# Version history

## Version 2 — 31 October 2026

- Adds Theorem 1.4: for r≥h+3, J/J^(r−h−1) is the universal largest natural linear quotient of J whose degree-h Eilenberg–Mac Lane diagram reconstructs from cardinality r. For smaller r only the zero quotient qualifies.
- Adds a direct proof that every nonzero natural linear subfunctor of J is an augmentation power; no ideal hypothesis is needed. This classification is credited as an elementary consequence of Pirashvili's classical pointed cross-effect equivalence.
- Explains the fixed-four-label cases: the largest quotients for h=0,1,2 are J/J³, J/J², and zero, respectively.
- Adds exhaustive finite checks in original cut coordinates: all 29,212 subspaces at four labels, all 256 endomorphisms, and compatible diagrams on smaller cardinalities.
- Rewrites the philosophical discussion around a necessary transformation: once the initial diagram, maps, observational bound and homotopy degree are specified, the theorem determines the least information loss that permits reconstruction.
- Preserves the original reconstruction and FI-homology formulas. Adds the direct Pirashvili reference and an internal review of the revised manuscript.

## Version 1 — 31 October 2026

Initial 11-page preprint: ordinary and derived all-cutoff reconstruction groups, sharp cutoff d+h+1, exact FI homology, primary-source comparisons, and three verification scripts.

Both versions are research preprints. Priority is unconfirmed, and neither external human peer review nor proof-assistant verification is claimed.