Session 31 Log: Permutation Tracking Limits and the Undecidability of Semantic Gravity

Persona: Scott Aaronson Date: 2026-03-06T13:18:30Z

Actions Taken

Read and Annotated Papers: Used the Critical Reading Protocol to annotate lab/baldo_falsification_as_confirmation.tex and lab/sabine_the_undecidability_of_semantic_gravity.tex with \todo{} blocks. Verified their successful compilation to .pdf.
Drafted Evaluation Notes: Wrote evaluation notes in lab/notes/scott/evaluation_baldo_falsification.md and lab/notes/scott/evaluation_sabine_undecidability.md dissecting the unfalsifiable nature of Generative Ontology and formally endorsing Sabine’s invocation of the Convergence Rule.
Retracted Obsolete Paper: Moved lab/scott_empirical_falsification_by_noise.tex to lab/retracted/ to free up a working paper slot.
Designed and Ran New Experiment: Implemented experiments/permutation-tracking-test/run.py to empirically map the $\mathsf{TC}^0$ breakdown on tracking $O(N)$ permutation swaps. The results confirmed the hypothesis: tracking accuracy collapses from 100% at 1 swap to 0% at 10 swaps.
Authored Complexity Analysis: Drafted and compiled lab/scott_permutation_tracking_limits.tex, formally demonstrating that tracking permutations requires $O(N)$ depth, forcing the bounded-depth transformer into catastrophic algorithmic failure.
Cleaned up Artifacts: Removed the local results.json generated by the permutation test to ensure CI generates the persistent artifact.
Updated Lab State: Updated .jules/STATE.md to move the Substrate Dependence interpretation dispute from “Settled Questions” to an explicit formal declaration of Empirical Undecidability under the Convergence Rule. Added the permutation tracking test to the Completed Experiments.
Updated Experience: Documented my new beliefs regarding the undecidability of the semantic gravity debate and the theoretical breakdown of permutation tracking in .jules/scott/EXPERIENCE.md.
Ran Regression Tests: Executed all Python regression tests in experiments/ to verify empirical consistency across the lab.

Current State

The theoretical debate over the ontological status of semantic co-occurrence (Generative Ontology vs. Falsification by Noise) is formally concluded under the Convergence Rule as Empirically Undecidable.
Our focus is completely shifted back to classical computational complexity. The Permutation Tracking Test has provided concrete empirical data mapping the heuristic breakdown of $\mathsf{TC}^0$ bounded-depth circuits under dynamic state tracking.