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Category Theory for AGI
Choose a familiar machine-learning pipeline and type every component as an object or morphism. State one equality of paths the system is intended to satisfy, then construct a countermodel in which every component runs but that equality fails. What information does the typed failure retain that a scalar task loss discards?
Functors for AGI
Choose two representations of the same learning workflow and propose a functor between them. List the objects, arrows, identities, and compositions it must preserve, then exhibit one plausible translation that fails functoriality. What empirical behavior would reveal that the failed preservation law matters rather than being merely notational?
Representable Functors and the Yoneda Lemma
Select a model state that cannot be inspected directly and define a family of probes into or out of it. Use the Yoneda perspective to state when two candidate states are observationally indistinguishable. Which additional probe would separate them, and what assumptions are required before the probes determine the state up to isomorphism?
Diagrams and Universal Constructions
Model one data-integration problem twice: first as an optimization objective and then as a limit, colimit, pullback, or pushout. Identify the universal property and construct a competing object that satisfies the local constraints but not universality. What practical guarantee is gained by the universal construction, and when is that guarantee too strong?
Categorical Deep Learning
Represent a small neural architecture as a compositional diagram, then specify an architecture change as a natural transformation or other typed map between diagrams. Which squares must commute for the change to preserve behavior? Can you give a counterexample showing why matching input and output tensor shapes alone does not establish compositional compatibility?
Diagrammatic Backpropagation
Choose a network with two paths that should implement the same transformation and train it with ordinary backpropagation and with an additional diagrammatic obstruction. Hold parameter count and optimization budget fixed; report task loss and pathwise residual separately. Does the structural signal improve generalization, diagnosis, or neither on a held-out compositional shift?
Geometric Transformers
Construct a relational dataset whose geometry is known before training. Compare a Geometric Transformer with a parameter-matched attention baseline while ablating positional, convolutional, and geometric components. Evaluate prediction and recovery of the registered geometry independently. Which component supplies information unavailable to the baseline, rather than merely increasing effective capacity?
Dynamic Compositionality
Define a learning diagram whose objects or commutativity obligations change over time. Specify how an observer distinguishes parameter drift, structural drift, and a genuine change of theory. Can a repair rule track the changing declaration without repeatedly forgetting previously admitted structure, and what counterexample would falsify the proposed stability criterion?
Information Regimes in Geometric Transformers
Create a task with registered local, relational, and global information requirements, then reveal these sources to a Geometric Transformer in controlled stages. Measure not only accuracy but the recovered relational geometry and calibration. Are observed regime transitions reproducible across seeds and widths, or can they be explained by ordinary changes in sample complexity?
Kan Extension and Topological Coend Transformers
Express one structured aggregation first as a Kan extension and then as a coend or quotient construction. State precisely which indexing category, equivalence relation, and universal property are being used. Under what conditions do the two presentations agree, and what observable failure occurs when the required identifications are imposed too aggressively?
Structured Language Modeling
Design a language task with an explicit compositional grammar and a held-out split requiring novel combinations rather than familiar strings. Compare a structured model with a capacity-matched language model on likelihood, grammatical validity, and recovery of the declared relations. Does better prediction coincide with better structural generalization, or can the two measures diverge?
Manifold Learning with Geometric Transformers
Generate observations from a manifold with known dimension, neighborhoods, and topology, then hide the generating coordinates. Compare Geometric Transformer embeddings with UMAP and another registered baseline using topology and neighborhood preservation as well as visualization. Which conclusions remain stable under changes of metric, sample density, and random seed?
Mean-Field Theory of Geometric Transformers
Choose one simplifying assumption used in a mean-field description of a Geometric Transformer and derive the corresponding limiting recurrence. Construct a finite-width or correlated-input regime in which the assumption fails. Which statistic should depart first from the theory, and how would you distinguish finite-sample error from a genuinely invalid approximation?
Depth Sweeps for Geometric Transformers
Preregister a depth sweep that holds training tokens, parameter budget, optimizer effort, and evaluation protocol as nearly fixed as possible. Measure task performance together with oversmoothing, geometry preservation, and compute. Is there evidence for a structural depth transition, or do the results reduce to parameter count, optimization difficulty, or selection among noisy runs?
Adjoint Functors
Identify a free-forgetful or extension-restriction pair in a learning problem and write the hom-set correspondence explicitly. Check naturality in both arguments rather than relying on an informal duality. What modeling decision is encoded by the unit and counit, and which causal or decision interpretation would require assumptions not supplied by the adjunction alone?
Causal Claims from Language
Design a held-out comparison between Democritus and UniCausal on a corpus neither system helped select. Register claim-span, direction, modality, and entity-resolution errors separately; require blinded human adjudication for disagreements. Which errors alter the induced causal structure rather than only the wording of an extracted claim?
Temporal Diffusion over Causal Trajectories
Hold out later filings and entire companies while estimating temporal causal trajectories from earlier documents. Compare the diffusion representation with a static embedding and a simple temporal baseline. Evaluate forecast calibration, change-point localization, and stability under paraphrase separately. Which apparent brand motions survive these controls and independent review of the source claims?
Building Agentic Systems using Kan Extension Transformers
Select a workflow with partially missing tools or evidence and compare a Kan-extension-based agent with a fixed workflow and an unconstrained planner. Register task completion, unsupported actions, provenance continuity, recovery from missing components, latency, and cost. Does universal extension improve reliable composition, or merely provide another planning heuristic?
Topos Causal Models
Build a small presheaf or sheaf model whose contexts expose different causal variables, and represent a proposed intervention as a declared structural operation. Separate what follows from the topos construction from what requires an intervention mechanism or identification assumption. Can two observationally compatible models disagree after intervention, and where is that disagreement represented internally?
Judo Calculus
Generate causal systems with known interventions, hidden variables, and finite-sample uncertainty, then compare Judo Calculus with a registered causal-discovery baseline. Score orientations, abstentions, and invalid causal claims separately. Which gains depend on information encoded in the categorical construction, and which disappear when the same information is supplied to the baseline?
Csql: Mapping Documents into Topos Causal Model Databases
Design a CSQL schema for a small scientific corpus containing contradictory, qualified, and temporally scoped causal claims. Write one query whose answer preserves document and span provenance, then perform a schema migration that adds a new claim type. What must commute so the migrated database does not silently change the meaning of earlier answers?
Homotopy in Language and Causal Inference
Construct two linguistic or causal derivations that have the same endpoints but different intermediate paths, and propose a 2-cell witnessing when they should count as equivalent. Which transformations should preserve the equivalence class? Can you give a counterexample where collapsing the paths loses scientifically relevant provenance or intervention information?
Model Categories for Causality and Language
For a finite family of causal or linguistic models, propose candidate weak equivalences, fibrations, and cofibrations and test the model-category axioms on explicit examples. Which choices encode meaningful invariance rather than convenient formal closure? Can you identify one lifting property whose causal interpretation fails without additional semantic assumptions?
Predictive State Representations in a Topos
Cover a partially observed dynamical system by local observation contexts and assign a predictive state to each context. Specify restriction maps and a gluing condition, then construct locally accurate states that fail to glue globally. What additional experiment or overlap registration would distinguish genuine hidden state from an incompatible family of local predictors?
Causal Density Functions
On a system where observational and interventional measures are both available, estimate their causal density comparison with uncertainty intervals and a null calibration. Vary intervention strength and support overlap. Which discrepancies can be reported without claiming causal identification, and under what conditions does the estimator become unstable or semantically uninterpretable?
Universal Decisions with Kan Extensions
Choose a finite decision problem and explicitly compute an extension followed by consistency along an information functor. State what each Kan extension guarantees and what it does not. Can two different action mechanisms induce the same observational restriction, and what extra datum is required before interpreting the left extension as an intervention?
Universal Reinforcement Learning
Express an MDP, POMDP, and predictive-state model as coalgebras of suitable endofunctors. Identify the morphisms that preserve behavior and the assumptions hidden by each encoding. Which additional stochastic coalgebraic constructor would describe a sequential system excluded by all three, and what learnable prediction would that extension make?
Deep URL with Geometric Transformers
Train Deep URL and matched recurrent or transformer baselines on several sequential systems, then evaluate transport to a held-out coalgebraic family. Keep observation and optimization budgets fixed. Does the categorical representation improve behavioral equivalence, sample efficiency, or out-of-family transfer, and which ablation isolates the responsible constructor?
Consciousness
Propose the weakest categorical structure that distinguishes a system with an inspectable self-model from one that merely predicts its own outputs. Give an observable consequence that could refute the distinction. Which part of the proposal is mathematics, which part is an operational definition, and which part remains philosophical interpretation?
Universal Imitation Games
Define an imitation game with multiple typed observers who can test behavior, explanation, intervention response, and transfer. Register which observations are hidden from each participant and how judgments are aggregated. Can a system pass all local observers yet fail a global consistency condition, and what would that reveal beyond a conventional Turing-style evaluation?
Formal Verification Map
Select one informal proposition from an earlier chapter and state a proof-assistant-ready theorem with all objects, morphisms, typeclass assumptions, and boundary conditions explicit. Identify the smallest reusable lemmas needed. Which part can be certified deductively, and which empirical or semantic premise must remain outside the formal kernel?
CLIFF Companion
Create a blinded question set containing direct facts, cross-chapter syntheses, deliberately false premises, and questions beyond the book. Compare CLIFF with an ungrounded language model on answer accuracy, citation precision, calibrated refusal, and correction behavior. Which failures come from retrieval, mathematical reasoning, or limitations in the source text itself?
Code Companion
Choose one empirical claim in the book and construct a complete reproduction trace from stable chapter tag to experiment-family record, code entry point, configuration, source data, generated artifact, and reported conclusion. Which links are public, staged, or unavailable, and does the executable evidence license the exact wording used in the chapter?
A Working Language of Compositionality
Draw an equivariant encoder square for a transformation relevant to an application you know, then tangent-lift the declaration. Identify the base and tangent path equalities and the observer that could detect each failure. Can base equivariance hold while tangent equivariance fails?
Axioms of Structural Learning
Choose one LINCS axiom and construct a minimal model satisfying the remaining axioms while violating the selected one. State whether the countermodel is mathematical, computational, or semantic. Which observable learning failure does the missing axiom permit, and does that evidence support independence or only reveal an incomplete formalization?
Learning by Repair
Design a small task in which parameter adjustment can reduce loss without repairing a registered compositional failure. Compare parameter-only training with the six-stage LINCS workflow while holding observations and compute fixed. Does declaration, localization, or admission add the decisive information, and what outcome would show that structural repair is unnecessary?
Learning Sketches and Factorization Obstructions
Present a familiar learning system as a finite sketch with one nontrivial path equation, limit cone, or factorization obligation. Construct an implementation that satisfies every local type check but violates that declaration. What is the smallest typed obstruction that localizes the failure, and which repairs preserve the rest of the sketch?
Tangent Learning Sketches
Start with a commuting base diagram and choose a Weil probe or tangent variation for each relevant object and arrow. Compute the lifted path residual and identify its null directions. Can the base declaration hold while the tangent declaration fails, and what observer semantics are needed before treating that failure as a learning signal?
Infinitesimal Causality
Construct a dynamical system with known intervention vector fields, including one commuting pair, one noncommuting pair, and a latent direction. Estimate Lie brackets from finite trajectories with uncertainty. Which closure residuals reliably distinguish mechanism interaction from estimation noise, and when must the observer abstain rather than infer causal structure?
Infinitesimal Categorical Databases
Design a categorical database schema whose base instance stores a scientific model and whose tangent instance stores admissible variations or intervention fields. Write one query that detects a bracket or constraint defect, then migrate the schema to represent its repair. Which integrity constraints must be transported for earlier query answers to remain meaningful?
Infinitesimal Decisions
Take a finite decision rule expressed through left and right Kan extensions and tangent-lift its information, action, and consistency maps. Separate variation of evidence from variation of the action mechanism. Under what assumptions can their comparison guide a decision repair, and which part remains merely universal rather than causal?
Deep Learning in Non-Compositional Sketches
Build a deep model with a declared structural equality that can be violated independently of task accuracy. Compare ordinary training, a scalarized obstruction penalty, and a typed DLINCS repair. Evaluate nonsmooth cases such as ReLU separately. Does retaining the obstruction's type improve localization or transfer beyond what an equally weighted auxiliary loss provides?
Quotients, Localization, Repair, and Admission
Construct a sketch with several apparent defects generated by one local obstruction. Compare quotienting, localization, and declaration extension as candidate repairs, stating exactly what each identifies or inverts. Which independent admission test prevents a formally compatible repair from erasing a scientifically meaningful distinction?
Reasoning by Structural Repair
Create reasoning problems that require maintaining a declaration, detecting a contradiction, and revising one warrant while preserving unrelated conclusions. Compare CoLT traces with ordinary chain-of-thought outputs using blinded correctness, localization, unsupported-repair, and auditability judgments. Does the typed trace prevent errors, or merely make them easier to diagnose after generation?
Geometric Causal Discovery
Construct a synthetic intervention family with a known latent confounder and finite-sample uncertainty. Compare bracket-residual screening with an observational baseline, but score candidate recall, false exclusions, calibration, and abstention separately. Under what intervention budget does the geometric diagnostic add information rather than merely re-rank correlations?
Repairing Kan-Extension Structure
Construct an extension task with registered missing blocks and incompatible overlaps, then compare LINCS-KET with unconstrained completion and a fixed Kan-extension implementation. Score completion, comparison-map defects, preservation of observed blocks, and abstention. Which failures require repairing the declaration rather than estimating a better extension within the original diagram?
Composable Neural Adapters
Train separately useful low-rank adapters whose two composition orders disagree on a held-out task. Compare naive merging, sequential application, and ALLORA repair while matching rank and inference cost. Does enforcing the declared square improve joint behavior without erasing individual skills, and how often should the method abstain from composition?
Skills over Lie Algebroids
Define a skill family with known useful, redundant, and interfering edits, then compare ordinary search with LASKO bracket-guided selection under the same evaluation budget. Measure accepted improvement, wasted trials, bracket-estimation cost, and transport to a new task. When do inexpensive local brackets predict genuinely useful global edits?
Infinitesimal Reinforcement Learning
Create an environment where two policies have similar returns but different tangent Bellman residuals under controlled transition perturbations. Compare GIRL with a return-matched RL baseline on adaptation and calibration. Does the tangent signal predict useful policy repair before reward reveals the change, and under what probe semantics does it become misleading?
Learning from Structured Preferences
Collect preferences over outputs that differ along independently registered criteria, including deliberately inconsistent comparisons. Compare scalar reward modeling with LINCS-RLHF while holding annotations fixed. Evaluate held-out preference prediction, contradiction localization, criterion preservation, and abstention. Does structural preference modeling reveal conflicts that a sufficiently flexible scalar model conceals?
Relational Manifold Recovery
Generate a relational manifold with registered neighborhoods, symmetries, missing edges, and out-of-distribution regions. Compare RADAR with geometric and nongeometric embedding baselines on topology recovery, uncertainty, and downstream transfer. Which repairs recover true relations rather than merely produce a visually smoother embedding?
Sheaf-Based Distributed Learning
Partition a learning problem among agents with overlapping but partially inconsistent observations. Compare independent training, centralized aggregation, and SID descent under matched communication. Measure local accuracy, overlap residuals, global gluing, and robustness to one corrupted agent. When does a sheaf obstruction identify disagreement that averaging cannot safely resolve?
Infinitesimal Argument Repair
Construct Toulmin arguments with localized missing warrants, incompatible evidence, and perturbations that should not change the claim. Compare textual revision with typed infinitesimal argument repair. Evaluate claim preservation, defect localization, unsupported warrant invention, and human-audited validity. Does tangent structure guide minimal repair, or only restate the argument schema?
Trustworthy Foundation Models
Choose a foundry task requiring extraction, synthesis, and evidence-backed recommendation across several documents. Compare an unconstrained agent with a Prometheus or Odyssey-style compositional workflow. Score answer quality, claim provenance, cross-stage consistency, correction behavior, latency, and cost. Which integrity declarations prevent consequential hallucinations rather than merely improve presentation?
A Pattern Language for LINCS Systems
Select an application not covered by the book and instantiate the pattern language with explicit declarations, probes, obstruction types, repair operators, observers, and admission tests. Which existing pattern transports unchanged, which must be specialized, and what evidence would justify adding a genuinely new pattern to the catalog?
Frontiers of Infinitesimal Learning
Design a declaration-acquisition protocol for a domain in which compositional obligations are not supplied in advance. Separate imitation, corrective apprenticeship, obstruction discovery, and admission. What evidence would show that the learner acquired a reusable sketch rather than memorized the demonstrations?
A Working Language for Creative Theory Change
Choose a domain with a familiar theory and distinguish its presentation, models, observations, probes, and admissible changes. Construct one failure that can be repaired within the theory and another that requires extending the presentation. What evidence prevents an arbitrary re-description from being counted as creative theory change?
Initial Sketch Acquisition
Give two learners the same expert demonstrations, but allow only one to receive corrective feedback on its attempted actions. Compare the sketches they recover on held-out tasks that change surface appearance while preserving function. Which relations are identifiable from observation alone, and which require apprenticeship?
Creativity as Theory Construction
Withhold a registered relation or constructor from a synthetic theory and supply observations that cannot be explained compactly without it. Compare fixed-language fitting, unrestricted search, and typed theory extension. Can the learner recover a reusable extension, transport previous models, and abstain when the evidence does not identify a unique construction?
The Differential Geometry of Creativity
Define assimilation and accommodation vector fields on a small theory manifold and compute their bracket or an appropriate abstract substitute. Quotient out motions already generated by the two side directions. Which persistent residual indicates an incomplete declaration rather than ordinary noncommutativity, numerical noise, or a poor choice of coordinates?
Skill Optimization for Creative Exploration
Specify a creative search policy as separately trainable proposal, diagnosis, and admission skills. Compare joint optimization with staged DIAL-SkillOpt training under a fixed trial budget. Measure successful extensions, wasted edits, false admissions, and transfer. Does Lie-bracket-guided selection improve discovery, or only concentrate search around already useful skills?
Algebraic Theories as Creative Targets
Present a small Lawvere theory, finite-limit theory, or PROP by generators and relations, then add one candidate generator with equations. Compute or describe model transport along the inclusion. What conservativity, irreducibility, and productivity tests would justify calling the extension a new theory rather than redundant notation?
Landscapes of Computational Creativity
Classify one historical or computational case using Boden's combinational, exploratory, and transformational modes and Piaget's assimilation-accommodation distinction. Identify what is lost in this translation rather than forcing a perfect correspondence. Which observable evidence would distinguish the competing classifications, and which judgment remains dependent on historical or cultural interpretation?
Combinational Creativity
Register a vocabulary of components and composition rules, then hide a family of valid but unseen combinations. Compare random recombination, language-model proposal, and typed compositional search on novelty, validity, usefulness, and diversity. Which results are genuinely new combinations rather than memorized variants or violations that happen to score highly?
Exploratory Creativity
Choose a fixed symbolic language with known simplifying symmetries and generate problems spanning its interior and boundary. Compare passive search with active infinitesimal probes. Does the probe policy discover valid expressions more efficiently, recognize when the language is insufficient, and avoid falsely claiming transformational creativity within a fixed hypothesis space?
Transformational Creativity
Create a registered world whose observations cannot be represented by the initial theory but become simple after adding one withheld generator. Compare DIAL with fixed-language and unrestricted program-synthesis baselines. Does it localize the obstruction, recover the missing generator, conserve prior results, and reject unsupported extensions?
DIAL as an Algorithmic Family
Select two DIAL-X instantiations and preregister a common ablation of assimilation, accommodation, core interaction, declaration extension, and admission. Which components are reusable across domains, which are application-specific, and does any claimed creative gain disappear when proposal budget and observer information are matched to simpler baselines?
Causal Learning with Infinitesimal Creativity
Generate intervention operators from a causal family containing registered, unsupported, and withheld alternatives with finite-sample uncertainty. Compare CLIC with BRIDGE/SKFM restricted to the original theory. Can CLIC actively seek counter-witnesses, propose the missing causal structure, calibrate uncertainty, and abstain when several extensions remain observationally equivalent?
Skill Optimization with Infinitesimal Creativity
Construct a skill world where improvement requires inventing one operation absent from the initial edit vocabulary. Compare OPTIC with LASKO and an unrestricted proposal model under the same evaluation budget. Does OPTIC distinguish optimization of known skills from justified skill-language extension, and can the invented operation transfer to a held-out task?
Reinforcement Learning with Infinitesimal Creativity
Use a partially observed environment whose transition theory changes and whose successful policy requires a new reusable schema. Compare RELIC with model-free RL, GIRL, and a schema learner given matched trajectories. Does RELIC acquire and admit the missing structure, improve planning on disjoint tasks, and avoid accommodation when assimilation suffices?
Composing Creative Workflows
Compose causal discovery, skill optimization, reinforcement learning, and formal admission in two different orders on a task requiring all four. Register interface contracts and boundary cases before allocating experiments. Does AGENTIC composition produce an extension none of the components can construct alone, and which order effects are explained by information flow rather than additional compute?
Mathematical Theory Construction
Withhold one algebraic constructor from a registered theory family and generate tasks whose shortest explanations require it. Compare DIAL-style extension with search in the fixed language and unrestricted program synthesis. Can the method recover the constructor, transport old models, and succeed on new tasks not used to propose it?
Simulator-Grounded Theory Extension
Hide one causal mechanism in a simulator and provide observations generated under several candidate interventions. Compare passive fitting with simulator-grounded theory extension under equal simulation budgets. Can the system propose a discriminating intervention, recover the withheld mechanism, reject attractive but wrong theories, and quantify when the simulator cannot identify the extension?
From Scientific Documents to Testable Theories
Give the system a time-bounded scientific corpus with explicit provenance and hold out later evidence. Require a theory sketch, unresolved contradictions, and a feasible discriminating experiment or grant-style research program. Can independent reviewers trace every premise, judge the experiment informative, and distinguish scientific synthesis from fluent proposal boilerplate?
Generative Visual Languages
Compare repair-after-generation with constraint-aware generation on the same scene sketches across several image providers. Register structural validity, preservation of protected content, abstention accuracy, edit count, latency, and cost. Does an acquired visual theory improve first-pass generation, or only make subsequent repair more reliable?
Artistic Theory Extension
Withhold a reusable rendering operator from an initial paintbrush language and supply artistic intents that require it across images or video. Compare DILATE with prompting, style transfer, and unconstrained latent optimization. Does the recovered operator form a coherent family, survive transport, preserve declared content, and offer a capability or efficiency gain beyond novelty alone?
Creativity as Compositional Theory Change
Choose a creative domain omitted from the book, such as music, choreography, materials, or social institutions. State what a theory, double variation, obstruction, and admission test could mean there, then identify where the DIAL analogy breaks. What alternative framework would better capture the missing aspect of creativity?
The Infant's Problem
Choose one domain of infant learning and write the smallest prior that makes its compositional discovery plausible. Which observations support assimilation within that prior, and which would force accommodation of the prior itself?
A Categorical and Homotopical Toolkit
Present a small doctrine by its objects, morphisms, models, and designated universal constructions. Build its nerve and exhibit one inner horn whose possible fillers encode genuinely different hypotheses.
The ORACLE Program
Formulate categorical identification for a familiar learning problem. Specify the presentation stream, query language, equivalence notion, and success criterion, then show why literal recovery of a hidden category is either too strong or not invariant.
Categorical Identification Under a Prior
Give two observationally indistinguishable worlds under one probe family and a refined probe that separates them. Which part of the resulting identification theorem is categorical, and which part is supplied by the prior?
Core Knowledge and Piagetian Construction
Translate one core-knowledge system into a categorical doctrine and propose a falsifying developmental observation. Does the doctrine constrain possible learning trajectories, or merely redescribe a mature competence?
Learning Inside Fixed Doctrines
Recast a learning method omitted from the chapter as UOCL within a fixed doctrine. Identify its objects of prediction, allowed tests, morphisms, and hidden closure assumptions; then name a world it cannot represent without doctrine change.
The UOCL Machine
Implement on paper one typed UOCL update for a finite presentation. Track transcript growth, hypothesis reindexing, query answers, contradictions, and repair so that every state transition has an auditable categorical type.
Persistent Categorical Identification
Specify when a learner has identified not only a category but compatible tangent structure. Which probes can distinguish two tangent structures on the same base category, and what persistence certificate survives reindexing?
Probably Approximately Categorically Correct Learning
Choose a finite categorical hypothesis class and define a probability-weighted discrepancy that respects the intended equivalence. Derive a PACC sample bound or show precisely which compositional dependence invalidates the classical PAC argument.
Online and Persistent Universal Decision Learning
Build a partial decision nerve for a three-step online problem and leave one inner horn unresolved. Compare two fillers and state what observer is needed before their difference can be interpreted as regret.
Convex and Regularized Universal Action
Instantiate tangent proximal FTRL in a concrete smooth category. Determine which convergence statement is inherited from the scalar realization and which coherence obligations must be proved separately after tangent lift.
Universal Bandit Decision Learning
Define a bandit doctrine in which feedback reveals only a selected component. What categorical substitute for importance weighting yields an unbiased or universally characterized reconstruction, and when is such reconstruction impossible?
Information Categories and Decentralized Decisions
Model a small asynchronous team without a global clock. Draw its information category, identify incomparable events, and compare the admissible policies with those obtained after imposing an artificial synchronous ordering.
Agentic Safety and Universal Online Decision Learning
Specify a temporal trust contract for a three-agent workflow with partial information. Exhibit one locally reasonable action that violates the global contract and design the smallest monitor or information refinement that prevents it.
Homotopy and Tangent Repair
Construct a repair problem with two homotopic local paths but incompatible finite realizations. State the horn-filling, tangent, and preservation conditions needed to admit one repair without confusing local coherence with global correctness.
Invariance, Amplification, and Approachability
Choose a weak invariant and a composition law under which repeated use amplifies rather than destroys it. Prove a closure or approachability statement, or construct a counterexample showing why the proposed invariant is not persistent.
Persistent Structure Across a Lifetime
Design a ledger for a reusable structure learned in one task and transported to two unrelated tasks. Record the interfaces, invariants, failed transports, and doctrine changes needed to distinguish genuine lifelong utility from repeated fine-tuning.
Synthesis and the Next ORACLE Theorem Ladder
Select one open rung in the ORACLE theorem ladder and state the weakest theorem worth proving. List its assumptions, observables, invariance class, failure modes, and the result required before advancing to the next rung.
Editorial boundary
These web-indexed prompts continue the research program beyond the immutable dated PDF and HTML editions. They are invitations to investigate, not claims that the proposed constructions or experiments already succeed.