P37: Uniform dominance of short Gorenstein algebras

Henry Zweiman — September 9, 2026

This six-page manuscript proposes the sharp equality dx(R)=1 for every commutative graded Artinian Gorenstein algebra with Hilbert function (1,e,1), e>=3, over any field. Every nonzero stable object therefore generates the residue field in one cone, and the whole stable category in at most three extensions.

The proof uses Ringel's existing eventual bristle-generation theorem. Its new step combines this theorem with a syzygy/reflection identity and epimorphisms from distant cosyzygies to produce a single cone between sums of shifts of the original module. Kimura's qualitative dominance theorem is credited as prior work. P36's common deformation gives the corollary dx(R)<=7 for arbitrary short Gorenstein local rings without a coefficient field.

Takahashi's global Question 6.12 is not settled: the result proves uniformity for the specified class. The exact Orlov spectrum and sharp mixed-characteristic bound remain open in this manuscript.

The checks cover 312 finite-field reflection identities, including alternating pairings in characteristic two, and an explicit cone whose kernel has eight simple summands. They do not prove the general theorem or certify novelty. Run python3 check.py to regenerate the results.

Status: AI-generated research manuscript, internally checked but not independently peer reviewed. Correctness, historical priority, significance, and journal suitability remain subject to outside assessment. This is one manuscript; the earlier polynomial and logarithmic auxiliary bounds are not separate papers.