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Discovering Compositional Worlds from Interaction

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Contents

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Front matter

  1. Prefaceora-0001
  2. Further readingora-0002
  3. Trail Mapora-0003

Foundations of Categorical Identification

0 The Infant’s Problem12 sections
  1. 0.1 Two complementary hypothesesora-0005
  2. 0.2 Structured experience creates reusable structuresec-structured-experience-curriculum
  3. 0.3 What is a compositional world?ora-0006
  4. 0.3.1 Two small examples of compositional compressionsec-small-compression-examples
  5. 0.4 Language as a discovered worldora-0007
  6. 0.5 Embodied action as a discovered worldora-0008
  7. 0.6 Social interaction as a discovered worldora-0009
  8. 0.7 Three worlds, one learnerora-0010
  9. 0.8 Particle physics as a discovered worldsec-particle-physics-discovered-world
  10. 0.9 A developmental contract for machine learningora-0011
  11. 0.10 What the categorical theory must provideora-0012
  12. Further Readingora-0013
1 A Categorical and Homotopical Toolkit22 sections
  1. 1.1 Categories and typed compositionora-0014
  2. 1.2 Doctrines: the structure known in advancesec-categorical-doctrines
  3. 1.3 Size, categories of categories, and indexed hypothesesora-0015
  4. 1.4 Universal properties, limits, and colimitsora-0016
  5. 1.5 Algebraic theories, sketches, and generative worldssec-toolkit-algebraic-theories
  6. 1.5.1 The LEGO theoryora-0017
  7. 1.6 Adjunctions and Kan extensionsora-0018
  8. 1.7 Functor categories and online informationora-0019
  9. 1.8 Simplicial sets and decision nervesora-0020
  10. 1.9 Horns, composition, and quasi-categoriesora-0021
  11. 1.10 Weak equivalence and localizationora-0022
  12. 1.11 Homotopy-coherent diagramsora-0023
  13. 1.12 Homotopy limits, colimits, and derived Kan extensionsora-0024
  14. 1.13 Stable infinity-categoriesora-0025
  15. 1.14 Repair spacesora-0026
  16. 1.15 Tangent categories and LINCSora-0027
  17. 1.15.1 Differential categories are not tangent categories with extra axiomsora-0028
  18. 1.16 Infinitesimal intrinsic modelsora-0029
  19. 1.16.1 Infinitesimal decision classesora-0030
  20. 1.17 Reading the derived skeletal-continuity theoremsec-reading-derived-continuity
  21. 1.18 What this chapter does not assumeora-0031
  22. Further readingora-0032
2 The ORACLE Program13 sections
  1. 2.1 The categorical priorora-0033
  2. 2.1.1 From bare categories to structured worldsora-0034
  3. 2.1.2 The generative-theory refinementora-0035
  4. 2.1.3 Running example: the collider declarationsec-collider-oracle-declaration
  5. 2.1.4 The tangent refinementora-0036
  6. 2.1.5 The Spelke sketchora-0037
  7. 2.1.6 A layered prior and three levels of repairora-0038
  8. 2.1.7 Identification under a core doctrineora-0039
  9. 2.2 Prediction before decisionora-0040
  10. 2.3 From universal imitation to ORACLEora-0041
  11. 2.4 The ORACLE–UOCL enrichment squareora-0042
  12. 2.5 Claim discipline and the theorem ladderora-0043
  13. Further readingora-0044
3 Categorical Identification Under a Prior16 sections
  1. 3.1 Presentation protocolsora-0045
  2. 3.2 The hypothesis fibrationora-0046
  3. 3.3 Universal query theoriesora-0047
  4. 3.4 The finite-transcript obstructionora-0048
  5. 3.5 A constrained positive regimeora-0049
  6. 3.6 Coinductive realizationora-0050
  7. 3.7 When prediction becomes decisionora-0051
  8. Further readingora-0052
  9. 3.8 Identification under a categorical priorora-0053
  10. 3.9 The fragment cover of the core sketchora-0054
  11. 3.10 Local identification does not automatically glueora-0055
  12. 3.11 Observers must be jointly separatingora-0056
  13. 3.12 Possible and impossible language worldsora-0057
  14. 3.13 Assimilation, ordinary accommodation, and doctrinal repairora-0058
  15. 3.14 The ORACLE readiness criterionora-0059
  16. Further readingora-0060

Categorical Core Knowledge

4 Core Knowledge and Piagetian Construction19 sections
  1. 4.1 Core knowledge as a doctrine of possible worldssec-core-knowledge-as-doctrine
  2. 4.2 Three meanings of categorical truthora-0062
  3. 4.3 The five-obligation testora-0063
  4. 4.4 Prior gain as a relative identification statementora-0064
  5. 4.5 Six systems, six categorical burdensora-0065
  6. 4.6 Cross-system composition is the decisive testora-0066
  7. 4.7 What would count against the hypothesis?ora-0067
  8. 4.8 Does existing machinery already solve ORACLE?sec-transformer-challenge
  9. 4.8.1 What would a Transformer have to demonstrate?ora-0068
  10. 4.8.2 Beyond tokens: JEPA and predictive groundingora-0069
  11. 4.8.3 Why a Kan extension is not yet an ORACLE learnersec-kan-insufficiency
  12. 4.9 Piagetian construction of a compositional worldora-0071
  13. 4.10 Development is not merely clock timeora-0072
  14. 4.11 Assimilation as conservative liftingora-0073
  15. 4.12 Two levels of accommodationora-0074
  16. 4.13 Equilibration as a repair processora-0075
  17. 4.14 Developmental composition of the core systemsora-0076
  18. 4.15 A grounded research programora-0077
  19. Further readingora-0070

Universal Online Categorical Learning

5 Learning Inside Fixed Doctrines18 sections
  1. 5.1 The doctrinal auditora-0080
  2. 5.2 System identification and predictive state representationsora-0081
  3. 5.3 Topos World Models from documentssec-topos-world-models-uocl
  4. 5.4 Automata and machine inference as coalgebra learningora-0082
  5. 5.4.1 Diversity representations: learning tests instead of statessec-rivest-schapire-diversity
  6. 5.4.2 Krohn–Rhodes decomposition: learning cascades instead of statessec-krohn-rhodes-uocl
  7. 5.4.3 Toward stochastic and coalgebraic prime decompositionsec-stochastic-krohn-rhodes
  8. 5.5 Reinforcement learning inside the MDP–Bellman doctrineora-0083
  9. 5.5.1 Options as persistent compositional objectssec-options-persistence
  10. 5.6 Causal discovery as identification in a causal categoryora-0084
  11. 5.7 Language learning inside grammar and sequence doctrinesora-0085
  12. 5.7.1 Krohn–Rhodes shortcuts and the identification gapsec-transformer-krohn-rhodes
  13. 5.8 JEPA, world models, and learned quotientsora-0086
  14. 5.8.1 RoboDreamer and compositional robot imaginationsec-robodreamer-uocl
  15. 5.8.2 PoE-World and the categorical frontiersec-poe-world-uocl
  16. 5.8.3 COMBO: composition under decentralized observationsec-combo-uocl
  17. 5.9 What the fixed-doctrine analysis establishesora-0087
  18. Further readingora-0088
6 The UOCL Machine11 sections
  1. 6.1 From a semantic section to a learning machineora-0089
  2. 6.2 State, probes, and dependent updateora-0090
  3. 6.3 Assimilation and accommodation as control flowora-0091
  4. 6.4 Active identification and separating probesora-0092
  5. 6.4.1 Running example: enlarging the collider probe familysec-collider-active-identification
  6. 6.5 Learning generators, relations, and modelssec-uocl-theory-learning
  7. 6.6 Executions realize ORACLE sectionsora-0093
  8. 6.7 Composing UOCL learnerssec-composing-uocl-learners
  9. 6.8 Limits, colimits, and multimodal fusionsec-uocl-limits-colimits
  10. 6.9 Failure modesora-0094
  11. Further readingora-0095
7 Persistent Categorical Identification9 sections
  1. 7.1 Non-anticipation and two forms of stabilizationora-0096
  2. 7.2 Coherent persistenceora-0097
  3. 7.3 Finite active identificationora-0098
  4. 7.4 Presentation invarianceora-0099
  5. 7.5 Tangent UOCL: learning how a world can varysec-tangent-uocl
  6. 7.5.1 Differential presentations of tangent worldsora-0100
  7. 7.6 The cost of categorical learningora-0101
  8. 7.7 Two enrichments of categorical learningora-0102
  9. Further readingora-0103
8 Probably Approximately Categorically Correct Learning12 sections
  1. 8.1 The PAC templateora-0104
  2. 8.2 The PACC declarationora-0105
  3. 8.2.1 Running example: PACC collider identificationsec-pacc-collider
  4. 8.3 Ordinary PAC learning is the discrete caseora-0106
  5. 8.4 Finite PACC boundsora-0107
  6. 8.5 Coverage is part of the categorical priorora-0108
  7. 8.6 Three strengths of categorical approximationora-0109
  8. 8.7 Toward a categorical dimension theoryora-0110
  9. 8.8 PACC evolvabilitysec-pacc-evolvability
  10. 8.9 Persistent and online PACCora-0111
  11. 8.10 What PACC does and does not promiseora-0112
  12. Further readingora-0113

Universal Online Decision Learning

9 Online and Persistent Universal Decision Learning21 sections
  1. 9.1 Claim discipline and the theorem ladderora-0115
  2. 9.2 Scientific questions for universal online decision learningora-0116
  3. 9.3 Prefix semantics, nerves, homotopy coherence, and repairsec-online-udl
  4. 9.4 Time, revelation, and the least available pastora-0117
  5. 9.5 Prefix-indexed UDL and non-anticipationora-0118
  6. 9.6 Online Kan invarianceora-0119
  7. 9.7 Growing diagrams and the Beck–Chevalley defectora-0120
  8. 9.8 Persistent online abstractionsora-0121
  9. 9.9 Decision nerves and compositional depthsec-decision-nerve
  10. 9.10 Simplicial online UDLora-0122
  11. 9.11 The online-to-offline comparisonsec-online-offline-comparison
  12. 9.12 Homotopy-coherent online UDLsec-homotopy-oudl
  13. 9.12.1 Horn fillers as repair spacesora-0123
  14. 9.12.2 Derived skeletal continuityora-0124
  15. 9.13 Observers and the origin of regretsec-observer-regret
  16. 9.14 Information-structured regret beyond the classical chainsec-intrinsic-regret
  17. 9.15 Running example: crossword solving as universal completionsec-crossword-limit
  18. 9.16 Running example: Sudoku and flow-based recurrent repairsec-sudoku-flow
  19. 9.16.1 Flow reasoning as learned fixed-point searchora-0125
  20. 9.16.2 Why Fixed-Point Forcing matters to UODLora-0126
  21. Further readingora-0127

Global-Clock Online Decision Learning

10 Convex and Regularized Universal Action11 sections
  1. 10.1 Universal decision learningora-0129
  2. 10.2 LINCS disciplineora-0130
  3. 10.3 The categorical OCO declarationora-0131
  4. 10.4 Regularized universal actionora-0132
  5. 10.5 The FTRL action normal formora-0133
  6. 10.6 Finite enriched accumulationora-0134
  7. 10.6.1 The categorical declarationora-0135
  8. 10.6.2 The first LINCS–UDL theoremora-0136
  9. 10.7 Quadratic FTRL and typed tangentsora-0137
  10. 10.8 Proximal FTRL and tangent repairora-0138
  11. Further readingora-0139
11 Universal Bandit Decision Learning7 sections
  1. 11.1 The bandit information channelora-0140
  2. 11.2 Pathwise loss and barycentric recoveryora-0141
  3. 11.3 Exploration as infinitesimal admissibilityora-0142
  4. 11.4 The expectation-level regret-transfer theoremora-0143
  5. 11.5 Entropic realization and the classical rateora-0144
  6. 11.6 Bandit convex optimization changes the repair targetora-0145
  7. Further readingora-0146

Information Beyond a Global Clock

12 Information Categories and Decentralized Decisions9 sections
  1. 12.1 Comparator sketches as descent datasec-comparator-sketches
  2. 12.2 The comparator spectrum on a finite horizonsec-comparator-spectrum
  3. 12.3 Approximate descent and observer transfersec-approximate-comparator-descent
  4. 12.4 Running example: asynchronous event categoriessec-asynchronous-event-categories
  5. Further readingora-0148
  6. 12.5 Decentralized teams, games, and nonclassical informationora-0149
  7. 12.6 Running example: asynchronous distributed minimizationsec-asynchronous-distributed-minimization
  8. 12.6.1 Q-learning as asynchronous coordinate computationora-0150
  9. Further readingora-0151
13 Agentic Safety and Universal Online Decision Learning8 sections
  1. 13.1 From model safety to information-structure safetyora-0152
  2. 13.2 Safety under a changing information shapeora-0153
  3. 13.3 Running example: stale authority under asynchronous executionsec-asynchronous-safety
  4. 13.4 Audit is an observer, not the executionora-0154
  5. 13.5 A contemporary incident as an information-structure failuresec-metr-agentic-incident
  6. 13.6 Temporal behavior types as trust contractssec-temporal-trust-contracts
  7. 13.7 Safety invariants beyond scalar regretora-0155
  8. Further readingora-0156

Coherent Repair and Structural Amplification

14 Homotopy and Tangent Repair10 sections
  1. 14.1 Registered repair problemsora-0158
  2. 14.2 Comparison-guided localizationora-0159
  3. 14.2.1 Running example: a collider repair laddersec-collider-repair-ladder
  4. 14.3 Assimilation, accommodation, and doctrinal changeora-0160
  5. 14.4 A repair ledgerora-0161
  6. Further readingora-0162
  7. 14.5 Tangent repair, stability, and convergenceora-0163
  8. 14.6 Tangent repair certificatesora-0164
  9. 14.7 A three-axis stability doctrineora-0165
  10. Further readingora-0166
15 Invariance, Amplification, and Approachability10 sections
  1. 15.1 The decision quotientora-0167
  2. 15.2 When invariance failsora-0168
  3. 15.3 Boosting as free convex completionora-0170
  4. 15.4 The weak-learning interfaceora-0171
  5. 15.5 When amplification becomes persistentora-0172
  6. 15.6 Duality, games, and approachabilityora-0174
  7. 15.7 Universal equilibrium objectsora-0175
  8. 15.8 Approachability as persistent consistencyora-0176
  9. 15.9 Approachability under changing informationora-0177
  10. Further readingora-0169

Persistent Structure Across a Lifetime

16 Persistent Structure Across a Lifetime11 sections
  1. 16.1 Environment morphisms and knowledge packagesora-0180
  2. 16.2 A transport theoremora-0181
  3. 16.2.1 Running example: transporting a collider theorysec-collider-transport
  4. 16.3 Transport defects and partial reuseora-0182
  5. Further readingora-0183
  6. 16.4 Persistent compositional knowledgeora-0184
  7. 16.5 Usefulness by future factorizationora-0185
  8. 16.6 The persistence coreora-0186
  9. 16.7 The compositional storehouseora-0187
  10. 16.8 Progress without a single task scoreora-0188
  11. Further readingora-0189
17 Synthesis and the Next ORACLE Theorem Ladder14 sections
  1. 17.1 What the current theory establishesora-0190
  2. 17.2 The first next rung: finite doctrine stabilizationora-0191
  3. 17.3 The open frontierora-0192
  4. 17.3.1 Program A: non-anticipating information structuresora-0193
  5. 17.3.2 Program B: obstruction-theoretic repairora-0194
  6. 17.3.3 Program C: functorial transportora-0195
  7. 17.3.4 Two cross-cutting inverse problemsora-0196
  8. 17.4 The ORACLE persistence conjectureora-0197
  9. 17.5 From the tetralogy to a research programora-0198
  10. 17.6 Coda: frontier models as collaboratorsora-0199
  11. Further readingora-0200
  12. A Claim ledgerora-0201
  13. Referencesreferences
  14. Glossary of Notationora-0202