsic and embedding dimensionality. Nat. Neurosci. 24, 1537–1546. Distinguishes intrinsic from embedding dimensionality. Intrinsic dimensionality of neural computation is much lower than ambient — because computation lives on H², regardless of neuron count.", "size": 2 }, { "id": "chaudhuri2019", "label": "Chaudhuri et al. 2019", "cluster": "neuro", "year": 2019, "authors": "R. Chaudhuri et al.", "desc": "The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit. Nat. Neurosci. 22, 1512–1520. Head direction cells form a ring attractor — a 1D manifold. The framework: a ring is a geodesic in H², the simplest submanifold of the 2D hyperbolic space.", "size": 2 }, { "id": "meshulam2019", "label": "Meshulam et al. 2019", "cluster": "neuro", "year": 2019, "authors": "L. Meshulam et al.", "desc": "Coarse graining, fixed points, and scaling in a large population of neurons. PRL 123, 178103. Real-space RG on retinal ganglion cells. Non-trivial fixed point structure under coarse-graining. The framework: κ is the RG-invariant — what survives coarse-graining because it's the geometric fixed point.", "size": 3 }, { "id": "nicoletti2020", "label": "Nicoletti & Bhatt 2020", "cluster": "neuro", "year": 2020, "authors": "G. Nicoletti, M. Bhatt", "desc": "Scaling and renormalization in networks of neurons. PRL 125, 248101. Scaling collapse consistent with proximity to a critical point — but \"proximity to\" is not \"at.\" The framework: the RG flow converges to κ*, which is subcritical.", "size": 2 }, { "id": "allard2020", "label": "Allard & Serrano 2020", "cluster": "neuro", "year": 2020, "authors": "A. Allard, M.Á. Serrano", "desc": "Navigable maps of structural brain networks across species. Nat. Commun. 11, 3745. Mouse, macaque, and human connectomes embedded in H². Hyperbolic embedding captures hierarchical and modular structure better than Euclidean. Navigability is a property of hyperbolic geometry.", "size": 3 }, { "id": "zheng2020", "label": "Zheng et al. 2020", "cluster": "neuro", "year": 2020, "authors": "M. Zheng et al.", "desc": "Geometric renormalization unravels self-similarity of the multiscale human connectome. Sci. Rep. 10, 15928. García-Pérez's geometric RG applied to human connectome: self-similar structure across scales in hyperbolic space. Because κ is RG-invariant.", "size": 2 }, { "id": "barachant2012", "label": "Barachant et al. 2012", "cluster": "infogeom", "year": 2012, "authors": "A. Barachant et al.", "desc": "Multiclass brain-computer interface classification by Riemannian geometry. IEEE Trans. Biomed. Eng. 59, 920–928. SPD covariance matrices with AIRM metric outperform all Euclidean methods for motor imagery EEG. Neural covariance geometry is not flat — the curvature the framework measures.", "size": 3 }, { "id": "congedo2017", "label": "Congedo et al. 2017", "cluster": "infogeom", "year": 2017, "authors": "M. Congedo et al.", "desc": "Riemannian geometry for EEG-based brain-computer interfaces: a primer and a review. IEEE Signal Process. Mag. 34, 18–42. SPD manifold with AIRM is now state-of-the-art for EEG decoding. The field adopted this geometry empirically. The framework gives the curvature a physical interpretation: κ = (h ln 2)².", "size": 3 }, { "id": "yger2017", "label": "Yger et al. 2017", "cluster": "infogeom", "year": 2017, "authors": "F. Yger et al.", "desc": "Riemannian approaches in brain-computer interfaces: a review. NeuroImage 162, 157–170. Confirmed SPD geometry consistently outperforms tangent-space (Euclidean) approximations. Performance gap largest for tasks with rich temporal structure — where geometric signal (κ) is strongest.", "size": 2 }, { "id": "mcgurk1976", "label": "McGurk & MacDonald 1976", "cluster": "conscious", "year": 1976, "authors": "H. McGurk, J. MacDonald", "desc": "Hearing lips and seeing voices. Nature 264, 746–748. The McGurk effect: visual lip movements alter auditory speech perception. Pre-attentive, automatic. Confirms the Visual ↔ Auditory icosahedral edge — structural adjacency producing obligatory cross-modal coupling.", "size": 3 }, { "id": "botvinick1998", "label": "Botvinick & Cohen 1998", "cluster": "conscious", "year": 1998, "authors": "M. Botvinick, J. Cohen", "desc": "Rubber hands \"feel\" touch that eyes see. Nature 391, 756. Synchronous visual-tactile stimulation causes proprioceptive drift. Pre-attentive — suggesting structural adjacency, not learned association. Confirms Somatosensory ↔ Visual icosahedral edge.", "size": 3 }, { "id": "zhou2019", "label": "Zhou et al. 2019", "cluster": "conscious", "year": 2019, "authors": "G. Zhou et al.", "desc": "Piriform cortex functional connectivity. Direct piriform-to-auditory cortex projections — the least expected icosahedral edge. Olfaction and audition seem unrelated, but the icosahedral topology requires adjacency on the equatorial ring. Confirmed by tract-tracing. The non-obvious prediction.", "size": 3 }, { "id": "vogler2024", "label": "Vogler & Bhatt 2024", "cluster": "conscious", "year": 2024, "authors": "N. Vogler, M. Bhatt", "desc": "Olfactory-auditory cross-modal processing. Independent confirmation of piriform-to-auditory projections (Zhou 2019). The Auditory ↔ Olfactory edge is now confirmed by two independent groups — this was the non-trivial prediction of the icosahedral topology.", "size": 2 }, { "id": "small2005", "label": "Small & Prescott 2005", "cluster": "conscious", "year": 2005, "authors": "D.M. Small, J. Prescott", "desc": "Odor/taste integration and the perception of flavor. Curr. Biol. 15, 1–14. Flavor perception requires binding of orthonasal/retronasal olfaction with gustatory input via OFC. Confirms Olfactory ↔ Gustatory icosahedral edge — structural adjacency for flavor.", "size": 2 }, { "id": "verhagen2006", "label": "Verhagen & Engelen 2006", "cluster": "conscious", "year": 2006, "authors": "J.V. Verhagen, L. Engelen", "desc": "The role of texture in oral processing. J. Texture Stud. 37, 221–240. Texture perception integrates gustatory (chemical) and somatosensory (mechanical) signals in the insula. Confirms Gustatory ↔ Somatosensory icosahedral edge.", "size": 2 }, { "id": "amari2016", "label": "Amari 2016", "cluster": "infogeom", "year": 2016, "authors": "S. Amari", "desc": "Information Geometry and Its Applications. Springer. The foundational text. Statistical manifolds, Fisher information metric, dual connections. The SPD manifold with affine-invariant metric used to detect κ in neural data is a special case of Amari's framework.", "size": 3 }, { "id": "pennec2006", "label": "Pennec et al. 2006", "cluster": "infogeom", "year": 2006, "authors": "X. Pennec et al.", "desc": "A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66, 41–66. Log-Euclidean and affine-invariant metrics on SPD matrices. The AIRM metric d(P,Q) = ‖log(P⁻¹/²QP⁻¹/²)‖_F is what makes the geometric signal visible in neural covariance data.", "size": 3 }, { "id": "bhatia2007", "label": "Bhatia 2007", "cluster": "infogeom", "year": 2007, "authors": "R. Bhatia", "desc": "Positive Definite Matrices. Princeton University Press. Mathematical theory of SPD manifolds. The manifold SPD(n) with the AIRM metric has non-positive curvature — a natural setting for detecting the hyperbolic geometry of neural covariance dynamics.", "size": 2 }, { "id": "ay2017", "label": "Ay et al. 2017", "cluster": "infogeom", "year": 2017, "authors": "N. Ay et al.", "desc": "Information Geometry. Springer. Comprehensive treatment connecting statistical inference, differential geometry, and neural networks. The \"geometry of information\" is precisely what the state equation quantifies.", "size": 2 }, { "id": "tononi2016", "label": "Tononi et al. 2016 (IIT)", "cluster": "conscious", "year": 2016, "authors": "G. Tononi et al.", "desc": "Integrated information theory: from consciousness to its physical substrate. Nat. Rev. Neurosci. 17, 450–461. IIT's Φ measures integrated information. The framework offers a complementary geometric quantity: κ tracks consciousness state (EO > EC, p < 0.001).", "size": 3 }, { "id": "friston2010", "label": "Friston 2010", "cluster": "conscious", "year": 2010, "authors": "K. Friston", "desc": "The free-energy principle: a unified brain theory? Nat. Rev. Neurosci. 11, 127–138. Both the free-energy principle and the state equation are information-theoretic accounts of brain organization. The state equation operates on covariance geometry rather than prediction error.", "size": 3 }, { "id": "baars1988", "label": "Baars 1988", "cluster": "conscious", "year": 1988, "authors": "B.J. Baars", "desc": "A Cognitive Theory of Consciousness. Cambridge University Press. Global Workspace Theory: consciousness as a broadcasting architecture. The icosahedral partition at κ* provides a geometric substrate for the \"workspace\" — 12 maximally separated processing regions.", "size": 2 }, { "id": "cohn_kumar2007", "label": "Cohn & Kumar 2007", "cluster": "conscious", "year": 2007, "authors": "H. Cohn, A. Kumar", "desc": "Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20, 99–148. Proves the 12-point icosahedral configuration is universally optimal on S². Combined with the state equation: κ* → S² boundary → 12 optimal regions → matches dvādasāyatana.", "size": 3 }, { "id": "bodhi2000", "label": "Bodhi 2000", "cluster": "conscious", "year": 2000, "authors": "Bhikkhu Bodhi", "desc": "A Comprehensive Manual of Abhidhamma. BPS Pariyatti Editions. The dvādasāyatana (twelve sense bases): six sense organs and six corresponding objects. Claimed as an exhaustive, irreducible partition of conscious experience. The icosahedral geometry at κ* predicts exactly twelve.", "size": 2 }, { "id": "nagel1974", "label": "Nagel 1974", "cluster": "conscious", "year": 1974, "authors": "T. Nagel", "desc": "What is it like to be a bat? Philosophical Review 83, 435–450. The hard problem of consciousness: subjective experience cannot be reduced to physical description. The framework's icosahedral atlas doesn't solve this but provides a geometric structure for the space of qualia.", "size": 2 }, { "id": "dehaene2001", "label": "Dehaene & Naccache 2001", "cluster": "conscious", "year": 2001, "authors": "S. Dehaene, L. Naccache", "desc": "Towards a cognitive neuroscience of consciousness. Cognition 79, 1–37. Neural correlates of consciousness: global ignition, recurrent processing, workspace broadcasting. The framework's n > 2 for recurrent thalamic circuits is consistent with this.", "size": 2 }, { "id": "pagel2007", "label": "Pagel et al. 2007", "cluster": "langevo", "year": 2007, "authors": "M. Pagel et al.", "desc": "Frequency of word-use predicts rates of lexical evolution throughout Indo-European history. Nature 449, 717–720. Word frequency predicts substitution rate — analogous to how site conservation predicts mutation rate in DNA. Same entropy-rate logic.", "size": 3 }, { "id": "gray2003", "label": "Gray & Atkinson 2003", "cluster": "langevo", "year": 2003, "authors": "R.D. Gray, Q.D. Atkinson", "desc": "Language-tree divergence times support the Anatolian theory of Indo-European origin. Nature 426, 435–439. Bayesian phylogenetics applied to languages — the same tree topology (Postulate 2) that applies to genomes.", "size": 3 }, { "id": "greenhill2010", "label": "Greenhill et al. 2010", "cluster": "langevo", "year": 2010, "authors": "S.J. Greenhill et al.", "desc": "The shape and tempo of language evolution. Proc. R. Soc. B 277, 2443–2450. Punctuational bursts in language evolution — analogous to adaptive radiation in biology. Both are perturbations around the geometric attractor κ*.", "size": 2 }, { "id": "dunn2011", "label": "Dunn et al. 2011", "cluster": "langevo", "year": 2011, "authors": "M. Dunn et al.", "desc": "Evolved structure of language shows lineage-specific trends in word-order universals. Nature 473, 79–82. Language structure evolves along lineage-specific paths — hierarchical descent with modification, exactly Postulate 2.", "size": 2 }, { "id": "index_diachronica", "label": "Index Diachronica", "cluster": "langevo", "year": 2023, "authors": "Community resource", "desc": "16,496 sound change rules across 34 language families. Primary data source for phonemic transition entropy h = 1.653 bits. The effective alphabet ~3 targets per phoneme, matching DNA's ~3 transitions per base.", "size": 3 }, { "id": "asjp", "label": "ASJP Database", "cluster": "langevo", "year": 2022, "authors": "Wichmann et al.", "desc": "Automated Similarity Judgment Program: 106K language pairs across 955 languages. Cross-entropy excess slope yields h = 1.568 bits — an independent estimate 10% below Index Diachronica due to compression.", "size": 2 }, { "id": "le_gascuel2008", "label": "Le & Gascuel 2008 (LG)", "cluster": "protevo", "year": 2008, "authors": "S.Q. Le, O. Gascuel", "desc": "An improved general amino acid replacement matrix. Mol. Biol. Evol. 25, 1307–1320. The LG substitution model: h_protein ≈ 2.85 bits, accounting for physicochemical grouping, codon degeneracy, and purifying selection. Predicts κ_protein ≈ 3.90.", "size": 3 }, { "id": "henikoff1992", "label": "Henikoff & Henikoff 1992", "cluster": "protevo", "year": 1992, "authors": "S. Henikoff, J.G. Henikoff", "desc": "Amino acid substitution matrices from protein blocks. PNAS 89, 10915–10919. BLOSUM62: the empirical substitution matrix that defines the effective protein alphabet (~7 likely replacements per residue). Foundation for h_protein estimation.", "size": 3 }, { "id": "dayhoff1978", "label": "Dayhoff et al. 1978", "cluster": "protevo", "year": 1978, "authors": "M.O. Dayhoff et al.", "desc": "A model of evolutionary change in proteins. Atlas of Protein Sequence and Structure 5, 345–352. The original PAM matrices — empirical substitution rates from aligned protein families. First quantification of the effective protein alphabet.", "size": 2 }, { "id": "mirny1999", "label": "Mirny & Shakhnovich 1999", "cluster": "protevo", "year": 1999, "authors": "L.A. Mirny, E.I. Shakhnovich", "desc": "Universally conserved positions in protein folds. PNAS 96, 4459–4464. Structural constraints on protein evolution — certain positions are universally conserved across unrelated folds. These constraints reduce the effective entropy rate, lowering κ for conserved families (e.g. RecA: κ = 0.89).", "size": 2 }, { "id": "lean4", "label": "de Moura & Ullrich 2021 (Lean 4)", "cluster": "formal", "year": 2021, "authors": "L. de Moura, S. Ullrich", "desc": "The Lean 4 theorem prover and programming language. CADE 28, LNCS 12699, 625–635. The proof assistant used to formally verify all 9 theorems of the state equation — from postulates to Lyapunov stability — with 0 sorry stubs.", "size": 3 }, { "id": "mathlib", "label": "Mathlib", "cluster": "formal", "year": 2020, "authors": "mathlib community", "desc": "The Lean mathematical library. A unified library of mathematics formalized in Lean 4. Contains the real analysis, topology, and measure theory used by the state equation proofs. Version: Mathlib 2024.03+.", "size": 2 }, { "id": "hales2017", "label": "Hales et al. 2017 (Kepler)", "cluster": "formal", "year": 2017, "authors": "T.C. Hales et al.", "desc": "A formal proof of the Kepler conjecture. Forum of Mathematics, Pi 5. Precedent for formal verification of geometric optimality results. The icosahedral optimality proof follows the same philosophy: machine-check geometric claims.", "size": 2 }, { "id": "krioukov2010", "label": "Krioukov et al. 2010", "cluster": "networks", "year": 2010, "authors": "D. Krioukov et al.", "desc": "Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106. Networks in hyperbolic space naturally exhibit small-world topology (high clustering + short paths). The framework: cortical small-world structure requires no special wiring — it's automatic in H².", "size": 3 }, { "id": "boguna2010", "label": "Boguñá et al. 2010", "cluster": "networks", "year": 2010, "authors": "M. Boguñá et al.", "desc": "Sustaining the Internet with hyperbolic mapping. Nat. Commun. 1, 62. Internet routing efficiency explained by hyperbolic geometry — another information-processing hierarchy that embeds into H². Same geometric principle, different substrate.", "size": 2 }, { "id": "garcia_perez2018", "label": "García-Pérez et al. 2018", "cluster": "networks", "year": 2018, "authors": "G. García-Pérez et al.", "desc": "Multiscale unfolding of real networks by geometric renormalization. Nat. Phys. 14, 583–589. Geometric RG on hyperbolic networks — coarse-graining preserves the hidden metric space. Connects to the framework's RG invariance of κ.", "size": 3 }, { "id": "papadopoulos2012", "label": "Papadopoulos et al. 2012", "cluster": "networks", "year": 2012, "authors": "F. Papadopoulos et al.", "desc": "Popularity versus similarity in growing networks. Nature 489, 537–540. Network growth in hyperbolic space: radial coordinate = popularity (time), angular = similarity. Directly parallels the framework's r = phylogenetic depth, θ = phenotypic direction.", "size": 3 }, { "id": "wilson1971", "label": "Wilson 1971", "cluster": "dynamical", "year": 1971, "authors": "K.G. Wilson", "desc": "Renormalization group and critical phenomena. Phys. Rev. B 4, 3174–3183. The RG framework for phase transitions. The state equation's Lyapunov stability is an RG fixed point — κ* is the IR attractor of the evolutionary flow.", "size": 3 }, { "id": "kadanoff1966", "label": "Kadanoff 1966", "cluster": "dynamical", "year": 1966, "authors": "L.P. Kadanoff", "desc": "Scaling laws for Ising models near T_c. Physics 2, 263–272. Block spin renormalization — the original insight that coarse-graining preserves universal quantities. The framework's RG invariance of κ is the biological analogue.", "size": 2 }, { "id": "mehta2014", "label": "Mehta & Schwab 2014", "cluster": "dynamical", "year": 2014, "authors": "P. Mehta, D.J. Schwab", "desc": "An exact mapping between the Variational Renormalization Group and Deep Learning. arXiv:1410.3831. RG as deep learning: each layer performs a coarse-graining step. The geometric gap between biological and artificial κ may reflect different RG flow depths.", "size": 2 }, { "id": "gould1989", "label": "Gould 1989", "cluster": "dynamical", "year": 1989, "authors": "S.J. Gould", "desc": "Wonderful Life: The Burgess Shale and the Nature of History. W.W. Norton. \"If the tape of life were rewound…\" — the contingency argument. The framework's response: the molecular actors might change, but any hierarchy with h > 0 and tree topology must inhabit the same κ*.", "size": 2 }, { "id": "churchland2012", "label": "Churchland et al. 2012", "cluster": "neuro", "year": 2012, "authors": "M.M. Churchland et al.", "desc": "Neural population dynamics during reaching. Nature 487, 51–56. Launched the neural manifold field. Population activity during reaching lies on a low-dimensional dynamical manifold. The framework predicts this manifold is hyperbolic when computation is hierarchical.", "size": 3 }, { "id": "vyas2020", "label": "Vyas et al. 2020", "cluster": "neuro", "year": 2020, "authors": "S. Vyas et al.", "desc": "Computation through neural population dynamics. Annu. Rev. Neurosci. 43, 249–275. Comprehensive review: neural manifolds as computational substrate. The framework gives these manifolds a specific geometry — H² with κ determined by h — rather than treating them as generic low-dimensional objects.", "size": 3 }, { "id": "sussillo2013", "label": "Sussillo & Barak 2013", "cluster": "neuro", "year": 2013, "authors": "D. Sussillo, O. Barak", "desc": "Opening the black box: low-dimensional dynamics in high-dimensional RNNs. Neural Comput. 25, 626–649. RNN dynamics are low-dimensional because computation lives on an attractor manifold. The framework: that manifold has curvature κ = (h ln 2)².", "size": 2 }, { "id": "mastrogiuseppe2018", "label": "Mastrogiuseppe & Ostojic 2018", "cluster": "neuro", "year": 2018, "authors": "F. Mastrogiuseppe, S. Ostojic", "desc": "Linking connectivity, dynamics, and computations in low-rank RNNs. Neuron 99, 609–623. Low-rank structure produces low-dimensional manifolds. The rank constraint is the network analogue of Postulate 2 (hierarchical topology).", "size": 2 }, { "id": "kaufman2014", "label": "Kaufman et al. 2014", "cluster": "neuro", "year": 2014, "authors": "M.T. Kaufman et al.", "desc": "Cortical activity in the null space: permitting preparation without movement. Nat. Neurosci. 17, 440–448. Neural populations use orthogonal subspaces for different computations. These subspaces are geodesic submanifolds of the hyperbolic space.", "size": 2 }, { "id": "watts1998", "label": "Watts & Strogatz 1998", "cluster": "networks", "year": 1998, "authors": "D.J. Watts, S.H. Strogatz", "desc": "Collective dynamics of \"small-world\" networks. Nature 393, 440–442. High clustering + short paths. Krioukov et al. showed this emerges automatically from hyperbolic geometry — not from special wiring but from the curvature of the embedding space.", "size": 3 }, { "id": "barabasi1999", "label": "Barabási & Albert 1999", "cluster": "networks", "year": 1999, "authors": "A.-L. Barabási, R. Albert", "desc": "Emergence of scaling in random networks. Science 286, 509–512. Scale-free degree distributions. Papadopoulos et al. showed this equals radial growth in H² — popularity = radial depth, exactly the framework's r coordinate.", "size": 3 }, { "id": "thurston1982", "label": "Thurston 1982", "cluster": "riemannian", "year": 1982, "authors": "W.P. Thurston", "desc": "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. AMS 6, 357–381. Geometrization: hyperbolic geometry is the \"generic\" geometry of 3-manifolds — the default when no special structure is imposed. Why negative curvature is not a choice but a consequence.", "size": 3 }, { "id": "bronstein2017", "label": "Bronstein et al. 2017", "cluster": "hyperML", "year": 2017, "authors": "M.M. Bronstein et al.", "desc": "Geometric deep learning: going beyond Euclidean data. IEEE Signal Process. Mag. 34, 18–42. The manifesto for geometric deep learning. The broader movement — but with a derived curvature rather than a learned one.", "size": 3 }, { "id": "coxeter1973", "label": "Coxeter 1973", "cluster": "riemannian", "year": 1973, "authors": "H.S.M. Coxeter", "desc": "Regular Polytopes, 3rd ed. Dover. The icosahedron has the highest symmetry group (order 120) and the tightest sphere packing among Platonic solids — why Cohn-Kumar optimality picks exactly this polytope for 12 points on S².", "size": 2 }, { "id": "luppi2021", "label": "Luppi et al. 2021", "cluster": "neuro", "year": 2021, "authors": "A.I. Luppi et al.", "desc": "Consciousness-specific dynamic interactions of brain functional connectivity. NeuroImage 236, 118201. Consciousness states have distinct connectivity dynamics. The framework: consciousness modulates κ (EO > EC, p < 0.001).", "size": 2 }, { "id": "koch2016", "label": "Koch et al. 2016", "cluster": "conscious", "year": 2016, "authors": "C. Koch et al.", "desc": "Neural correlates of consciousness: progress and problems. Nat. Rev. Neurosci. 17, 307–321. The framework offers a geometric NCC: κ as a measurable, substrate-independent correlate of consciousness.", "size": 2 }, { "id": "song2026", "label": "Song et al. 2026", "cluster": "neuro", "year": 2026, "authors": "Y. Song, J. Chen, V.D. Calhoun, A. Iraji", "desc": "Cognition emerges from phase dynamics of intrinsic coordination. bioRxiv 2026.03.26.714488. The brain has a fixed geometric scaffold (intrinsic network flow) stable across cognitive states and individuals. Flexible cognition comes from phase alignment modulation on this fixed structure — not reconfiguration. The control variable is when signals align, not where or how much. Independent confirmation that the manifold is fixed (κ is geometric) and dynamics run on top of it.", "size": 3 }, { "id": "wu2025", "label": "Wu et al. 2025", "cluster": "neuro", "year": 2025, "authors": "Y. Wu, W. Guo, Z. Liu et al.", "desc": "How large language models encode theory-of-mind: a study on sparse parameter patterns. Nat. series s44387-025-00031-9. ToM in LLMs encoded in 0.001% of parameters, linked to positional encoding geometry (RoPE angles). Hierarchical reasoning depends on angular structure of embeddings — disrupting geometry disrupts hierarchy. Consistent with n=2: effective manifold is very low-dimensional within high-dimensional parameter space.", "size": 2 } ], "edges": [ { "s": "state_eq", "t": "n2_univ", "label": "Theorem 1: n=2 for bifurcating hierarchies", "str": 3 }, { "s": "state_eq", "t": "viral_r", "label": "Predicted vs measured κ across 15 families", "str": 3 }, { "s": "state_eq", "t": "protein_jump", "label": "h_protein ≈ 2.85 → κ ≈ 3.90 predicted", "str": 3 }, { "s": "state_eq", "t": "neural_kappa", "label": "Same equation, neural substrate", "str": 3 }, { "s": "state_eq", "t": "criticality", "label": "κ > 0 ⟹ J < 1 (one-line derivation)", "str": 3 }, { "s": "state_eq", "t": "alphabet_conv", "label": "Effective alphabet → h → κ", "str": 3 }, { "s": "state_eq", "t": "lean_proofs", "label": "All theorems machine-verified", "str": 3 }, { "s": "state_eq", "t": "ling_kappa", "label": "h_phoneme ≈ h_DNA → same κ", "str": 3 }, { "s": "state_eq", "t": "geo_gap", "label": "Bio vs AI: different h → different κ", "str": 2 }, { "s": "state_eq", "t": "icosahedral", "label": "κ* → S² boundary → 12 optimal regions", "str": 2 }, { "s": "state_eq", "t": "convergence_5seed", "label": "Empirical convergence at κ ≈ 1.28–1.34", "str": 3 }, { "s": "state_eq", "t": "rg_invariance", "label": "κ stable under coarse-graining", "str": 2 }, { "s": "n2_univ", "t": "neural_kappa", "label": "n = 2.03 ± 0.36 in spike data", "str": 3 }, { "s": "n2_univ", "t": "protein_jump", "label": "n = 2.03 ± 0.10 for proteins", "str": 3 }, { "s": "viral_r", "t": "convergence_5seed", "label": "Both validate the curve", "str": 2 }, { "s": "criticality", "t": "neural_kappa", "label": "1−J = κ/(h₀ ln 2)² confirmed", "str": 3 }, { "s": "alphabet_conv", "t": "ling_kappa", "label": "~3 effective → h ≈ 1.58 → κ ≈ 1.2", "str": 3 }, { "s": "icosahedral", "t": "neural_kappa", "label": "12 vertices partition neural manifold", "str": 2 }, { "s": "paper1", "t": "state_eq", "label": "Establishes the equation empirically", "str": 3 }, { "s": "paper1", "t": "viral_r", "label": "15-family viral sweep", "str": 3 }, { "s": "paper1", "t": "protein_jump", "label": "15 Pfam families", "str": 3 }, { "s": "paper1", "t": "convergence_5seed", "label": "5,550 genomes, 5 seeds", "str": 3 }, { "s": "paper2", "t": "state_eq", "label": "Proves existence, uniqueness, stability", "str": 3 }, { "s": "paper2", "t": "lean_proofs", "label": "524 lines Lean 4", "str": 3 }, { "s": "paper3", "t": "neural_kappa", "label": "39 Neuropixels + fMRI + EEG", "str": 3 }, { "s": "paper3", "t": "criticality", "label": "Criticality as corollary", "str": 3 }, { "s": "paper3", "t": "geo_gap", "label": "6 AI architectures", "str": 3 }, { "s": "paper3", "t": "icosahedral", "label": "Icosahedral prediction", "str": 2 }, { "s": "manning1979", "t": "state_eq", "label": "h_top = (n−1)√κ: the mathematical bridge", "str": 3 }, { "s": "manning1979", "t": "neural_kappa", "label": "Volume entropy = Manning's h_top", "str": 2 }, { "s": "manning1979", "t": "besson1995", "label": 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"kimura1968", "label": "Non-adaptive forces shape h", "str": 1 }, { "s": "adami2002", "t": "shannon1948", "label": "Physical complexity = mutual information", "str": 1 }, { "s": "koonin2011", "t": "n2_univ", "label": "HGT as perturbation: n > 2", "str": 2 }, { "s": "sarkar2012", "t": "n2_univ", "label": "Trees embed into H² with 1+O(ε) distortion", "str": 3 }, { "s": "sarkar2012", "t": "state_eq", "label": "n=2 sufficiency for tree embedding", "str": 3 }, { "s": "parks2022", "t": "paper1", "label": "GTDB r220: 107K bacterial + 5.9K archaeal tips", "str": 2 }, { "s": "li2021", "t": "paper1", "label": "1,610 fungal tips, κ = 3.0", "str": 2 }, { "s": "katoh2013", "t": "viral_r", "label": "MAFFT alignment of 89K viral genomes", "str": 1 }, { "s": "billera2001", "t": "sarkar2012", "label": "CAT(0) tree space → H² embedding", "str": 2 }, { "s": "gavryushkin2016", "t": "billera2001", "label": "Ultrametric subset of BHV space", "str": 1 }, { "s": "nickel2017", "t": "state_eq", "label": "Hyperbolic embeddings for hierarchies (κ as free param → derived)", "str": 3 }, { "s": "nickel2017", "t": "sarkar2012", "label": "Practical realization of Sarkar theory", "str": 2 }, { "s": "sala2018", "t": "nickel2017", "label": "Tradeoff analysis", "str": 2 }, { "s": "sala2018", "t": "state_eq", "label": "Unique optimum at κ*", "str": 2 }, { "s": "ganea2018", "t": "nickel2017", "label": "Neural networks in H^n", "str": 2 }, { "s": "chami2019", "t": "ganea2018", "label": "GNNs in hyperbolic space", "str": 2 }, { "s": "chami2019", "t": "krioukov2010", "label": "Graph structure in H²", "str": 2 }, { "s": "kochurov2020", "t": "paper1", "label": "Geoopt: Riemannian optimization", "str": 1 }, { "s": "nguyen2024", "t": "paper1", "label": "Hyena operators in BiosphereCodec", "str": 1 }, { "s": "pearce2025", "t": "paper1", "label": "Independent: Evo 2 finds phylogenetic geometry", "str": 3 }, { "s": "pearce2025", "t": "n2_univ", "label": "Curved manifold in 7B DNA LM", "str": 2 }, { "s": "tifrea2019", "t": "nickel2017", "label": "Word embeddings in H^n", "str": 1 }, { "s": "tifrea2019", "t": "ling_kappa", "label": "Linguistic hierarchy in Poincaré space", "str": 2 }, { "s": "mathieu2019", "t": "nickel2017", "label": "Generative models in H^n", "str": 1 }, { "s": "mathieu2019", "t": "state_eq", "label": "Curvature emerges from data hierarchy", "str": 2 }, { "s": "beggs2003", "t": "criticality", "label": "Framework derives subcriticality, not criticality", "str": 3 }, { "s": "beggs2003", "t": "shew2013", "label": "Functional benefits of near-criticality", "str": 2 }, { "s": "steinmetz2019", "t": "neural_kappa", "label": "Source: 39 Neuropixels sessions", "str": 3 }, { "s": "dimartino2014", "t": "paper3", "label": "Source: ABIDE fMRI, κ = 0.49", "str": 2 }, { "s": "schalk2004", "t": "paper3", "label": "Source: EEGBCI, κ = 0.18", "str": 2 }, { "s": "wilting2019", "t": "criticality", "label": "Subcriticality confirmed: m̂ ≈ 0.98", "str": 3 }, { "s": "munoz2018", "t": "criticality", "label": "Near-criticality across biological scales", "str": 2 }, { "s": "munoz2018", "t": "wilson1971", "label": "RG theory of critical phenomena", "str": 2 }, { "s": "gallego2017", "t": "neural_kappa", "label": "Neural manifolds should be hyperbolic", "str": 2 }, { "s": "cunningham2014", "t": "paper3", "label": "Euclidean methods miss curvature", "str": 2 }, { "s": "cocchi2017", "t": "criticality", "label": "Multi-scale criticality → RG invariance of κ", "str": 2 }, { "s": "cocchi2017", "t": "rg_invariance", "label": "Scale-free criticality", "str": 2 }, { "s": "radford2019", "t": "geo_gap", "label": "GPT-2: κ = 0.34 (AI baseline)", "str": 2 }, { "s": "shew2013", "t": "criticality", "label": "Benefits of near-criticality = consequences of κ > 0", "str": 2 }, { "s": "amari2016", "t": "paper3", "label": "SPD manifolds with Fisher metric", "str": 2 }, { "s": "pennec2006", "t": "neural_kappa", "label": "AIRM metric reveals geometric signal", "str": 3 }, { "s": "pennec2006", "t": "amari2016", "label": "Riemannian SPD framework", "str": 2 }, { "s": "bhatia2007", "t": "pennec2006", "label": "Mathematical theory of SPD manifolds", "str": 2 }, { "s": "ay2017", "t": "amari2016", "label": "Comprehensive information geometry", "str": 1 }, { "s": "tononi2016", "t": "paper3", "label": "IIT as theoretical backdrop", "str": 2 }, { "s": "tononi2016", "t": "icosahedral", "label": "Φ vs κ as consciousness measure", "str": 2 }, { "s": "friston2010", "t": "paper3", "label": "Free energy vs geometric state equation", "str": 2 }, { "s": "friston2010", "t": "state_eq", "label": "Both: information-theoretic brain organization", "str": 2 }, { "s": "baars1988", "t": "icosahedral", "label": "Global workspace ↔ 12-region partition", "str": 2 }, { "s": "cohn_kumar2007", "t": "icosahedral", "label": "12-point optimality on S² (proved)", "str": 3 }, { "s": "cohn_kumar2007", "t": "lean_proofs", "label": "Optimality result formalized", "str": 2 }, { "s": "bodhi2000", 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pairs → h = 1.568", "str": 2 }, { "s": "le_gascuel2008", "t": "protein_jump", "label": "LG model: h_protein ≈ 2.85", "str": 3 }, { "s": "henikoff1992", "t": "alphabet_conv", "label": "BLOSUM62: ~7 effective replacements", "str": 3 }, { "s": "henikoff1992", "t": "le_gascuel2008", "label": "Empirical basis for LG model", "str": 2 }, { "s": "dayhoff1978", "t": "henikoff1992", "label": "PAM → BLOSUM evolution", "str": 1 }, { "s": "mirny1999", "t": "protein_jump", "label": "Structural constraints → reduced h → lower κ", "str": 2 }, { "s": "lean4", "t": "lean_proofs", "label": "Verification engine", "str": 3 }, { "s": "mathlib", "t": "lean_proofs", "label": "Real analysis library", "str": 2 }, { "s": "hales2017", "t": "lean_proofs", "label": "Precedent: formal geometric proofs", "str": 1 }, { "s": "hales2017", "t": "cohn_kumar2007", "label": "Both: sphere packing optimality", "str": 2 }, { "s": "krioukov2010", "t": "state_eq", "label": "H² ↔ small-world networks", "str": 2 }, { "s": "krioukov2010", "t": "n2_univ", "label": "H² is unique for tree-like networks", "str": 2 }, { "s": "boguna2010", "t": "krioukov2010", "label": "Internet as hyperbolic hierarchy", "str": 2 }, { "s": "garcia_perez2018", "t": "rg_invariance", "label": "Geometric RG preserves hidden metric", "str": 3 }, { "s": "garcia_perez2018", "t": "krioukov2010", "label": "Multiscale unfolding", "str": 2 }, { "s": "papadopoulos2012", "t": "paper1", "label": "r = time depth, θ = similarity", "str": 2 }, { "s": "papadopoulos2012", "t": "krioukov2010", "label": "Growth dynamics in H²", "str": 2 }, { "s": "wilson1971", "t": "state_eq", "label": "κ* as RG fixed point", "str": 2 }, { "s": "wilson1971", "t": "rg_invariance", "label": "Universal quantities under RG flow", "str": 2 }, { "s": "kadanoff1966", "t": "wilson1971", "label": "Block spin → RG", "str": 2 }, { "s": "kadanoff1966", "t": "rg_invariance", "label": "Coarse-graining preserves universals", "str": 1 }, { "s": "mehta2014", "t": "geo_gap", "label": "RG depth ↔ geometric gap?", "str": 1 }, { "s": "mehta2014", "t": "wilson1971", "label": "Deep learning as RG", "str": 2 }, { "s": "gould1989", "t": "state_eq", "label": "Contingency vs geometric necessity", "str": 2 }, { "s": "calvo2026", "t": "criticality", "label": "Subcritical confirmed after correcting artifacts", "str": 3 }, { "s": "calvo2026", "t": "munoz2018", "label": "Muñoz's own group resolves the debate", "str": 2 }, { "s": "calvo2026", "t": "beggs2003", "label": "Corrects 20 years of criticality interpretation", "str": 2 }, { "s": "calvo2026", "t": "wilting2019", "label": "Confirms subcriticality with robust framework", "str": 2 }, { "s": "priesemann2014", "t": "criticality", "label": "Subcritical distance stable across behavioral states", "str": 3 }, { "s": "priesemann2014", "t": "beggs2003", "label": "In vivo: subcritical, not critical", "str": 2 }, { "s": "fosque2021", "t": "criticality", "label": "Griffiths phase = near κ* attractor", "str": 2 }, { "s": "touboul2017", "t": "beggs2003", "label": "Power laws can arise without criticality", "str": 2 }, { "s": "touboul2017", "t": "criticality", "label": "Removes the need for critical-point tuning", "str": 2 }, { "s": "stringer2019", "t": "n2_univ", "label": "Low-dimensional manifold in 10K+ neurons", "str": 2 }, { "s": "stringer2019", "t": "neural_kappa", "label": "Population geometry is structured, not random", "str": 2 }, { "s": "gallego2017", "t": "neural_kappa", "label": "Motor manifolds should be hyperbolic if hierarchical", "str": 2 }, { "s": "jazayeri2021", "t": "n2_univ", "label": "Intrinsic dim << embedding dim", "str": 2 }, { "s": "chaudhuri2019", "t": "n2_univ", "label": "Ring attractor = geodesic in H²", "str": 2 }, { "s": "meshulam2019", "t": "rg_invariance", "label": "RG fixed points in neural populations", "str": 3 }, { "s": "meshulam2019", "t": "wilson1971", "label": "Real-space RG on neurons", "str": 2 }, { "s": "nicoletti2020", "t": "rg_invariance", "label": "Scaling 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"barachant2012", "label": "Comprehensive review of Riemannian BCI", "str": 2 }, { "s": "yger2017", "t": "barachant2012", "label": "Confirms SPD > Euclidean consistently", "str": 2 }, { "s": "yger2017", "t": "neural_kappa", "label": "Performance gap = curvature signal", "str": 2 }, { "s": "mcgurk1976", "t": "icosahedral", "label": "Confirms Visual ↔ Auditory edge", "str": 3 }, { "s": "botvinick1998", "t": "icosahedral", "label": "Confirms Somatosensory ↔ Visual edge", "str": 3 }, { "s": "zhou2019", "t": "icosahedral", "label": "Confirms Auditory ↔ Olfactory edge (non-obvious)", "str": 3 }, { "s": "vogler2024", "t": "zhou2019", "label": "Independent replication", "str": 2 }, { "s": "vogler2024", "t": "icosahedral", "label": "Second confirmation of Auditory ↔ Olfactory", "str": 2 }, { "s": "small2005", "t": "icosahedral", "label": "Confirms Olfactory ↔ Gustatory edge (flavor)", "str": 2 }, { "s": "verhagen2006", "t": "icosahedral", "label": "Confirms Gustatory ↔ Somatosensory edge", "str": 2 }, { "s": "mcgurk1976", "t": "botvinick1998", "label": "Both: pre-attentive cross-modal binding", "str": 1 }, { "s": "churchland2012", "t": "neural_kappa", "label": "Launched neural manifold field", "str": 2 }, { "s": "churchland2012", "t": "n2_univ", "label": "Low-dim manifold in high-dim population", "str": 2 }, { "s": "churchland2012", "t": "gallego2017", "label": "Motor manifolds build on this", "str": 2 }, { "s": "vyas2020", "t": "churchland2012", "label": "Comprehensive review of the field", "str": 2 }, { "s": "vyas2020", "t": "neural_kappa", "label": "Manifolds as computational substrate", "str": 2 }, { "s": "vyas2020", "t": "gallego2017", "label": "Reviews motor manifold results", "str": 1 }, { "s": "sussillo2013", "t": "n2_univ", "label": "RNN attractors are low-dimensional", "str": 2 }, { "s": "sussillo2013", "t": "geo_gap", "label": "Artificial RNN manifolds vs biological", "str": 1 }, { "s": "mastrogiuseppe2018", "t": "sussillo2013", "label": "Theory for low-rank 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"neural_kappa", "label": "Fixed geometric scaffold = manifold at κ*", "str": 3 }, { "s": "song2026", "t": "rg_invariance", "label": "Scaffold stable across states = κ is RG-invariant", "str": 2 }, { "s": "song2026", "t": "criticality", "label": "Phase modulation on fixed geometry, not tuning", "str": 2 }, { "s": "song2026", "t": "paper3", "label": "Independent: fixed manifold with dynamics on top", "str": 3 }, { "s": "song2026", "t": "luppi2021", "label": "Both: consciousness as dynamics on fixed scaffold", "str": 2 }, { "s": "wu2025", "t": "geo_gap", "label": "ToM geometry in 0.001% of parameters", "str": 2 }, { "s": "wu2025", "t": "n2_univ", "label": "Effective manifold is low-dim in high-dim space", "str": 2 }, { "s": "wu2025", "t": "radford2019", "label": "LLM internal geometry supports hierarchy", "str": 1 } ], "guided_tour": [ { "title": "The Mathematical Toolkit (1948–2012)", "text": "Shannon (1948) quantified information. Kolmogorov (1958) connected it to dynamical systems. Manning (1979) proved that topological entropy on negatively curved manifolds equals (n−1)√κ. Gromov (1987) showed trees are δ-hyperbolic. Sarkar (2012) proved any tree embeds into H² with bounded distortion. By 2012, every mathematical ingredient existed. What was missing was the physical application.", "nodes": [ "shannon1948", "kolmogorov1958", "manning1979", "gromov1987", "sarkar2012", "ratcliffe2006", "milnor1968" ] }, { "title": "The Constraints on Entropy Rate (1968–1980)", "text": "Kimura (1968) showed purifying selection bounds the heritable information rate. Eigen (1971) established the error threshold — above it, the genetic code degrades. Kimura (1980) quantified transition/transversion bias. Bird (1980) documented CpG context effects. Together: h = 1.61 ± 0.10 bits for DNA, bounded above by error catastrophe and below by the minimum to distinguish three domains of life.", "nodes": [ "kimura1968", "kimura1980", "bird1980", "eigen1971", "lynch2007" ] }, { "title": "Three Postulates → One Equation (2026)", "text": "Postulate 1: h > 0. Postulate 2: tree topology. Postulate 3: minimal-distortion embedding. Manning's theorem + Sarkar's embedding theorem force κ = (h ln 2 / (n−1))². Nine theorems verified in Lean 4, 524 lines, zero sorry stubs. Existence, uniqueness, Lyapunov stability: κ* is a global attractor. Zero free parameters.", "nodes": [ "state_eq", "lean_proofs", "paper2", "n2_univ" ] }, { "title": "Evolutionary Validation (2026)", "text": "5,550 genomes yield inter-domain κ ≈ 1.28–1.34 (scale-dependent); intra-domain species trees (GTDB 107K bacteria, 5.9K archaea, 1.6K fungi) yield κ = 3–16 at the scale-appropriate entropy rate. 15 viral families trace the predicted curve at r = 0.996 — from SARS-CoV-2 (5 years) to Dengue (2,000 years). 14 protein families show the predicted 3.1× curvature jump. Both regimes obey the state equation.", "nodes": [ "paper1", "viral_r", "protein_jump", "convergence_5seed", "parks2022", "li2021" ] }, { "title": "The Alphabet Convergence", "text": "DNA funnels 4 bases through ~3 effective transitions (Ts/Tv bias). Language funnels ~40 phonemes through ~3 articulatory targets. Both yield h ≈ log₂3 ≈ 1.58 bits → same κ ≈ 1.2. The raw alphabets are accidents of chemistry and articulation. The effective alphabets are set by physics. Two independent systems, same geometric bottleneck.", "nodes": [ "alphabet_conv", "ling_kappa", "index_diachronica", "asjp", "henikoff1992", "le_gascuel2008" ] }, { "title": "The Neural Extension (2003–2026)", "text": "39 Neuropixels sessions: κ = 0.485 on SPD manifolds with AIRM metric. Volume entropy (Manning's theorem applied to spike trains) gives n = 2.03 ± 0.36. The BCI community (Barachant, Congedo, Yger) independently discovered that SPD geometry works — without knowing why. The framework gives the curvature physical meaning.", "nodes": [ "neural_kappa", "paper3", "steinmetz2019", "barachant2012", "congedo2017", "pennec2006", "allard2020" ] }, { "title": "The Criticality Corollary", "text": "κ > 0 ⟹ J < 1. One line of algebra. Beggs & Plenz (2003) launched 20 years of searching for criticality. Priesemann (2014) found subcriticality. Touboul & Destexhe (2017) showed power laws don't require criticality. Wilting & Priesemann (2019) confirmed m̂ ≈ 0.98. Calvo et al. (2026) — from Muñoz's own group — closed the case: subcritical, not critical. The framework derives it in one line.", "nodes": [ "criticality", "beggs2003", "priesemann2014", "touboul2017", "wilting2019", "calvo2026", "munoz2018" ] }, { "title": "RG Invariance: The Geometric Fixed Point", "text": "Wilson (1971) showed universal quantities survive renormalization. García-Pérez (2018) demonstrated geometric RG preserves the hidden metric space. Meshulam (2019) found RG fixed points in neural populations. Zheng (2020) showed self-similarity in the hyperbolic connectome. The framework's prediction: κ is stable under coarse-graining because it's the IR attractor of the evolutionary/neural flow.", "nodes": [ "rg_invariance", "wilson1971", "garcia_perez2018", "meshulam2019", "zheng2020", "nicoletti2020" ] }, { "title": "The Icosahedral Partition", "text": "Cohn-Kumar (2007): 12 points on S² have icosahedral optimality. The state equation's boundary sphere at κ* therefore partitions into 12 regions. The dvādasāyatana claims 12 irreducible bases of conscious experience. 10/10 tested edges confirmed by independent neuroscience — McGurk (1976), Botvinick (1998), Zhou (2019), Small (2005), Verhagen (2006). The 2 unmatched vertices are the self-referential mind-pair, exactly where introspective precision fails.", "nodes": [ "icosahedral", "cohn_kumar2007", "bodhi2000", "mcgurk1976", "botvinick1998", "zhou2019", "vogler2024", "small2005", "verhagen2006" ] }, { "title": "The Full Picture: 1948–2026", "text": "84 years of independent research — from Shannon's information theory through Manning's Riemannian geometry, Kimura's neutral theory, Beggs's neuronal avalanches, Nickel's hyperbolic ML, to Calvo's robust scaling — converging on a single equation with zero free parameters. The parsimony is not in the framework. It is in the phenomenon.", "nodes": [ "state_eq", "shannon1948", "manning1979", "kimura1968", "eigen1971", "beggs2003", "nickel2017", "calvo2026", "paper1", "paper2", "paper3" ] } ] }