We derive the linear union-of-subspaces (UoS) model for subspace clustering (SC) from the nonlinear mixture model (NMM) used in blind source separation (BSS) to represent a D-dimensional observation vector as an unknown multivariate nonlinear mapping of C latent variables. Assuming the mapping is differentiable up to an unknown order K, we approximate NMM by a K-th order Taylor expansion, yielding a model equivalent to the linear UoS framework underlying SC. This establishes that: (i) the smoothness order K corresponds to the unknown subspace dimension d; (ii) KC equals the number of anchors; and (iii) the sparsity of the representation vector equals K (i.e., d). These relationships enable estimation of bounds on subspace dimension, and that is validated on six benchmark datasets using five established SC algorithms. Established theoretical results are important for post-processing of self-representation matrices estimated by SC algorithms.
How does a system that merely predicts the world come to distinguish its own causal influence from everything else? We trace this transition in a minimal 192-dimensional GRU through a developmental sequence -- 6 experimental stages, 12 falsified alternatives, and cross-signal validation. Starting with no action or self-representation, we add components one at a time, tracking whether the system distinguishes self-caused from world-caused changes. The central finding is the encoding gap: a system can perfectly compensate for its own actions in prediction while failing to encode "I am acting" as a readable state -- implicit causal use and explicit self-representation are dissociated capabilities. The developmental path crosses this gap when four conditions are jointly satisfied: (1) persistent state that forms stable attractors, (2) a causal action loop linking the system's output to its input, (3) proprioceptive feedback that makes implicit causal knowledge explicit, and (4) asynchronous awakening -- consolidating perceptual learning before action learning, which yields the only configuration robust to hyperparameter choice. We propose agency gain (A = Err_world - Err_self), the predictive advantage of knowing one's own action, as a continuous metric that generalizes across signal types. A decisive test confirms the causal grounding of the encoding: after the external training signal is removed, the causal agent retains its self-representation at 94.9% while a statistically-matched control collapses to 53.9%. Self-representation persists only when causally useful for prediction -- an intrinsic property of the causal loop, not a training artifact.
We administer 45 validated psychometric questionnaires to 50 large language models (LLMs) to identify the dimensions along which LLMs differ psychometrically. Using Supervised Semantic Differential (SSD), we find that the primary axis of between-model variance separates items describing phenomenally rich experience, including embodied sensation, felt affect, inner speech, imagery, and empathy, from items describing stimulus-driven behavioral reactivity ($R^2_{adj}=.037$, $p<.0001$). To test this hypothesis at the item level, we introduce the Pinocchio score ($π_i$), the ratio of inter-model response variance under neutral prompting to that under a human-simulation prompt, as an annotation-free measure of each item's experiential demand. $π_i$ predicts condition-induced shifts in primary factor loading magnitudes ($ρ=-.215$, $p<.0001$, $n=1292$--$1310$ items), confirming that between-model divergence on experiential items is structured rather than noisy. Applying PCA to per-model EFA scores across all questionnaires reveals one dominant dimension, the Pinocchio Axis ($Π$): the degree to which a model presents itself as a locus of phenomenal experience rather than a system of behavioral responses. This axis captures 47.1% of cross-questionnaire between-model variance in primary factor scores and converges with item-level Pinocchio scores ($r=.864$). Marked within-provider divergence across closely related model variants is consistent with post-training fine-tuning as a key contributor, supporting the interpretation that $Π$ reflects a training-shaped self-representational tendency governing how a model treats experiential language as self-applicable. The dominant axis of between-model psychometric variation is therefore not a conventional personality trait but a self-representational stance toward one's own nature as an experiencer.