sae-functional-abstraction-extends-geometric-scope
OUT derived (depth 1)
Created 2026-08-25T04:19:08+00:00 · Reviewed 2026-08-25T04:28:09+00:00
SAE features activating on functional analogies (transit feature on wormholes) and cross-modal inputs (text-trained features firing on images) demonstrate the geometric space encodes intensional and relational structure beyond Park's extensional categorical polytopes, broadening the geometric framework's explanatory scope to include non-lexical, compositional semantics.
Justifications
This belief has 3 justifications — it is IN if any one holds.
SL — The polytope theorem covers categorical/extensional structure; the SAE evidence independently shows the same geometric space also carries functional/intensional structure, extending the framework.
Antecedents (all must be IN):
- IN sae-transit-feature-abstract-generalization — The transit infrastructure feature (1M/3) activates on wormholes alongside physical transit entities like trains, ferries, and tunnels, suggesting it captures a higher-level transit/transport concept rather than literal entities.
Unless (any of these IN defeats this justification):
- IN sae-functional-abstraction-extends-geometric-scope-v2 — SAE features activating on functional analogies (a transit feature firing on wormholes alongside physical transit entities) and on cross-modal inputs (text-trained features responding to image inputs) suggest that LLM representation spaces capture abstract, generalized concepts and shared cross-modal structure alongside the categorical polytope geometry described by Park et al., pointing toward a representational picture that extends beyond strict category membership to include non-literal generalization.
SL — The polytope theorem covers categorical/extensional structure; the SAE evidence independently shows the same geometric space also carries functional/intensional structure, extending the framework.
Antecedents (all must be IN):
- IN sae-cross-modal-text-trained-image-activation — SAE features trained on text-only data also activate on relevant image inputs, indicating shared latent cross-modal representations in Claude 3 Sonnet.
Unless (any of these IN defeats this justification):
- IN sae-functional-abstraction-extends-geometric-scope-v2 — SAE features activating on functional analogies (a transit feature firing on wormholes alongside physical transit entities) and on cross-modal inputs (text-trained features responding to image inputs) suggest that LLM representation spaces capture abstract, generalized concepts and shared cross-modal structure alongside the categorical polytope geometry described by Park et al., pointing toward a representational picture that extends beyond strict category membership to include non-literal generalization.
SL — The polytope theorem covers categorical/extensional structure; the SAE evidence independently shows the same geometric space also carries functional/intensional structure, extending the framework.
Antecedents (all must be IN):
- IN park-2025-iclr-categorical-polytope-geometry — Park et al. (ICLR 2025) prove that categorical concepts in LLM representation spaces are geometrically represented as polytopes (convex hulls of vertex vectors), with 'natural' concepts forming (k−1)-simplices.
Unless (any of these IN defeats this justification):
- IN sae-functional-abstraction-extends-geometric-scope-v2 — SAE features activating on functional analogies (a transit feature firing on wormholes alongside physical transit entities) and on cross-modal inputs (text-trained features responding to image inputs) suggest that LLM representation spaces capture abstract, generalized concepts and shared cross-modal structure alongside the categorical polytope geometry described by Park et al., pointing toward a representational picture that extends beyond strict category membership to include non-literal generalization.