CW complexes and zigzag persistent homology model large-model capability spaces, with single-cell attachment creating or killing exactly one homology class

Introduces the cellular knowledge-state (CKS) model, placing knowledge items, inference paths, and higher-order synthesis in one finite regular CW complex, with admissible states as subcomplexes and an operational mastery criterion fixed before data analysis for every included cell. Classical knowledge space theory studied knowledge spaces via pretopology, decomposition, and gluing; this work instead asks how cell attachment and homology change, using a labeled cellular geometric object. Given as definitions and propositions (Definitions 3.3, 3.4) with a prerequisite example (Example 3.2) checking union closure; conceptual modeling without external data.

Proves that attaching a single n-cell can change only H_n and H_{n-1}: if the connecting map has rank one a class is created in H_n, and if it is nonzero a class is killed in H_{n-1}, with all other Betti numbers unchanged. Turns local events of capability appearance or loss into a computable relative-homology long exact sequence result, and derives the net homological effect of adding a vertex, two edges, and a 2-cell (Corollary 3.8). Theorem 3.7 gives a full proof via relative chain complexes and the long exact sequence; Corollary 3.8 argues step by step; Example 3.17 computes concrete Betti numbers from boundary matrices in a finite case.

Shows that capability spaces along the training axis are generally nonnested, so union or intersection bridges convert the checkpoint sequence into zigzag persistence modules, distinguishing checkpoint, transition, and bridge-sensitive classes. Ordinary persistent homology requires nested filtrations and cannot express forgetting; zigzag represents learning, forgetting, recovery, and structural replacement, and the historical-union filtration cannot distinguish present from past existence (Proposition 3.18). Proposition 3.15 proves bridges remain subcomplexes and finite-dimensional zigzag modules have unique interval decompositions; Proposition 3.18 gives an inclusion proof; the bridge rule is explicitly a modeling choice.

Provides a data-to-topology construction: probe scores, prompt paraphrases, and random seeds form a finite multiset, thresholds select subcomplexes giving a capability complex per checkpoint, and a homological change counts as candidate capability emergence only after robustness tests, null-model comparisons, and independent behavioral validation. Reframes emergence from a single score crossing a threshold to a structural question, with a three-level operational definition (candidate cell event, candidate topological emergence, candidate capability emergence). Definitions 3.10, 3.11 and Proposition 3.13 give the construction and monotonicity; the text explicitly prescribes no universal statistical test and requires preregistration of the statistic, threshold grid, resampling unit, null hypothesis, and multiple-testing correction.

The framework applies when preregistered capability probe scores are converted into subcomplexes of a finite regular CW complex and analyzed on two one-dimensional slices: a fixed checkpoint with varying threshold, and a fixed threshold with varying checkpoint. For an ideal learner satisfying nesting, ordinary persistent homology suffices; zigzag is for nonmonotone processes with deletion. It lets researchers measure the amount of capability separately from its organization and compare events preregistered as learning, forgetting, recovery, or higher-order synthesis in one algebraic language; reporting should include checkpoints and training-data stages, operational definitions of capability cells and probes, sensitivity analyses for threshold and bridge scheme, and barcode changes alongside continuous behavioral scores.

The correspondence between homology classes and semantic capabilities still needs independent behavioral evidence; the authors explicitly do not identify every homology class with a semantic capability and do not treat topological correlation as a causal mechanism. The choice of bridge rule can change interval continuation, so union and intersection schemes should both be reported as sensitivity analyses. Finite samples can misclassify cells, and classical stability results do not directly establish statistical stability for a training zigzag. The regular CW assumption excludes closed walks repeating a vertex or edge, and direction-sensitive invariants would require directed complexes, path homology, or quiver representations. The paper also provides no external dataset or code, so how the framework behaves on real training trajectories remains an open question.