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Machine Learning from Enforcing Compositionality

A Categorical Framework for Structural Diagnosis and Repair

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Contents

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Front matter

  1. Prefacelin-0001
  2. Why category theory?lin-0002
  3. From causal models to models of learninglin-0003
  4. The observability caveatlin-0004
  5. Compositionality is not compressionlin-0005
  6. The recurring workflowlin-0006
  7. What the book developslin-0007
  8. How to read the booklin-0008
  9. The boundary of the claimlin-0009
  10. A Trail Map of LINCSlin-0010
  11. The view from abovelin-0011
  12. The foundations traillin-0012
  13. The recurring field procedurelin-0013
  14. The applications traillin-0014
  15. Choose a routelin-0015
  16. Signposts for the journeylin-0016

Categorical Foundations

0 A Working Language of Compositionality19 sections
  1. 0.1 Categories: typed compositionlin-0018
  2. 0.2 Ordinary syntax and enriched semanticssec-enriched-categories
  3. 0.3 Graphs, paths, and diagramssec-graphs-paths-diagrams
  4. 0.4 Functors: compositional translationslin-0019
  5. 0.5 Natural transformations: coherent changelin-0020
  6. 0.6 Reading coherent change in two dimensionslin-0021
  7. 0.7 Universal constructionslin-0022
  8. 0.7.1 Products, pullbacks, and equalizerslin-0023
  9. 0.7.2 Coproducts, pushouts, and quotientslin-0024
  10. 0.8 Adjunctions and Kan extensionslin-0025
  11. 0.9 Sketches: declarations before objectiveslin-0026
  12. 0.10 Presheaves, sheaves, and local-to-global reasoninglin-0027
  13. 0.11 Tangent structure and infinitesimal languagesec-tangent-structure-language
  14. 0.12 Internal logic: infinitesimals, subobjects, and interventionssec-internal-logic
  15. 0.13 Symbolic and neural reasoningsec-symbolic-neural-lincs
  16. 0.14 From structural failure to learning signallin-0028
  17. 0.15 A compact worked examplelin-0029
  18. 0.16 How to read the rest of the booklin-0030
  19. Further readinglin-0031
1 Axioms of Structural Learning12 sections
  1. 1.1 What an axiomatization must distinguishlin-0032
  2. 1.2 The universal obstruction axiomlin-0033
  3. 1.3 Nonsmooth refinement: from a derivative to a response spacelin-0034
  4. 1.4 The semantic corelin-0035
  5. 1.5 Structural laws governing the obstructionlin-0036
  6. 1.6 From semantic axioms to a repair calculuslin-0037
  7. 1.7 Six rule schemaslin-0038
  8. 1.8 An admission rule is not an equational rulelin-0039
  9. 1.9 A minimal worked examplelin-0040
  10. 1.10 Soundness, completeness, and the research frontierlin-0041
  11. 1.11 What this adds to the LINCS workflowlin-0042
  12. Further readinglin-0043
2 Learning by Repair8 sections
  1. 2.1 From loss minimization to structural diagnosislin-0044
  2. 2.2 An actionable model of the learning systemlin-0045
  3. 2.3 The six-stage workflowlin-0046
  4. 2.4 A small factorization examplelin-0047
  5. 2.5 Repair is a gated transformationlin-0048
  6. 2.6 One pattern, different domainslin-0049
  7. 2.7 What changes in the chapters aheadlin-0050
  8. Further readinglin-0051
3 Learning Sketches and Factorization Obstructions10 sections
  1. 3.1 From computational graphs to learning sketcheslin-0052
  2. 3.2 Sketches as categorical theorieslin-0053
  3. 3.3 Models and the factorization testlin-0054
  4. 3.4 Three declarations, three characteristic failureslin-0055
  5. 3.5 A running equivariance sketchlin-0056
  6. 3.6 Observation and scalarizationlin-0057
  7. 3.7 Presentation, transport, and sketchabilitylin-0058
  8. 3.8 A sketch as a model of learninglin-0059
  9. 3.9 The declaration cardlin-0060
  10. Further readinglin-0061
4 Tangent Learning Sketches11 sections
  1. 4.1 Tangent structurelin-0062
  2. 4.2 Tangent learning sketcheslin-0063
  3. 4.3 Infinitesimal non-compositionalitylin-0064
  4. 4.4 Why infinitesimal?lin-0065
  5. 4.5 The running equivariance examplelin-0066
  6. 4.6 Probe semanticslin-0067
  7. 4.7 Weil probes and higher orderlin-0068
  8. 4.8 Interactions and bracketslin-0069
  9. 4.9 Natural LINCS: intrinsic tangent repairlin-0070
  10. 4.10 From tangent diagnosis to learninglin-0071
  11. Further readinglin-0072
5 Infinitesimal Causality10 sections
  1. 5.1 A Three-Level Hierarchylin-0073
  2. 5.2 Statistical tangent semanticslin-0074
  3. 5.3 Smooth intervention protocolslin-0075
  4. 5.4 Bracket residuals and integrabilitylin-0076
  5. 5.5 Invariance and dependencelin-0077
  6. 5.6 Two counterexamples that fix the interpretationlin-0078
  7. 5.7 Linearized copy and discardlin-0079
  8. 5.8 The three IC compatibility testslin-0080
  9. 5.9 From IC to discoverylin-0081
  10. Further readinglin-0082
6 Infinitesimal Categorical Databases11 sections
  1. 6.1 From functorial databases to tangent instanceslin-0083
  2. 6.2 Instance lift, schema expansion, and constraint stabilitylin-0084
  3. 6.3 Vector fields as natural database sectionslin-0085
  4. 6.4 Why the tangent functor does not supply a Lie bracketlin-0086
  5. 6.5 A bracket-enriched CSQL schemalin-0087
  6. 6.6 Queries and data migration at tangent levellin-0088
  7. 6.7 From Democritus to BRIDGE and SKFMlin-0089
  8. 6.8 The six-stage database workflowlin-0090
  9. 6.9 Appearances, variations, and abductive repairlin-0091
  10. 6.10 Scope and mathematical boundarieslin-0092
  11. Further readinglin-0093
7 Infinitesimal Decisions11 sections
  1. 7.1 Decision making as universal extensionlin-0094
  2. 7.2 A tangent site for decisionslin-0095
  3. 7.3 Differentiating universal extensionlin-0096
  4. 7.4 The infinitesimal decision obstructionlin-0097
  5. 7.5 Decision-relevant quotientslin-0098
  6. 7.6 Nonsmooth and set-valued decisionslin-0099
  7. 7.7 The six-stage decision workflowlin-0100
  8. 7.8 Decision modalitieslin-0101
  9. 7.9 Relation to GIRLlin-0102
  10. Design lessonlin-0103
  11. Further readinglin-0104
8 Deep Learning in Non-Compositional Sketches13 sections
  1. 8.1 Why a deep specialization is neededlin-0105
  2. 8.2 Internal model spaces and parametrized mapslin-0106
  3. 8.3 Deep learning sketcheslin-0107
  4. 8.4 Transport through backpropagationlin-0108
  5. 8.5 Tangent lifting of parametrized moduleslin-0109
  6. 8.6 Natural DLINCS updateslin-0110
  7. 8.7 A four-level auditlin-0111
  8. 8.8 Weil probes beyond first orderlin-0112
  9. 8.9 The DLINCS compilerlin-0113
  10. 8.10 Five recurring deep shapeslin-0114
  11. 8.11 Scaling and implementation choiceslin-0115
  12. 8.12 What Deep LINCS addslin-0116
  13. Further readinglin-0117
9 Quotients, Localization, Repair, and Admission14 sections
  1. 9.1 Quotient before measuringlin-0118
  2. 9.2 Localizing factorization failureslin-0119
  3. 9.3 Compatibility is not effectivitylin-0120
  4. 9.4 The repair languagelin-0121
  5. 9.5 Repair as structural surgerylin-0122
  6. 9.6 Functorial and descent-compatible repairlin-0123
  7. 9.7 Observation profiles before scalarizationlin-0124
  8. 9.8 Admission is a separate decision problemlin-0125
  9. 9.9 Validated scalarizationlin-0126
  10. 9.10 Uncertainty and stochastic consistencylin-0127
  11. 9.11 Certificates and provenancelin-0128
  12. 9.12 Iterated repair and coalgebraic stabilizationlin-0129
  13. 9.13 The reusable contractlin-0130
  14. Further readinglin-0131
10 Reasoning by Structural Repair12 sections
  1. 10.1 Reasoning as the evolution of a maintained modellin-0132
  2. 10.2 The anatomy of a CoLT transitionlin-0133
  3. 10.3 Constructive epistemic statuslin-0134
  4. 10.4 Base, tangent, and transverse reasoninglin-0135
  5. 10.5 Local reasoning and global effectivitylin-0136
  6. 10.6 An audit record, not private chain of thoughtlin-0137
  7. 10.6.1 Chain-of-Evidence as an admission architecturelin-0138
  8. 10.7 SCoLT: the Toulmin specializationlin-0139
  9. 10.8 A compact SCoLT tracelin-0140
  10. 10.9 Minimal implementation contractlin-0141
  11. 10.10 Failure modes and open obligationslin-0142
  12. Further readinglin-0143

Applications

11 Geometric Causal Discovery10 sections
  1. 11.1 Three data regimes, three outputslin-0145
  2. 11.2 The BRIDGE learning sketchlin-0146
  3. 11.3 Influence and closure are separate observerslin-0147
  4. 11.4 Falsifying a latent interpretationlin-0148
  5. 11.5 Nonclosure as a discovery triggerlin-0149
  6. 11.6 SKFM: amortizing the geometrylin-0150
  7. 11.7 Repair and admissionlin-0151
  8. 11.8 What the experiments establishlin-0152
  9. 11.9 Design lessonlin-0153
  10. Further readinglin-0154
12 Repairing Kan-Extension Structure11 sections
  1. 12.1 The declared Kan diagramlin-0155
  2. 12.2 Base and tangent obstructionslin-0156
  3. 12.3 The repair languagelin-0157
  4. 12.4 Why a joint norm failedlin-0158
  5. 12.5 Blockwise admissionlin-0159
  6. 12.6 A small language-model testlin-0160
  7. 12.7 From commutators to a Čech calculationlin-0161
  8. 12.8 Frontier-model case study: Kimi K3lin-0162
  9. 12.9 Admission contractlin-0163
  10. 12.10 Design lessonlin-0164
  11. Further readinglin-0165
13 Composable Neural Adapters11 sections
  1. 13.1 Adapters as local intervention fieldslin-0166
  2. 13.2 Why the bracket controls orderlin-0167
  3. 13.3 The ALLORA objectivelin-0168
  4. 13.4 Training and repairlin-0169
  5. 13.5 Controlled and language-model evidencelin-0170
  6. 13.6 Capability and safety compositionlin-0171
  7. 13.7 Pretrained GPT-2 probelin-0172
  8. 13.8 Structured extension: LICKETlin-0173
  9. 13.9 Admission and limitationslin-0174
  10. 13.10 Design lessonlin-0175
  11. Further readinglin-0176
14 Skills over Lie Algebroids11 sections
  1. 14.1 Why a Lie algebroid?lin-0177
  2. 14.2 The workflow learning sketchlin-0178
  3. 14.3 Anchor and closure defectslin-0179
  4. 14.4 Bracket screening as repair searchlin-0180
  5. 14.5 Controlled benchmarklin-0181
  6. 14.6 Scaling the validation budgetlin-0182
  7. 14.7 From anonymous anchors to agent infrastructurelin-0183
  8. 14.8 Application probeslin-0184
  9. 14.9 Repair admissionlin-0185
  10. Design lessonlin-0186
  11. Further readinglin-0187
15 Infinitesimal Reinforcement Learning12 sections
  1. 15.1 The Bellman factorizationlin-0188
  2. 15.2 The tangent Bellman residuallin-0189
  3. 15.3 IL–GTD–MPlin-0190
  4. 15.4 Why the advantage quotient is necessarylin-0191
  5. 15.5 Natural GIRLlin-0192
  6. 15.6 Localization over Bellman coverslin-0193
  7. 15.7 Six layers of controllin-0194
  8. 15.8 The retained experimental sequencelin-0195
  9. 15.9 Safe recoverylin-0196
  10. 15.10 Global-to-local reconstructionlin-0197
  11. Design lessonlin-0198
  12. Further readinglin-0199

Relational Learning and Reasoning

16 Learning from Structured Preferences9 sections
  1. 16.1 The scalar-reward declarationlin-0201
  2. 16.2 Projection is not repairlin-0202
  3. 16.3 Tangent transport and reward gaugelin-0203
  4. 16.4 Localize before poolinglin-0204
  5. 16.5 Statistical admissionlin-0205
  6. 16.6 Relational fallbacklin-0206
  7. 16.7 What the experiments establishlin-0207
  8. Design lessonlin-0208
  9. Further readinglin-0209
17 Relational Manifold Recovery8 sections
  1. 17.1 A database is a diagramlin-0210
  2. 17.2 Foundries and the apexlin-0211
  3. 17.3 Infinitesimal relational descentlin-0212
  4. 17.4 What is exactlin-0213
  5. 17.5 Controlled relational recoverylin-0214
  6. 17.6 MovieLens as a conjunctive gatelin-0215
  7. Design lessonlin-0216
  8. Further readinglin-0217
18 Sheaf-Based Distributed Learning10 sections
  1. 18.1 Compatibility and effectivitylin-0218
  2. 18.2 Why a gluing loss is not the declarationlin-0219
  3. 18.3 The linear predictive-state modellin-0220
  4. 18.4 Tangent descentlin-0221
  5. 18.5 Transverse repairlin-0222
  6. 18.6 Restricted repairs and integrabilitylin-0223
  7. 18.7 The SID controllerlin-0224
  8. 18.8 Four foundry proofs of conceptlin-0225
  9. Design lessonlin-0226
  10. Further readinglin-0227
19 Infinitesimal Argument Repair6 sections
  1. 19.1 The Toulmin learning sketchlin-0228
  2. 19.2 Base and tangent argument obstructionslin-0229
  3. 19.3 Quotient, localization, and repairlin-0230
  4. 19.4 E\(_0\): deterministic structural perturbationslin-0231
  5. 19.5 Limits and evidential ladderlin-0232
  6. Further readinglin-0233
20 Trustworthy Foundation Models12 sections
  1. 20.1 From a model to a foundrylin-0234
  2. 20.2 The Odyssey–Prometheus division of laborlin-0235
  3. 20.3 Safety as an admission problemlin-0236
  4. 20.4 The six-stage workflow at foundry scalelin-0237
  5. 20.5 Three foundry case studieslin-0238
  6. 20.5.1 A GLP-1 health-research atlaslin-0239
  7. 20.5.2 A corporate 10-K foundrylin-0240
  8. 20.5.3 Product and review foundrieslin-0241
  9. 20.6 A compositional account of AI safetylin-0242
  10. 20.7 What the architecture does not guaranteelin-0243
  11. 20.8 Toward foundry-scale learninglin-0244
  12. 20.9 Further readinglin-0245

Synthesis

21 A Pattern Language for LINCS Systems7 sections
  1. 21.1 The declaration cardlin-0247
  2. 21.2 Six recurring patternslin-0248
  3. 21.3 The actionable-model patternlin-0249
  4. 21.4 The cross-application maplin-0250
  5. 21.5 A minimal experiment contractlin-0251
  6. 21.6 Certificates and reproducibilitylin-0252
  7. Further readinglin-0253
22 Frontiers of Infinitesimal Learning29 sections
  1. 22.1 Compositionality and compressionlin-0254
  2. 22.1.1 The jump changes the declarationlin-0255
  3. 22.1.2 Can infinitesimal learning jump?lin-0256
  4. 22.1.3 Extending the workflow from repair to discoverylin-0257
  5. 22.1.4 Generative closure and functorial leapslin-0258
  6. 22.1.5 Duality as discovery between theorieslin-0259
  7. 22.1.6 AM and degrees of mathematical noveltylin-0260
  8. 22.2 What is fixed, and what may be learned?lin-0261
  9. 22.2.1 Theory repair as sketch augmentationlin-0262
  10. 22.2.2 Appearance, reality, and theoretical inventionlin-0263
  11. 22.3 Adaptive sites and learned coverslin-0264
  12. 22.4 Higher interaction signatureslin-0265
  13. 22.5 Homotopical repairlin-0266
  14. 22.6 Completion and coalgebraic stabilizationlin-0267
  15. 22.7 Reflective repair and reverse transportlin-0268
  16. 22.8 Topology of repair spaceslin-0269
  17. 22.9 Statistical and language-valued tangent siteslin-0270
  18. 22.10 Mechanizing the declaration-to-certificate pathlin-0271
  19. A staged research agendalin-0272
  20. From causal models to actionable learning modelslin-0273
  21. The continuing LINCS questionlin-0274
  22. Further readinglin-0275
  23. Referencesreferences
  24. Glossary of Notationlin-0276
  25. Categorical declarationslin-0277
  26. Obstruction, observation, and scalarizationlin-0278
  27. Tangents, probes, and infinitesimal structurelin-0279
  28. Repair and admissionlin-0280
  29. Local-to-global, causal, and decision notationlin-0281