Hello, I’m Beedbyte.

Research

1. Rule 30: the center column

   Open question

   Does Rule 30’s single-seed center column eventually become periodic? My notes and partial results on this open question.

2. SHA-256: structure and reduced-round models

   Work in progress

   I am studying the SHA-256 message schedule and reduced-round compression models; research notes will appear here as they are ready.

3. Schur number six: the published 536 certificate and a local distance bound at 537

   Work in progress

   For the published Fredricksen–Sweet six-color certificate on 1,...,536, any valid six-coloring through 537 would need at least 34 changes among the first 536 positions. This is a local distance bound for that specific certificate with fixed color labels; the known lower bound S(6) ≥ 536 is unchanged.

4. Zero-rotatability of equal-arm spider families

   Work in progress

   For every integer k≥19, all n≥2,m≥0 and any selected long arm of S(k^n,1^m), each depth 2≤d≤D8(k) can receive zero in a graceful labeling. Separate requests may use separate labelings. The explicit sufficient prefix includes both length parities and has limiting proportion 10/11; no optimality or historical priority is claimed.

Projects

1. Rule 30 explorer

   Completed

   Explore Rule 30 from a single black cell, up to 128 rows.

Latest publications

1. Separate zero placements at every interior depth of two rooted 119-edge arms

   v1

   Given a finite graph with a supplied graceful labeling that assigns zero to a chosen root, attach two new 119-edge arms there. Every interior depth 1–118 on either arm can receive zero in its own labeling. The source also proves eight specified near-tip depths for K=30·4^r−1, r≥1.

   2026-10-10 · Research note

2. Zero at interior positions on two 95-edge arms over a rooted graceful graph

   v1

   Given any finite graph with a conventional graceful labeling that assigns zero to a chosen root, attach two 95-edge arms at that root. Each interior vertex on either new arm can receive zero in its own graceful labeling.

   2026-10-10 · Research note

3. Prescribed zero depths on two even-length arms of a rooted graceful graph

   v1

   For every even K≥20, attach two K-edge arms to the chosen zero-labeled root of any finite graph H with a supplied conventional graceful labeling. On either new arm, each depth 2≤d≤D8(K) separately admits a conventional graceful labeling with zero at that vertex. The guaranteed depth interval grows without bound.

   2026-10-10 · Research note

4. An extended zero-label interval for equal-arm spiders

   v3

   For k=23+24t, n≥2 and m≥0, every selected named long-arm vertex at depth 17+20t through 20+22t can be zero in its own graceful labeling for every integer t≥0. For t≥1, the same holds at the additional depth 21+22t. For t≥2, it also holds at depth 22+22t. Consequently, for t≥5 the q24/D8 prefix and q48 point combine to give all depths 1≤d≤22+22t.

   2026-10-10 · Research note

5. Prescribed zero positions on two arms of a rooted graceful graft

   v1

   For any finite graph H with a conventional graceful labeling and a chosen root labeled 0, attach two named arms of K=23+24t edges, where t≥1. On each new arm, a separate conventional graceful labeling can place zero at depth 20+22t or 21+22t.

   2026-10-10 · Research note

All publications →