Homology profiles and graded Betti tables of squarefree monomial ideals in five and six variables
Abstract
Ripke and Yoon recently determined that exactly 134 distinct graded Betti tables occur among the 208 nonzero proper squarefree monomial ideals in five variables up to relabelling of the variables, and closed their paper with several questions, two of which we answer here: whether those 134 types are classified by the homology profiles of induced subcomplexes, and how many distinct graded Betti tables occur for general n. The homology profile — the multiset, for each cardinality j, of the reduced homology dimension vectors of the induced subcomplexes on j vertices — is a strictly finer invariant than the graded Betti table and a strictly coarser one than the relabelling class: in five variables the 208 classes realize exactly 189 profiles and 134 tables, so of the 74 coincidences of Betti tables the loss of positional information named by Ripke and Yoon accounts for 19 and the summation for the remaining 55. Exactly 97 of the 134 table types come from a single profile. In six variables, the first open cell of the enumerative question, the 16 351 nonzero proper classes realize exactly 8233 homology profiles and exactly 1604 graded Betti tables, over every field; the enumeration is certified by orbit counting against the Dedekind number M(6)=7 828 354, itself computed rather than quoted. Every count stated here is machine-checked in Lean 4, with Hochster's formula quoted from the literature and one further step, named where it is used, argued rather than formalized.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
0f4944aa67ef430cc7b286ac39e516a2104160ff54673ae5ccb3883e0378984a
Claim ledger
Stated results
sqfbetti-01known2026-08-23
n = 3, 4, 5: antichain / relabelling-class / nonzero-proper counts 20/10/8, 168/30/28, 7581/210/208
sqfbetti-02known2026-08-23
n = 5: exactly 134 distinct graded Betti tables – the source's Proposition 8.1
sqfbetti-03known2026-08-23
n = 5: the source's multiplicity table for the 134 types – 96 realized once, 18 twice, 11 three times, 3 four times, 5 five times, 1 six times
sqfbetti-04routine2026-08-23
Euler-Poincare for the defined homology dimensions, proved in general with no native axiom
sqfbetti-05routine2026-08-23
Three consequences of Hochster's formula checked against combinatorics: row 1 counts the minimal generators by degree, row 0 is beta_(0,0) = 1, and the boundary maps compose to zero with no negative homology dimension
sqfbetti-06known2026-08-23
The source's Section 9 closed forms: the squarefree Veronese and almost squarefree Veronese Betti tables, including its Example 9.1
sqfbetti-07known2026-08-23
Peeva's Example 12.4 in both characteristics, and its ideal identified with the RP² triangulation up to relabelling
sqfbetti-08known2026-08-23
Five variables: the graded Betti table is the same over ZMod 2 and over Q, on all 208 classes
sqfbetti-n1routine2026-08-23
negative control: dropping the H-tilde₋₁ term gives 130, not 134, and the whole gap is one five-fold collapse of the linear ideals (x₁..xₘ)
sqfbetti-n2routine2026-08-23
negative control, the more dangerous one: dropping the empty face outright returns the RIGHT count 134 with ALL 208 tables wrong
sqfbetti-n3routine2026-08-23
negative control: the nonzero-proper hypothesis is not vacuous – keeping the zero and unit ideals gives 136, and 134 is neither 133 nor 135 nor 208
sqfbetti-09routine2026-08-23
The Finset (Finset (Fin n)) model: the Stanley-Reisner complex really is a simplicial complex, and the bitmask layer computes it
sqfbetti-10candidate2026-08-23
n = 6, the source's own open cell: 1604 distinct graded Betti tables, in every characteristic – COMPUTED IN C ONLY, not kernel-checked
This ledger entry is reported in prose and is not bound to a Lean theorem.sqfbetti-11known2026-08-28
Five vertices: every one of the 2114 face sets carries a verified rank certificate, so every boundary-matrix rank is the same over every field
sqfbetti-12known2026-08-28
Six vertices: of all 2²0 triangle sets and all 65 666 face sets of the other cardinalities, exactly twelve boundary matrices are uncertified – one S₆ orbit – and their Smith normal form is (1⁹, 2)
sqfbetti-13routine2026-08-28
Exactly 12 of the 7 828 354 squarefree monomial ideals in six variables – one of the 16 353 relabelling classes – have a characteristic-dependent graded Betti table at the face-set level; every other one has the same table over every field
sqfbetti-14routine2026-08-28
A rank certificate pins the rank over EVERY field: proved for an arbitrary ring homomorphism into a field in mathlib's Matrix.rank, kernel-clean
sqfbetti-n4routine2026-08-28
negative controls for the rank checker: it rejects a fabricated Z, a rank one too small and a rank one too large, and accepts the true one – and the sweep that certifies all 2114 five-vertex face sets fails on the six-vertex RP² complex
sqfbetti-15candidate2026-08-30
Five variables, the source's first open question answered: the 208 relabelling classes carry 189 distinct homology profiles of induced subcomplexes and only 134 distinct graded Betti tables, so the loss the source names accounts for 19 of the 74 coincidences and the summation for the other 55
sqfbetti-16routine2026-08-30
The labelled homology function sigma |-> dim H _*(Delta restricted to sigma) is a complete invariant of the relabelling class: 8, 28, 208 labelled profiles for 8, 28, 208 classes at n = 3, 4, 5
sqfbetti-17candidate2026-08-30
Six variables, the source's closing question at its first open cell, now KERNEL-CHECKED: 1604 distinct graded Betti tables and 8233 distinct homology profiles among the 16 351 nonzero proper relabelling classes
sqfbetti-18known2026-08-30
The Dedekind numbers M(3) = 20, M(4) = 168, M(5) = 7581 and M(6) = 7 828 354 computed inside Lean rather than cited
sqfbetti-19routine2026-08-30
The 16 353 supplied masks are a complete, irredundant set of S₆-orbit representatives of the squarefree monomial ideals in six variables, proved by orbit counting
sqfbetti-n5routine2026-08-30
negative controls for the homology-profile layer: four distinct classes share one profile (too large refuted concretely), the profile is a relabelling invariant on all 208 x 120 pairs, its packing is lossless, and it sums to the family's own Betti table
sqfbetti-n6routine2026-08-30
negative controls for the six-variable layer: the new array sort is the family's sort, the canonicality and orbit tests are non-vacuous, and the table this file counts is Compute.bettiTable
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- The closing question of *Characteristic Independence of Betti Numbers of Monomial Ideals in Five Variables*, Noah Ripke and Phillip Yoon, arXiv:2607.10639 (math.AC, v1 12 Jul 2026, v2 read here on 2026‑08‑23 and identical on every claim used), §10, verbatim:
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7