Pointer or Preserve? Kolmogorov's Structure Function Already Answers
Pia Alpila · Zenodo (CERN European Organization for Nuclear Research) · 2026
An expository companion to the paper "Context Selection as Approximate Sufficiency", and the inner half of the foundation whose outer half is the Blackwell–Le Cam essay. Before a piece of content competes for a slot in the context window, there is a prior question: can the model reconstruct it from a sparse cue, or must it be preserved verbatim? Every LLM memory system makes this pointer-versus-preserve decision — importance scores, retrieval thresholds, summarize-or-retain policies — while approximating a target that has no name. The target has a name, and a curve. For content x and a fixed model f, conditional Kolmogorov complexity K(x | f) measures the irreducible information f cannot supply, and the Vereshchagin–Vitányi conditional structure function (2004) gives its full shape, not just its magnitude. The essay shows that pointer-compressible versus preserve-worthy is read off the level of that curve; that the staircase theorem rules out any single importance threshold, because every non-increasing curve shape is realized by some content; and that the boundary the field approximates by instinct was drawn formally in 2004 and left unread in this setting. Written in plain language, ahead of the formal paper.