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Infinitesimal Creativity

Learning by Double Involution

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Contents

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Front matter

  1. Prefaceifc-0001
  2. A Trail Map of Infinitesimal Creativityifc-0002

Foundations of Infinitesimal Creativity

0 A Working Language for Creative Theory Change15 sections
  1. 0.1 Progressive abstraction as creative reconstructionifc-0004
  2. 0.2 Categories: typed compositionifc-0005
  3. 0.3 Diagrams and compositional promisesifc-0006
  4. 0.4 Functors and natural transformationsifc-0007
  5. 0.5 Universal constructions: defining by relationsifc-0008
  6. 0.6 Algebraic theories, sketches, and modelsifc-0009
  7. 0.7 Lexical creativity: extending a languageifc-0010
  8. 0.8 Presheaves, sheaves, and contextual agreementifc-0011
  9. 0.9 Tangent categories and local possibilityifc-0012
  10. 0.10 Weil algebras as probe shapesifc-0013
  11. 0.11 Classical double Lie algebroids: a concrete realizationifc-0014
  12. 0.12 Two axes: Boden and Piagetifc-0015
  13. 0.13 The geometry example as a creative traceifc-0016
  14. 0.14 How to read the rest of the bookifc-0017
  15. Further Readingifc-0018
1 Initial Sketch Acquisition13 sections
  1. 1.1 The initialization problemifc-0019
  2. 1.2 Four routes into a domainifc-0020
  3. 1.3 Explanation-based learning as a precedentifc-0021
  4. 1.4 Learning apprentices and continuous knowledge acquisitionifc-0022
  5. 1.5 What must be acquired from a demonstration?ifc-0023
  6. 1.6 Coaching as typed counter-witness selectionifc-0024
  7. 1.7 Conservation as developmental sketch acquisitionifc-0025
  8. 1.8 From acquisition to double involutionifc-0026
  9. 1.8.1 The readiness boundaryifc-0027
  10. 1.9 An initial acquisition contractifc-0028
  11. 1.10 A controlled acquisition benchmarkifc-0029
  12. 1.11 What this chapter does not assumeifc-0030
  13. Further Readingifc-0031
2 Creativity as Theory Construction11 sections
  1. 2.1 Three different achievementsifc-0032
  2. 2.2 Assimilation, accommodation, and cognitive developmentifc-0033
  3. 2.3 Two interacting classes of repairifc-0034
  4. 2.4 Boden’s three modesifc-0035
  5. 2.5 Four evaluation media, five admission regimesifc-0036
  6. 2.6 From LINCS repair to creative transformationifc-0037
  7. 2.7 The status-bearing extension dossierifc-0038
  8. 2.8 What counts as success?ifc-0039
  9. 2.9 Personal and historical creativityifc-0040
  10. 2.10 From a target to a mechanismifc-0041
  11. Further Readingifc-0042
3 The Differential Geometry of Creativity18 sections
  1. 3.1 The ambient differential universeifc-0043
  2. 3.2 Weil algebras as probe shapesifc-0044
  3. 3.3 The missing bridge: from tangent semantics to double algebroidsifc-0045
  4. 3.4 Tensor products of sketches as a semantics for doublingifc-0046
  5. 3.4.1 The proposed internal DIAL targetifc-0047
  6. 3.4.2 Formal status and empirical burdenifc-0048
  7. 3.5 A Weil profile of a theory statesec-weil-profile
  8. 3.6 Double-infinitesimal repair geometryifc-0049
  9. 3.7 Curve objects and the dynamics of theory revisionifc-0050
  10. 3.7.1 Two flows rather than oneifc-0051
  11. 3.7.2 From flow to a change of theoryifc-0052
  12. 3.8 Assimilation and accommodation in differential formifc-0053
  13. 3.9 The three modes in differential formifc-0054
  14. 3.10 Differential theory extensionsifc-0055
  15. 3.11 A differential creativity processifc-0056
  16. 3.12 Beyond two directions: the higher-DIAL conjectureifc-0057
  17. 3.13 What the differential account does not claimifc-0058
  18. Further Readingifc-0059
4 Skill Optimization for Creative Exploration14 sections
  1. 4.1 From a script to a creative skillifc-0060
  2. 4.2 The Lie algebroid of creative movesifc-0061
  3. 4.3 Brackets expose interactionifc-0062
  4. 4.4 Assimilation and accommodation in skill geometryifc-0063
  5. 4.5 Latent skill variationifc-0064
  6. 4.6 Two coupled optimization problemsifc-0065
  7. 4.7 DIAL-SkillOpt: training a discovery skillsec-dial-skillopt
  8. 4.7.1 Typed state and actionsifc-0066
  9. 4.7.2 One discovery episodeifc-0067
  10. 4.7.3 The training loopifc-0068
  11. 4.7.4 A preregistered evaluation designifc-0069
  12. 4.8 Boden’s modes as changes of control geometryifc-0070
  13. 4.9 Scope and experimental consequencesifc-0071
  14. Further Readingifc-0072
5 Algebraic Theories as Creative Targets14 sections
  1. 5.1 Lawvere’s functorial semanticsifc-0073
  2. 5.2 Sketches present theoriesifc-0074
  3. 5.3 From products to finite limitsifc-0075
  4. 5.4 PROPs and monoidal theoriesifc-0076
  5. 5.5 From presentations to Grothendieck toposesifc-0077
  6. 5.6 The creative target packagesec-creative-target-package
  7. 5.7 From a double repair theory to a finite theory changeifc-0078
  8. 5.8 Kan extensions as the transport semantics of DIALifc-0079
  9. 5.8.1 Transport does not choose the creative extensionifc-0080
  10. 5.8.2 A Beck–Chevalley interpretation of double interactionifc-0081
  11. 5.8.3 Left–right asymmetry and admissionifc-0082
  12. 5.9 Developmental towers of theoryifc-0083
  13. 5.10 Boden’s modes revisitedifc-0084
  14. Further Readingifc-0085
6 Landscapes of Computational Creativity20 sections
  1. 6.1 A map by creative objectifc-0086
  2. 6.2 Conceptual spaces and transformationsifc-0087
  3. 6.3 Creativity as a neurocognitive processifc-0088
  4. 6.4 Assimilation and accommodation across the landscapeifc-0089
  5. 6.5 Prediction, explanation, and structural possibilityifc-0090
  6. 6.5.1 Chomskyan structure and Piagetian developmentifc-0091
  7. 6.5.2 An empirical target, not a settled verdictifc-0092
  8. 6.6 Novelty, information, and surpriseifc-0093
  9. 6.7 Discovery as heuristic searchifc-0094
  10. 6.7.1 BACON: discovery from numerical regularityifc-0095
  11. 6.8 Local novelty, integrability, and persistenceifc-0096
  12. 6.9 Artifacts, process, and evaluationifc-0097
  13. 6.10 Imagination and counterfactual worldsifc-0098
  14. 6.11 Cellular automata and emergent conceptual structureifc-0099
  15. 6.12 Evolution as a creative constructorifc-0100
  16. 6.13 AI Feynman as a controlled testbedifc-0101
  17. 6.14 A four-level symbolic-discovery challengeifc-0102
  18. 6.15 Large language models in scientific discoveryifc-0103
  19. 6.15.1 Biological foundation models, world models, and co-scientistsifc-0104
  20. Further Readingifc-0105
7 Combinational Creativity9 sections
  1. 7.1 From juxtaposition to compositionifc-0106
  2. 7.2 Interface discoveryifc-0107
  3. 7.3 Assimilation and accommodationifc-0108
  4. 7.4 Differential tests for interactionifc-0109
  5. 7.5 Combination inside a tensor-product theoryifc-0110
  6. 7.6 Combinations across the AGENTIC testbedsifc-0111
  7. 7.7 A combinational experiment protocolifc-0112
  8. 7.8 Failure modes and admissionifc-0113
  9. Further Readingifc-0114
8 Exploratory Creativity10 sections
  1. 8.1 The boundary of a conceptual spaceifc-0115
  2. 8.2 Assimilation as explorationifc-0116
  3. 8.3 Local exploration with Weil probesifc-0117
  4. 8.4 Exploration as an integrable pathifc-0118
  5. 8.5 Active probe selectionifc-0119
  6. 8.6 Novelty, surprise, and interestingnessifc-0120
  7. 8.7 From exploration to obstructionifc-0121
  8. 8.8 Exploration across the testbedsifc-0122
  9. 8.9 The exploratory audit recordifc-0123
  10. Further Readingifc-0124
9 Transformational Creativity11 sections
  1. 9.1 What can be transformed?ifc-0125
  2. 9.2 Declaration interventionsifc-0126
  3. 9.3 Accommodation and equilibrationifc-0127
  4. 9.4 From infinitesimal evidence to a finite jumpifc-0128
  5. 9.4.1 Transforming a factor, interchange, or domain theoryifc-0129
  6. 9.5 Transport, forgetting, and conceptual lossifc-0130
  7. 9.6 An obstruction-to-extension workflowifc-0131
  8. 9.7 Transformations across the testbedsifc-0132
  9. 9.8 Evaluating transformational creativityifc-0133
  10. 9.9 The boundary of the claimifc-0134
  11. Further Readingifc-0135

Algorithmic Realizations

10 DIAL as an Algorithmic Family19 sections
  1. 10.1 The missing algorithmic layerifc-0137
  2. 10.2 The four-level implementation invariantifc-0138
  3. 10.3 Where creativity entersifc-0139
  4. 10.3.1 The creative deltaifc-0140
  5. 10.4 Three targets for every DIAL-X methodifc-0141
  6. 10.4.1 The artifact dossiersec-dial-artifact-dossier
  7. 10.4.2 Five evidential statusessec-dial-evidential-statuses
  8. 10.5 The DIAL-X contractifc-0142
  9. 10.5.1 The target cardifc-0143
  10. 10.6 The generic DIAL-X algorithmifc-0144
  11. 10.6.1 Four control actionsifc-0145
  12. 10.7 DIAL-GIRL: the target is an admitted decision stateifc-0146
  13. 10.8 DIAL-ALLORA: the target is an admitted adapter compositionifc-0147
  14. 10.9 CLIC: creative causal learningifc-0148
  15. 10.10 OPTIC: creative skill optimizationifc-0149
  16. 10.11 Further members of the DIAL-X familyifc-0150
  17. 10.12 How DIAL-X sharpens an experimentifc-0151
  18. 10.13 Boundary of the algorithmic claimifc-0152
  19. Further Readingifc-0153
11 Causal Learning with Infinitesimal Creativity7 sections
  1. 11.1 The CLIC target cardifc-0154
  2. 11.2 Exact closure and typed repairifc-0155
  3. 11.3 Finite operators and propagated uncertaintyifc-0156
  4. 11.4 Active counter-witness interventionsifc-0157
  5. 11.5 Explicit alternative posteriorsifc-0158
  6. 11.6 From local repair to causal admissionifc-0159
  7. Further Readingifc-0160
12 Skill Optimization with Infinitesimal Creativity12 sections
  1. 12.1 The OPTIC target cardifc-0161
  2. 12.2 From vector-field brackets to involution algebroidsifc-0162
  3. 12.3 The OPTIC contractifc-0163
  4. 12.4 An executable OPTIC episodeifc-0164
  5. 12.5 Evidence for skill factorizationifc-0165
  6. 12.5.1 Exact mixed-witness calibrationifc-0166
  7. 12.5.2 The compound-skill failureifc-0167
  8. 12.5.3 Factored subskillsifc-0168
  9. 12.6 What has been learned—and what has notifc-0169
  10. 12.7 The open construction problemifc-0170
  11. 12.8 CLIC, OPTIC, and RELICifc-0171
  12. Further Readingifc-0172
13 Reinforcement Learning with Infinitesimal Creativity13 sections
  1. 13.1 What is creative about reinforcement learning?ifc-0173
  2. 13.2 The RELIC target cardifc-0174
  3. 13.3 From GIRL to RELICifc-0175
  4. 13.4 Schema learning in an embodied text worldifc-0176
  5. 13.5 The ALFWorld benchmark contractifc-0177
  6. 13.6 Competence of persistent structural artifactsifc-0178
  7. 13.7 Why novelty and one-step guidance are insufficientifc-0179
  8. 13.8 Admitting a supplied temporal composition (B3)sec-relic-b3
  9. 13.9 A DIAL reading of the resultifc-0180
  10. 13.10 Constructing the temporal composition (B4)sec-relic-b4
  11. 13.11 Residual failure and declaration scopeifc-0181
  12. 13.12 Constructing a missing stageifc-0182
  13. Further Readingifc-0183
14 Composing Creative Workflows13 sections
  1. 14.1 Three maintained theoriesifc-0184
  2. 14.2 The exchange objectifc-0185
  3. 14.3 A small algebra of workflow compositionifc-0186
  4. 14.4 Three modes of meta-creativityifc-0187
  5. 14.5 Cross-workflow obstructionsifc-0188
  6. 14.6 Canonical compositionsifc-0189
  7. 14.7 An integrated episodeifc-0190
  8. 14.8 An integrated testbedifc-0191
  9. 14.9 Ablations and interaction effectsifc-0192
  10. 14.10 A controlled demonstration: active evidence across the workflowsec-agentic-controlled-demonstration
  11. 14.11 What surprising outcomes would meanifc-0193
  12. 14.12 From orchestration sketch to workflow theoryifc-0194
  13. Further Readingifc-0195

AGENTIC Testbeds and Open Domains

15 Mathematical Theory Construction26 sections
  1. 15.1 The AGENTIC mathematical contractifc-0197
  2. 15.2 Historical and generated worldsifc-0198
  3. 15.3 A hierarchy of mathematical constructionifc-0199
  4. 15.4 An AGENTIC mathematical episodeifc-0200
  5. 15.5 Admission records and formal certificatesifc-0201
  6. 15.6 Persistenceifc-0202
  7. 15.7 Early calibration: theory extension with Leaifc-0203
  8. 15.8 A worked episode: inventing an intrinsic propertyifc-0204
  9. 15.9 From interaction laws to latent presentationstab-ipc-math-ladder
  10. 15.10 Symbolic laws and active mathematical experimentstab-af-math-ladder
  11. 15.11 Cellular-automaton theory inductionifc-0205
  12. 15.12 An integrated benchmarkifc-0206
  13. 15.13 A grand challenge: constructing a coalgebraic theory of learningifc-0207
  14. 15.13.1 Assimilation and accommodation over behavior functorsifc-0208
  15. 15.13.2 The admitted artifactifc-0209
  16. 15.13.3 The DIAL-URL evidence sequenceifc-0210
  17. 15.13.4 The synthetic laboratory behind the laddertab-dial-url-environments
  18. 15.13.5 The concrete sequential worldsifc-0211
  19. 15.13.6 What the grammar constructors mean experimentallyifc-0212
  20. 15.13.7 In what sense is this theory construction?ifc-0213
  21. 15.13.8 Finite-trajectory accommodationtab-dial-url1
  22. 15.13.9 Active coalgebraic discriminationtab-dial-url2
  23. 15.13.10 From named families to functor expressionsifc-0214
  24. 15.13.11 Discovering a withheld composition lawtab-dial-url4
  25. 15.14 Pattern repair before causal explanationifc-0215
  26. Further Readingifc-0216
16 Simulator-Grounded Theory Extension17 sections
  1. 16.1 The AGENTIC simulator contractifc-0217
  2. 16.2 Executable semanticsifc-0218
  3. 16.2.1 From relations to mechanismsifc-0219
  4. 16.3 Three validity layersifc-0220
  5. 16.4 Double involution in an executable worldifc-0221
  6. 16.5 The integrated loopifc-0222
  7. 16.6 Three Prometheus executable substratesifc-0223
  8. 16.6.1 Neural projection expansion in singing miceifc-0224
  9. 16.6.2 Hydroclimate and the Harappan transformationifc-0225
  10. 16.6.3 Colored airborne plastics and radiative forcingifc-0226
  11. 16.7 A worked theory-repair traceifc-0227
  12. 16.8 Ontology extensionifc-0228
  13. 16.9 An evidence hierarchyifc-0229
  14. 16.10 AGENTIC-SIM: the integrated benchmarkifc-0230
  15. 16.11 What would count as scientific creativity?ifc-0231
  16. 16.12 Open problemsifc-0232
  17. Further Readingifc-0233
17 From Scientific Documents to Testable Theories33 sections
  1. 17.1 The transition from simulation to literatureifc-0234
  2. 17.2 Two directions between theory and evidenceifc-0235
  3. 17.3 Documents as implicit sketch presentationsifc-0236
  4. 17.4 The AGENTIC corpus contractifc-0237
  5. 17.4.1 How to read the evidence sequenceifc-0238
  6. 17.5 From reports to local modelsifc-0239
  7. 17.6 Why a corpus is not a simulatorifc-0240
  8. 17.7 The Theory Extension Packageifc-0241
  9. 17.8 Controlled evidence for corpus-to-theory constructionifc-0242
  10. 17.8.1 Source discipline (CTTE–0)ifc-0243
  11. 17.8.2 Entity registration (CTTE–1)ifc-0244
  12. 17.8.3 Context-cover construction (CTTE–2)ifc-0245
  13. 17.8.4 Active overlap acquisition (CTTE–3)ifc-0246
  14. 17.8.5 Missing-presentation construction (CTTE–4)ifc-0247
  15. 17.8.6 Cross-domain transport (CTTE–5)ifc-0248
  16. 17.9 A corpus-to-theory evidence hierarchyifc-0249
  17. 17.10 The GLP-1 challengeifc-0250
  18. 17.10.1 Treatment status as a missing context (GLP1–PILOT–0)ifc-0251
  19. 17.10.2 Discovering the missing context (GLP1–PILOT–1)ifc-0252
  20. 17.11 A grant proposal as a capstone artifactifc-0253
  21. 17.11.1 Structural compilation without semantic admissionifc-0254
  22. 17.11.2 Semantic admission as a type systemifc-0255
  23. 17.11.3 Compiling prose from a typed intermediate programifc-0256
  24. 17.11.4 Scientific invariants across a research programifc-0257
  25. 17.11.5 Auditable rhetorical realizationifc-0258
  26. 17.11.6 Sanitized realization and surface separationifc-0259
  27. 17.11.7 Structure-first rhetorical realizationifc-0260
  28. 17.11.8 Local metric and semantic boundsifc-0261
  29. 17.11.9 Controlled paraphrase and evaluator failureifc-0262
  30. 17.11.10 From an AI-authored study to an AI-constructed programifc-0263
  31. 17.12 Admission beyond the corpusifc-0264
  32. 17.13 Toward an integrated open-domain benchmarkifc-0265
  33. Further Readingifc-0266
18 Generative Visual Languages27 sections
  1. 18.1 Diffusion is one infinitesimal directionifc-0267
  2. 18.2 The second direction: proto-accommodation and semantic extensionifc-0268
  3. 18.3 A candidate double geometryifc-0269
  4. 18.4 From a new token to a new languageifc-0270
  5. 18.5 ARTISTIC as an AGENTIC systemifc-0271
  6. 18.6 The ARTISTIC procedureifc-0272
  7. 18.7 Textual inversion as staged calibrationifc-0273
  8. 18.8 A visual-language benchmarkifc-0274
  9. 18.9 An evidence architecture for visual assuranceifc-0275
  10. 18.10 Lexical calibration and typed realizationifc-0276
  11. 18.11 Learning the declarationifc-0277
  12. 18.12 Transfer to natural photographsifc-0278
  13. 18.13 Intervals, abstention, and active acquisitionifc-0279
  14. 18.14 From image repair to structural assuranceifc-0280
  15. 18.14.1 View-conditioned contractsifc-0281
  16. 18.14.2 A sealed Firefly cricket challengeifc-0282
  17. 18.14.3 Three kinds of Nano Banana repairifc-0283
  18. 18.14.4 Strong generators change the opportunityifc-0284
  19. 18.14.5 The generator interfaceifc-0285
  20. 18.14.6 Movie Mode as a semantic audit traceifc-0286
  21. 18.14.7 Three practical axesifc-0287
  22. 18.15 Visual theory invention in Bongard problemsifc-0288
  23. 18.15.1 Five evidence regimes for concept inductionifc-0289
  24. 18.16 What the experiments establishifc-0290
  25. 18.17 Measurements and failure modesifc-0291
  26. 18.18 Open mathematicsifc-0292
  27. Further Readingifc-0293
19 Artistic Theory Extension18 sections
  1. 19.1 What CAN makes creativeifc-0294
  2. 19.2 The state space is a space of artistic theoriesifc-0295
  3. 19.3 Two local artistic directionsifc-0296
  4. 19.4 From noncommutativity to structural obstructionifc-0297
  5. 19.5 The Weil profile of an obstructionifc-0298
  6. 19.6 Constructing a candidate techniqueifc-0299
  7. 19.7 Transporting the old artistic theoryifc-0300
  8. 19.8 The admission firewallifc-0301
  9. 19.9 The hierarchy from CAN to DILATEifc-0302
  10. 19.10 Calibration of the mixed-order diagnosticifc-0303
  11. 19.11 A paintbrush over the space of generative dynamicsifc-0304
  12. 19.12 Observer-induced metric and discrete transportifc-0305
  13. 19.13 Registered paintbrush evidenceifc-0306
  14. 19.14 Theory routing across natural videosifc-0307
  15. 19.15 The withheld-generator experimentifc-0308
  16. 19.16 Baselines and ablationsifc-0309
  17. 19.17 Open mathematical obligationsifc-0310
  18. Further Readingifc-0311
20 Creativity as Compositional Theory Change14 sections
  1. 20.1 The argument in one viewifc-0312
  2. 20.2 What the present evidence supportsifc-0313
  3. 20.3 Why two interacting directions matterifc-0314
  4. 20.4 What DIAL does not explainifc-0315
  5. 20.5 Music as an open compositional domainifc-0316
  6. 20.5.1 A controlled benchmark proposalifc-0317
  7. 20.6 Other domains not explored hereifc-0318
  8. 20.7 Evaluation beyond noveltyifc-0319
  9. 20.8 Open obligationsifc-0320
  10. 20.9 Conclusionifc-0321
  11. Further Readingifc-0322
  12. Referencesreferences
  13. 22 Glossary of Notationifc-0323
  14. AA