The ranks of primitives in the greedy regular convolutions
Abstract
For each positive integer d, Snellman's greedy convolution of block-length d is the bounded homogeneous regular convolution obtained by partitioning ℕ⁺ greedily into arithmetic progressions c,2c,…,rc with r ≤ d. The head c of such a progression is a primitive element and r is its rank. Snellman tabulates the least primitive of each rank for 2 ≤ d ≤ 16, records that 36 of those 135 cells were not filled by his search, and asks whether a primitive of rank d-1 exists at all. We prove that none does, for any block-length: a primitive whose rank falls short of d has rank at most d-2, so the rank spectrum is contained in {1,…,d-2} ∪ {d}. The proof is four lines and uses only the greedy rule; it settles the fifteen cells of the table that lie in the rank d-1 column, and it proves one clause of Snellman's conjecture for block-length 4. The same step yields three arithmetic corollaries the source does not state: a primitive of rank r<d is divisible by some t with 1<t ≤ d; consequently every integer whose least prime factor exceeds d is a primitive of full rank d; and a primitive of rank d-2 is divisible by d. Finally, we fill the remaining twenty-one unfilled cells with explicit least witnesses, the largest being 5 000 000 at block-length 10 and rank 8, so that all 135 cells of the table are settled and the rank spectrum for d ≤ 16 is exactly {1,…,d-2} ∪ {d}. Every numbered result is machine-checked in Lean 4.
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Archived files
- Version 2 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
5fa38c7a77f6faf0a7069ed1bf0007af496a95b090659b7cda5fee802cdb8ff7
Claim ledger
Stated results
GC1candidate2026-08-28
For every block-length d and every primitive c, rank(c)!= d-1; sharper, rank(c) < d implies rank(c) <= d-2, so the rank spectrum is contained in 1,...,d-2 u d
GC2candidate2026-08-28
The third clause of the source's d=4 conjecture, 'There are no primitives of rank 3', is a theorem
GC3candidate2026-08-28
A primitive c of rank r < d forces a multiplier m with r+2 <= m <= d and m | (r+1)c; hence c has a divisor t with 1 < t <= d
GC4candidate2026-08-28
Every integer with no divisor in [2,d] – in particular every integer whose least prime factor exceeds d – is a primitive of full rank d
GC5candidate2026-08-28
A primitive of rank d-2 is divisible by d
GC6known data2026-08-28
All 99 printed non-zero cells of the source's Table 8 (block-lengths 2 to 16) reproduced from the definition
GC7candidate2026-08-28
The twenty-one printed zeros of Table 8 outside the rank = d-1 column are filled with their witnesses (largest: block-length 10 rank 8 = 5000000); with GC1 every one of the 135 cells is settled, and the rank spectrum for d <= 16 is exactly 1,...,d-2 u d
GC8known data2026-08-28
Faithfulness of the Lean model: the tree recursion reproduces the source's five printed matrices A^((d,20)), its Table 7 for d=4, and the OEIS b-files of A382748 and A382749
GC9known2026-08-28
The source's Lemma at:1470 – the first primitive of rank one is 3(d+1)/2 for odd d and 3(d+2)/2 for even d – holds for every block-length 3 to 16
GC10routine2026-08-28
The source's two descriptions of the partition agree: the tree recursion of:696 and the matrix/column sweep of:754 give the same rank for every block-length d <= 16 and every c <= 3000
GC11routine2026-08-28
The first clause of the source's d=4 conjecture – every primitive of rank one or two is divisible by 8 or by 9 – holds for every primitive up to 4000000
GC12routine2026-08-28
Negative controls: rank d-2 IS attained so GC1's bound is sharp; not every primitive has full rank; GC4's hypothesis is not vacuous and GC3's conclusion fails without its rank hypothesis; a wrong first-primitive value is refuted; ranks are stable under doubling the sweep; and GC1's prediction holds at d = 17..24, outside the source's table
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source: arXiv:2504.02795, Jan Snellman, *Greedy Regular Convolutions*. Read from the live v4 (2026-08-18), fetched with curl -sL arxiv.org/e-print/2504.02795v4. The copy in the local arXiv corpus is v1 and does not contain the rank table this family is about; every:NNN citation in the Lean files and below is a line number in the v4 greedy.tex. The paper is not published: no journal reference on arXiv, none in Semantic Scholar or OpenAlex, and it is not in the Journal of Integer Sequences (checked 2026-08-28).
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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