22 Glossary of Notation

A declared conceptual space or algebraic theory.

A current realization or instance of a declared theory.

A versioned collection of scientific documents and structured sources in Chapter 17; distinguished from the theory realization \(D\).

The provenance-bearing presentation extracted from \(\mathsf D\): candidate objects, arrows, contexts, equations, evidence types, and source-span links.

A source-bearing sketch obtained by reconciling the local presentation \(P_{\mathsf D}\).

The ontology- and extractor-relative family of candidate source-bearing sketches, support values, and provenance maps compiled from a document collection. It is not assumed to be a unique functor from prose.

The admission interface compiled from a document-grounded theory: an observation protocol, intervention, simulator, manipulable model, or locked evidence query with discriminating outcomes.

A family of observers through which failures or consequences become detectable.

A maintained representational package and its differential refinement \((\mathbb S,\mathfrak D,\mathcal M,F,\Lambda ,\mathcal G,\mathcal A)\).

Initial sketch-acquisition evidence: explicit instruction, expert demonstrations, corrective feedback, and the learner’s own interventions.

An acquired provisional representational package, a current realization typed by it, its evidential support and provenance record, and its unresolved or unsupported alternatives.

Open-loop sketch acquisition from observed expert demonstrations, without corrections conditioned on the learner’s own attempts.

Closed-loop sketch acquisition in which a teacher observes the learner’s attempts and supplies adaptive corrections, contrasts, explanations, or counterexamples.

A double repair package: a differential representational package together with a candidate DIAL object and, when supplied, its classical cochain realization.

Respectively, the preservation doctrine, semantic universe, generative operations, and admission procedures.

The complete DIAL-X algorithmic contract: target, creative mandate when applicable, maintained realization, two repair theories and their operations, observers, mixed witness and localization record, finite-realization test, proposal language, less-disruptive control suite, package comparison, admission rule, and memory. It is distinct from the preservation doctrine \(\mathfrak D\).

The observer-relative localization record for the mixed witness \(\Theta _X\). It is distinct from the semantic-transport functor \(K\), and may be empty when the registered observers do not support a meaningful localization.

A proposed extension or transformation of \(\mathbb S\).

A candidate presentation-extension map, \(J:\mathbb S\to \mathbb S^+\), together with an interpretation of its new generators and relations. It records the presentation component of a proposed package change, not by itself the semantic, observer, doctrine, or admission comparison, and it remains proposed until transport and consequences are independently admitted.

The package comparison for DIAL-X, recording changes to the presentation, semantics, doctrine, observers, generative operations, proposal grammar, or admission interface. It is the identity for an unchanged package; a sketch map \(J_X\) is one component when the presentation changes.

A localized obstruction motivating a possible theory change.

A finite-realization record or integration observer testing whether a candidate local repair persists and composes coherently beyond its diagnostic neighborhood.

The status-bearing extension dossier containing prior and proposed representational packages and realizations, obstruction, finite-realization record, componentwise package comparison \(\Upsilon \), transport, predictions, and epistemic status. A theory map \(J\) appears when the presentation or generative language changes.

Leung’s monoidal theory of algebraic probe shapes classifying ordinary tangent structure. It is distinct from the cochain Weil algebra of a Lie algebroid.

The strong monoidal functor realizing Weil probes as endofunctors of a conceptual universe.

The proposed internal interchange defect between two tangent-categorical algebroid directions. It is meaningful before choosing a classical cochain realization.

A supplied cochain realization sending the proposed internal interchange defect to its classical graded-commutator observer, \(\mathsf{Real}(\Theta _{\mathrm{int}})=\Omega _{\mathrm{cre}}\). It is distinct from the frozen DIAL-X reference space \(\mathcal R_X\).

The proposed category of Double Involution ALgebroids internal to a suitable tangent category \(\mathcal M\); constructing it and proving its realization properties are open objectives of this book.

An algorithmic specialization of DIAL to a base system \(X\), defined by an operational target, structural declaration, two typed repair operations, mixed witness, finite-realization observer, optional creative mandate, proposal language, less-disruptive controls, package comparison, admission contract, and persistent memory.

A training method for an external, typed discovery skill. A frozen agent executes the skill over DIAL states and curve-object flows; a separate optimizer accepts bounded edits by strict held-out validation improvement; a locked split supplies final independent admission.

Causal Learning with Infinitesimal Creativity: a DIAL-X realization that uses uncertain intervention geometry, active counter-witnesses, explicit unsupported alternatives, and typed causal package comparison. A causal-sketch map is required only when the presentation changes.

Optimizing Programs Through Infinitesimal Creativity: a DIAL-X realization that combines involution-algebroid skill geometry, versioned external programs, obstruction-triggered workflow construction, and independent execution admission.

Reinforcement Learning with Infinitesimal Creativity: a DIAL-X realization that augments policy learning with obstruction-triggered construction, independent admission, persistence, and transport of decision structure.

Acquiring Generalizable Novel Theories with Infinitesimal Creativity: the proposed integrated architecture that composes CLIC, OPTIC, and RELIC through typed interfaces, structural packets, and independent component and interface admission.

Acquiring Representational Theories through Infinitesimal Semantic Transport in Image Creation: the AGENTIC text-to-image realization that couples generative transport to visual-language diagnosis, repair, bounded extension tests, and independent structural admission.

The conceptual and experimental bridge from CAN-style statistical deviation to DIAL-based artistic theory extension. Its archived bracket and paintbrush experiments retain this prefix for provenance.

Double Involution Learning for Artistic Theory Extension: the proposed paintbrush-discovery architecture that projects mixed obstructions away from registered visual operations, proposes reusable trajectory techniques, transports artistic intent through local observer geometry, and submits visual-theory extensions to typed admission.

The artistic obstruction quotient on a regular model stratum: local visual-model directions modulo those generated by registered assimilatory and proto-accommodative operations. When anchor rank varies, a stratified or sheaf-theoretic replacement is required.

The quotient map that removes directions already generated by the registered assimilatory and proto-accommodative anchors. It is distinct from the query component \(q\) of a structural packet.

The artistic closure obstruction \(q_{\mathrm{art}}([\rho _A(a),\rho _B(b)])\), retaining only the mixed bracket component that the current visual theory cannot express. On a constant-rank stratum it is tensorial in \(a\) and \(b\).

A candidate artistic-generator bundle, its anchor into observable model-space directions, and a local section. The proposal condition is \(q_{\mathrm{art}}(\rho _C(c))\simeq \Omega _{\mathrm{art}}(a,b)\); this quotient equality does not itself choose a canonical lift or type the new generator.

The unit comparing an old artistic model with the restriction of its proposed left-Kan-extended transport. Conservativity requires this map to meet the registered equivalence standard; existence of the Kan extension alone is insufficient.

A natural family of lifted quotient residuals that remains nontrivial, directionally and locally coherent, and observer-robust across a registered family of Weil probes.

The meta-sketch presenting legal interfaces among the current CLIC causal sketch, OPTIC skill sketch, and RELIC decision sketch.

The typed exchange object \((\tau ,\mathfrak R,\Upsilon ,\omega ,U,\gamma ,h,q,c,s,\kappa ,v)\) carrying packet type, prior representational package, proposed package comparison, localized obstruction, uncertainty, finite-realization record, registered-control record, query, status-bearing admission record, epistemic status, interface contract, and version between creative workflows. A changed sketch map \(J\) is stored as the presentation component of \(\Upsilon \).

The descriptive three-way factorial interaction among CLIC, OPTIC, and RELIC; evidence of complementarity or interference, but not by itself a certificate of creativity.

A declared reinforcement-learning problem, including state and action types, transition and reward interfaces, discount convention, and observer.

The RELIC target card \((Y_{\mathrm{dec}},\mathbb S_{\mathrm{dec}},\mathcal C_{\mathrm{dec}})\): deployable decision object, structural decision declaration, and epistemic certificate family.

The three-level target of DIAL-X: deployable object \(Y_X\), structural declaration \(\mathbb S_X\), and epistemic certificate family \(\mathcal C_X\).

A versioned DIAL-X artifact dossier: operational construction, package comparison, transport record, status-bearing admission record produced from independent evidence, and persistent provenance or replay state. The sketch map \(J_X\), when present, is the presentation component of \(\Upsilon _X\). A rejected or unsupported proposal can retain a dossier without being admitted.

The creative mandate of DIAL-X when creativity is claimed: Boden mode, frozen reference space, novelty observer, and surprise or explanatory-gain measure. Value remains part of the independent epistemic certificate.

The named difference between LINCS-X and Creative DIAL-X: the frozen language and reference repertoire, the visible obstruction, the artifact or rule that may be constructed, and the independent test of value. Combinational and exploratory deltas may remain within the language; transformational deltas change it.

The preregistered interface declaring a DIAL-X target, side semantics, mixed witness, observability gate, finite-realization test, creative mandate when applicable, proposal language, less-disruptive controls, package comparison, admission evidence, and persistence policy.

The declared conditions under which the internal side and core objects admit enough duality to reproduce, or explicitly replace, Mackenzie’s finite-rank double-vector-bundle duality.

A double vector bundle organizing two interacting classes of infinitesimal repair over a common theory-state space.

The interpreted side bundles of assimilatory and proto-accommodative repair directions. These meanings are supplied by the application.

The core of \(\mathbb D_{\mathrm{rep}}\), containing mixed directions invisible from either side bundle alone.

Meinrenken–Pike’s bigraded cochain Weil algebra associated with a classical repair double vector bundle; not Leung’s classifying theory \(\mathsf{Weil}_1^{\mathrm{tan}}\).

Candidate horizontal and vertical differentials representing the two local repair processes.

The classical cochain realization \([d_{\mathrm{asm}},d_{\mathrm{acc}}]_{\mathrm{gr}}\) of the interchange defect; its vanishing characterizes compatibility of the candidate classical double Lie algebroid structure.

A contextual curve object supplying the abstract time parameter for differential equations in a tangent category, with distinguished initial point \(c_0\) and vector field \(c_1\). For algebroid transport, the book separately declares the required linear-completeness doctrine; Burke–MacAdam’s earlier complete curve object packages a restricted version of that doctrine into its definition.

Realized vector fields for assimilation and proto-accommodation on a common theory-state object, when the relevant anchors, sections, and integrability data have been supplied.

Partial or complete curve-object flows integrating the two realized DIAL directions.

The nine-component residual of the matched-pair conditions for the two 2-term representations up to homotopy induced by a chosen splitting \(\Sigma \). Its coordinates depend on the splitting, while its vanishing does not.

The skill-bundle component of transport across a proposed theory change, covering the semantic base map \(\bar J:M\to M^+\). It is distinct from the partial flows denoted by \(\Phi \).

The comparison between transporting an \(A\)-probe and probing after a finite theory extension.

The category of admissible typed Markdown skill programs and interface-preserving revisions within one fixed language version. Grammar or interface changes require registered maps between versioned artifact categories.

The typed subcollections whose artifacts compile, respectively, to local skill sections and to policies over admissible typed actions.

The partial semantic compiler from local skill artifacts to sections in environment \(E\).

The partial semantic compiler from workflow artifacts to policies in environment \(E\); either compiler may reject an ill-typed program.

A Lie algebroid whose base contains theory states and whose fibers contain locally available skill directions.

The local skill section obtained as \(\operatorname {compile}^{\mathrm{sec}}_E(m)\) in a declared environment.

The anchor mapping a skill direction to its observable infinitesimal effect on a theory state.

Latent skill directions with no immediate visible displacement, though they may affect traces or later compositions.

The visual-language sketch in Chapter 18; its sorts and arrows present visual objects, attributes, relations, scenes, transformations, and their compositional constraints.

The conditioned probability-flow transport from diffusion time \(s\) to \(t\) under visual-language state \(\ell \), when existence, uniqueness, and invertibility conditions hold.

The category of models of theory \(\mathbb T\) in a semantic category \(\mathcal C\), with the preservation doctrine understood.

A sketch presenting a theory by generators, equations, and designated universal constructions.

A proposed preservation doctrine and the declared comparison with the old doctrine, including the compatibility needed to compare their model categories.

The structural extension subrecord containing a proposed presentation, doctrine, registered model semantics, theory map, doctrine comparison, and restriction specification. It refines fields of the status-bearing dossier \(\Xi \); it does not contain or control independent admission evidence.

Restriction of expanded models along a theory or sketch map \(\iota :\mathbb S\to \mathbb S^+\), when the doctrine comparison makes that reduct functor well-defined.

Combinational, exploratory, and transformational creativity.

Infinitesimal Probes of Creativity; the controlled mathematical benchmark family.

The AI-Feynman-inspired symbolic-law construction family.

The DIAL specialization of Universal Reinforcement Learning.

Simulator-Grounded Theory Extension.

Corpus-to-Theory Extension: the controlled document-grounded ladder that progressively withholds supplied entity, context, query, mechanism-language, and domain structure.

Textual-inversion calibration and natural-image transport phases in the ARTISTIC evidence ladder.

Independent assignment of an epistemic status to a proposed extension using an appropriate, named certificate. Admission is relative to that contract; it is not automatically proof, external validity, or general scientific acceptance.

Compatibility of an observed outcome with a bounded claim under stated controls. Support does not by itself establish the claim outside the registered environment.

A consequence derived from stated assumptions in a specified formal system. Proof establishes that formal consequence, not the empirical adequacy, novelty, or value of the surrounding theory.

A typed language of objects, relations, laws, observers, and admissible constructions.

A conceptual structure that can be communicated, criticized, reused, and extended by other reasoners.

Economy achieved by a compact theory whose objects and laws generate explanations, predictions, interventions, and new questions; not merely a short encoding of observed data.

The unit whose novelty is being assessed, such as an artifact, equation, experiment, search trajectory, or presented theory.

Recovery of a symbolic expression from numerical observations, usually balancing empirical fit and expression complexity.

Difference relative to a declared repertoire, description language, observer, or conceptual space.

The unexpectedness of an event or description relative to a specified probabilistic or information-theoretic model.

An explicit change to a theory together with an account of transport, preservation, and loss.

An infinitesimal probe of a possible representational reorganization inside the current double repair package; not yet an admitted theory change.

A tangent-categorical presentation of Lie-algebroid structure, supplying the established first-order bridge between tangent categories and classical Lie algebroids.

Burke–MacAdam’s braid relation for the involution on an anchored bundle’s prolongation. It is the internal coherence from which Jacobi is recovered on sections, and is not identical to Drinfel’d’s classical \(r\)-matrix Yang–Baxter equation.

A representation of a Lie algebroid on a two-term complex, encoded by a boundary map, compatible connections, and curvature data satisfying coherent differential identities.

The nine compatibility conditions which, after choosing a linear splitting, characterize a classical double Lie algebroid.

Double Involution ALgebroid: the proposed internal double-algebroid structure used in this book, consisting of two involution-algebroid directions, a dualizability doctrine, and an interchange comparison. Its relationship with Mackenzie’s classical double Lie algebroids and their matched-pair homotopy presentation remains a bridge requirement, not a theorem assumed here.

A conjectural \(n\)-fold involution-algebroid object with \(n\) typed repair directions, DIAL structure on each two-dimensional face, and coherent higher interchange comparisons. The case \(n=2\) is ordinary DIAL.

The proposed two-parameter hierarchy separating tangent depth \(k\), which probes higher-order behavior of fixed repair directions, from repair arity \(n\), which counts independently typed directions.

Proposed enriched limit sketch or classifying tangent-categorical theory for one involution-algebroid direction.

Candidate tensor-product presentation of two internal involution-algebroid directions and their interchange. Identifying its smooth finite-rank models with Mackenzie double Lie algebroids is a research objective, not an assumed theorem.

Candidate enriched tensor-sketch presentation of \(n\) internal involution-algebroid directions. Its existence and classical realization are part of the higher-DIAL conjecture.

A presentation whose models are iterated models: under the required limit and enrichment hypotheses, \(\operatorname {Mod}(\mathbb A\otimes \mathbb B,\mathcal M)\simeq \operatorname {Mod}(\mathbb A,\operatorname {Mod}(\mathbb B,\mathcal M))\). Its symmetry presents order-independence but does not imply Lie integrability.

A domain-specific finite-realization observer: a declared mathematical or empirical test asking whether an infinitesimal repair composes into a coherent finite path. Examples include monodromy criteria, approximate holonomy, multi-step rollout, recurrence, intervention closure, and continuation across covers; these examples are not interchangeable.

The ordered distinction among presenting a tensor-product repair theory, diagnosing an interchange obstruction in one of its models, integrating a finite repair trajectory, and transporting models along a separately proposed domain-theory extension.

A double vector bundle with compatible horizontal and vertical VB-algebroid structures; equivalently here, one whose Weil-algebra differentials super-commute.

The architecture that diagnoses representational inadequacy through interacting local repairs, tests their persistent finite realization, and records creative jumps as admitted finite package changes. A sketch extension is claimed only when the presentation or generative language changes.

A finite sketch map, semantic transport, and coherent comparison maps relating its old and new Weil actions. These data specify a candidate; admitted differential accommodation additionally requires coherent finite realization, transport, and independent testing of new consequences.

A versioned, typed program declaring a skill’s inputs, outputs, invariants, probes, and admission rule.

A categorical algebraic theory whose models are finite-product-preserving functors.

A small finitely complete category whose models preserve finite limits.

A one-generated strict symmetric monoidal category; its algebras are symmetric monoidal functors.

A category equivalent to sheaves on a small site, equivalently a left-exact reflective localization of a presheaf category.