# SGL₆(F₂) has a Hamilton cycle — the case n = 6 of Orel's Open Problem 16 (the non-vertex-transitive side, positive side)

**Written by**: a working record[^1] (2026-09-08)
**Work it is based on**: a working record (a working session[^2] — up to 99.9637% in 998 seconds),
a working record (a working session — **changed the design and closed it**, the independent verifier), the present working record (the normalised hash, the cross-check against the printed paper, this document)
**Status**: **this is a document submitted for examination, not a declaration that a record has been updated. It does not claim "world first".**
**In the ledger[^3] it has been assigned a record number[^4] by the examination of a working session**, but the record number is the ledger-internal fact that "the cycle has become an independently verified certificate";
it is not a claim of priority in the literature (§7.2).
This document performs no new search. All figures are copied from two runs of the present working record
(`n960_canon.out`, 2026-09-08, 6.0 seconds; `n960_verify_rerun.out`, the same day, 35.5 seconds).
**Whether to report this outside, and if so how it should be worded, is the operator's decision.**

> **English summary is in §8.**

---

## 1. The claim

### 1.1 The claim (in one line)

> **The graph SGL₆(F₂) (888,832 vertices; not regular: degrees 31 and 63) has a Hamilton cycle.**
> The evidence is the 888,832 lines of `sgl6_cycle_mat.txt`,
> and the check finishes in **35.5 seconds** with a single Python program that uses the standard library only (`verify_sgl6.py`).

### 1.2 What is not claimed (stated first)

- **This is not "better than Orel".** Orel found Hamilton cycles in the three cases on 280, 448 and 13,888 vertices
  with the Concorde TSP solver, and stated the general case as an open problem. What is used here is Pósa's rotation plus extension, a **heuristic**,
  and a heuristic can only be used on the positive side. The same tool cannot say "there is none".
- **No construction for general n is given.** The second half of Open Problem 16
  ("is there a general way, valid for each n, to construct it?") is not addressed.
- **This is a data point about one single n.** Together with HGL₄(F₄) (a record number, `HGL4-NOTE.md`)
  it only means that each of Orel's two families now has one point on the positive side.
- **A comparison of costs is a comparison of procedures, not a comparison of objects.**
  The uniformly random Pósa of a working record stopped at 99.9637% after 998 seconds, and the
  **one-step lookahead** of a working record closed it from that state in 53.3 seconds.
  **We do not write "SGL₆ is easy / hard"** (Wall 39[^5]).
  Furthermore, **53.3 seconds is not "53.3 seconds in total"** — the 998 seconds of a working record
  come before it (§9-3).
- **The denial of regularity and of vertex-transitivity is taken from Orel Prop. 12.**
  What this verifier confirmed independently goes only as far as "the set of degrees of 100 randomly
  chosen vertices is {31, 63}".
- **We do not write "world first".** What can be written is only "**within the search range of this
  ledger, no existing source was found**". The range searched, the queries, and the **searches not
  performed** are listed in §7.2.

---

## 2. Definitions (verbatim from the printed paper)

Orel, *On generalizations of the Petersen graph and the Coxeter graph*,
**Electronic Journal of Combinatorics 22(4) (2015), #P4.27**
(Submitted: Sep 27, 2013; Accepted: Nov 4, 2015; Published: Nov 13, 2015.
MSC 2010: 05C50, 15B33, 15B57).

### 2.1 Definition of the graphs (verbatim, corresponding to p.4 of the printed paper)

> Let
> H_n(F₄), HGL_n(F₄), S_n(F₂), SGL_n(F₂)　　(2)
> denote, in the same order, the sets of all hermitian n × n matrices over F₄, invertible
> hermitian n × n matrices over F₄, symmetric n × n matrices over F₂, and invertible
> symmetric n × n matrices over F₂. Each of these four sets is a vertex set of a graph, where
> {A, B} is an edge if and only if rk(A − B) = 1.　　(3)
> We slightly abuse the notation and denote these graphs with the same symbols (2).

And then (in the same paragraph):

> Graphs HGL₂(F₄) and SGL₃(F₂) are the Petersen and the Coxeter graph, respectively,
> as it can be seen in Figures 1 and 2.

**⟹ SGL₆(F₂) is the fourth step of the Coxeter graph** (n = 3, 4, 5, 6 give 28, 448, 13,888, 888,832 vertices).
Since the characteristic is 2 we have `rk(A − B) = rk(A + B)`, and the verifier looks at the latter.

### 2.2 Open Problem 16 (verbatim, corresponding to p.14 of the printed paper)

> There are only five known connected vertex-transitive graphs without a Hamilton
> cycle: the complete graph on two vertices, the Petersen graph, the Coxeter graph, and
> two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with
> a triangle (see the survey paper [9] and the references therein). It is believed by many
> mathematicians that no other exist. Since the graphs HGL_n(F₄) and SGL_{2n−1}(F₂) are
> vertex-transitive by Proposition 12 and they generalize the Petersen and the Coxeter
> graph in a natural way, it seems worthwhile to check if these graphs have a Hamilton cycle.
> Unfortunately we did not succeed in this task. With an application of the Concorde TSP
> Solver [5] we were able to find a Hamilton cycle in HGL₃(F₄), SGL₄(F₂), and SGL₅(F₂),
> which are graphs on 280, 448, and 13888 vertices, respectively. **We state the general case,
> which include non-vertex-transitive graphs SGL_{2n}(F₂), as an open problem.**
>
> **Open Problem 16.** Do graphs HGL_n(F₄) and SGL_n(F₂) contain a Hamilton cycle? If
> the answer is positive, is there a general way, valid for each n, to construct it?

**⟹ SGL₆(F₂) is the smallest open case on "the non-vertex-transitive side".**
The sentence set in bold is the place where Orel **deliberately included** the SGL with even n in the problem.
**The symbol "SGL₆(F₂)" itself is not printed in the paper** —
what is printed is the question for general n, and the three that Concorde could bring down (280 / 448 / 13,888).
**That n=6 is "open" comes out of the difference between these two statements** (it is not in the list that Concorde could bring down).
This reading is this document's own; Orel does not print n=6 as open by name. **Do not mix the two.**

### 2.3 Cardinality and degree (the formulas of the printed paper)

Cardinality (corresponding to p.11 of the printed paper; by [2, Lemma 9.5.9]):

> The cardinality of SGL_n(F₂) is computed in [2, Lemma 9.5.9], which tells that
> |SGL_n(F₂)| = 2^{(n²−1)/4} Π_{j=1}^{(n+1)/2} (2^{2j−1} − 1)　　if n is odd,
> |SGL_n(F₂)| = 2^{((n+1)²−1)/4} Π_{j=1}^{n/2} (2^{2j−1} − 1)　　if n is even.

In the same paragraph, the number of alternate matrices:

> … precisely Π_{j=0}^{n/2−1} 2^{2j}(2^{n−2j−1} − 1)(2^{n−2j} − 1)/(2^{2j+2} − 1)
> = 2^{((n−1)²−1)/4} Π_{j=1}^{n/2} (2^{2j−1} − 1) = |SGL_n(F₂)| / 2ⁿ
> matrices in SGL_n(F₂) are alternate.

Degree and transitivity (**Proposition 12**, verbatim):

> **Proposition 12.** Let n ≥ 2. Graph HGL_n(F₄) is arc-transitive of degree
> (2ⁿ − (−1)ⁿ)(2^{n−1} − (−1)^{n−1}) / 3.　　(23)
> **If n is odd, then SGL_n(F₂) is arc-transitive. If n is even, then SGL_n(F₂) is not
> vertex-transitive.** It is not edge-transitive unless n = 2. **Nonalternate and alternate
> matrices have vertex degree 2^{n−1} − 1 and 2ⁿ − 1, respectively.**

(**How to read the superscripts was settled from the printed page of the proof of Prop. 12** — the OCR
drops exponents, so from the single line of Prop. 12 alone one cannot distinguish
`2^{n−1} − 1 and 2ⁿ − 1` from `2^{n−1} − 1 and 2^{n−1}`.
On the side of the proof there is, verbatim,
> If A ∈ SGL_n(F₂) is alternate, … any nonzero x generate a neighbor A + xxᵀ.
> Hence **the degree of A equals 2ⁿ − 1**. For nonalternate A ∈ SGL_n(F₂) … the degree of A
> equals … **= 2^{n−1} − 1**.

so **alternate is 2ⁿ − 1 and non-alternate is 2^{n−1} − 1**. For n=6 this gives **63 and 31**.)

**The values recomputed independently by the present working record** (verbatim from `n960_canon.out` [1];
integer arithmetic only):

```
    n=2: |SGL_n(F_2)| = 4   交代行列 = 1
    n=3: |SGL_n(F_2)| = 28
    n=4: |SGL_n(F_2)| = 448   交代行列 = 28
    n=5: |SGL_n(F_2)| = 13,888
    n=6: |SGL_n(F_2)| = 888,832   交代行列 = 13,888
    n=7: |SGL_n(F_2)| = 112,881,664
    n=8: |SGL_n(F_2)| = 28,897,705,984   交代行列 = 112,881,664
```

(The block above is left exactly as the program emits it; 交代行列 = alternate matrices.)

- n=3 → **28** (Coxeter), n=4 → **448**, n=5 → **13,888** — **these agree with the three that Orel printed in the paper.**
- **n=6 → 888,832**, of which alternate **13,888** (= 888,832 / 2⁶) and non-alternate **874,944**.
- Degrees: non-alternate 2⁵ − 1 = **31**, alternate 2⁶ − 1 = **63**.
  **This agrees with the verifier's measurement (the set of degrees of 100 randomly chosen vertices = {31, 63})** (see [6] of §5.2).
  The proportion of alternate matrices, 13,888 / 888,832 = **1/64 = 1.5625%**, also agrees with
  `|SGL_n(F₂)| / 2ⁿ` from the printed paper.

**The verifier does not take this cardinality from the printed paper: it sweeps all 2²¹ = 2,097,152
symmetric matrices by elimination and produces 888,832 independently** (see [2] of §5.2).

(The OCR of the printed paper is the OCR text in the working directory; the PDF is `orel_ejc22.pdf`
in the same place. The verbatim passages above were copied from the OCR, with the mathematical symbols
(≥, −, superscripts) restored to what the printed page means.
Only the glyphs of the symbols were rewritten; the word order and the wording were not changed.)

---

## 3. How the cycle was constructed (how it was found)

**Two stages: a working record (a working session) and a working record (a working session).**
Neither Concorde nor a SAT solver is used.

1. **Encoding**: the set S₆(F₂) of all symmetric matrices is a vector space over F₂ of dimension 6·7/2 = **21**.
   `|S₆(F₂)| = 2²¹ = 2,097,152`. **Addition is exactly the bitwise XOR of the codes.**
2. **Neighbourhood**: adjacency is determined by the rank of the difference alone, so once the set `S` of
   rank-1 symmetric matrices (|S| = 2⁶ − 1 = **63**) has been built, the neighbours of a vertex `v` are
   obtained simply by filtering `v XOR S` for invertibility (no adjacency list is held).
   That is, **SGL₆(F₂) is the subgraph of a Cayley graph on S₆(F₂) induced on the invertible vertices**.
   (The whole graph is **not** a Cayley graph — because of the induction. Orel says the same thing on p.14 of the paper.)
3. **First stage (a working record, random Pósa)**: when the path can no longer be extended from an
   endpoint, a neighbour of the endpoint is chosen at random and the path is reversed.
   **998 seconds gave a coverage of 99.9637% (888,509 / 888,832; 323 left out). It does not close.**
4. **Second stage (a working record, changing the design)**: instead of increasing the budget, the
   procedure was split into three and each was measured separately.

   | Design | Contents | Vertices inserted |
   |---|---|---|
   | (i) insertion | if two of the neighbours of v are adjacent on the path, insert v between them | **0 / 323** |
   | (ii) double exchange | if v~p_i, v~p_j and p_{i+1}~p_{j+1}, reverse the interval [i+1,j] and insert | **0 / 323** |
   | **(iii) one-step-lookahead rotation at the endpoint** | choose the rotation so that the new endpoint p_{i+1} **falls in the neighbourhood of a vertex left out** | **323 / 323** |

   Pósa's rotation is "choose a neighbour p_i of the endpoint e on the path, cut the edge (p_i, p_{i+1})
   and add (e, p_i)", and **the new endpoint becomes p_{i+1}**. **Rather than choosing p_i at random,
   choose it so that p_{i+1} falls in the neighbourhood of a vertex left out** — reading one step ahead.
   The endpoint has 31 neighbours on the path, and the neighbourhood of the leftover set has
   323×31 ≈ 10⁴ elements, so the probability of a hit in one rotation is
   31 × (10⁴/888,832) ≈ 35%. A few rotations per leftover vertex are enough.

   **(iii) inserted all 323 in 53.3 seconds with 6,906 rotations, and then closed the path into a cycle in a further 3.5 seconds with 439 rotations.**

   | | A working record (random, 998 s) | A working record (lookahead, 53 s) |
   |---|---|---|
   | Left out | 8,116 → 2,453 → 323 | **323 → 0** |
   | Rotations | 128,081 | **6,906** |
   | Total reversed length | 56.4 billion | **2.98 billion** (**a factor of 19**) |

   **That (i) and (ii) do not work is not a property of SGL₆ but a matter of numbers** —
   the expected number of degree-31 vertices whose neighbours are adjacent on a path of length 888,509 is
   323 × 2·31·31/888,509 ≈ **0.70**, and even for the double exchange there are 465 pairs, so the
   expectation stops at about 5.
   **The measurements are 0 in both cases, consistent with this count.**

**On the reproducibility of this search**: Pósa involves randomisation, so the same cycle will not
necessarily appear again. **What has to be reproduced is not the cycle itself but the fact that a cycle
exists**, and that is what `sgl6_cycle_mat.txt` and `verify_sgl6.py` are delivered for. So that one can
decide whether a cycle found by someone else is the same one,
**a normalised hash independent of rotation and reflection** is given in §6.

---

## 4. Deliverables

| File | Contents | Size |
|---|---|---|
| **`sgl6_cycle_mat.txt`** | The cycle itself. **One line = one vertex = the 6×6 symmetric matrix over F₂**, printed row by row as six digits of 0/1 (six such groups, space-separated) | 888,832 lines / 37,330,944 bytes |
| **`verify_sgl6.py`** | The independent verifier. **Depends only on the Python 3 standard library** (no numpy). It reads `sgl6_cycle_mat.txt` and nothing else | 154 lines |
| `verify_sgl6.out` / `n960_verify_rerun.out` | The output of the runs above (a working session, 36.8 seconds / the present working record, **35.5 seconds**) | — |
| `sgl6_cycle.txt` | The sequence of 21-bit codes in decimal (in storage order; for reference. **The verifier does not read it**) | 888,832 lines / 6,648,840 bytes |
| `n960_canon.py` / `n960_canon.out` | Recomputation of the closed formulas of the printed paper, the normalised hash, and the cross-check that the two files are the same sequence | — |
| `n940_sgl6.py` | The search program (a separate machine, uses numpy). **Not a verifier** | 284 lines |

**Usage**:

```bash
python3 verify_sgl6.py sgl6_cycle_mat.txt      # no argument = same-named file
# exit code 0 = pass, 1 = fail
```

---

## 5. The verification procedure (what `verify_sgl6.py` does)

Only **two things** are checked. These two exhaust the claim.

> **(i)** The 888,832 lines of the file are distinct and **exactly exhaust** SGL₆(F₂),
> the set of all invertible symmetric 6×6 matrices.
> **(ii)** The differences of cyclically adjacent lines **all have rank 1**.

### 5.1 The decision procedure is on a different path from the search program

| | The search program `n940_sgl6.py` | The verifier `verify_sgl6.py` |
|---|---|---|
| Machine | **a separate machine** | this machine |
| Dependencies | numpy 2.3.5 (sweeps all 2²¹ at once with arrays) | **standard library only** |
| Input | 21-bit codes (in memory) | **`.txt` (the matrices themselves)** |
| Population | exhaustive, likewise | **rebuilt exhaustively** (Orel's formula is not used) |
| Adjacency | whether the XOR of the codes lies in the rank-1 set | **decode the matrices and measure the rank of A+B directly by elimination** |

**The logic of the adjacency test (this is the crux)**: the verifier does not ask "does the difference
belong to the rank-1 set S"; it **measures the rank of the difference A + B by elimination and sees
whether it is 1**. The hole of "only a sufficient condition" that the examination of a working session
pointed out about a working record (the first version of HGL₄) — that `D ∈ S ⟹ rk D = 1` can be used but
the converse is sometimes needed — **is simply absent in this form from the start**.
On top of that, it also checks exhaustively that `{x xᵀ : x ≠ 0}` coincides with the set of all rank-1
symmetric matrices ([4]).

### 5.2 Result of the run (verbatim from `n960_verify_rerun.out`; the present working record ran it again itself)

```
[0] 入力 sgl6_cycle_mat.txt : 888832 行   (2.2s)
[1] 対称である行 = 888832 / 888832
[1] 可逆である行（掃き出し） = 888832 / 888832   (8.3s)
[2] |S_6(F_2)| = 2^21 = 2097152
[2] |SGL_6(F_2)| = 888832   (29.5s)
[3] 行数 = |SGL_6(F_2)| : True
[3] 全行が相異なる      : True
[3] 行の集合 = SGL_6(F_2) : True
[4] 階数 1 の対称行列の個数 = 63   (2^6 − 1 = 63)
[4] {x x^T : x ≠ 0} = 階数 1 の対称行列 : True
[5] 閉路の辺の総数（末尾→先頭を含む） = 888832
[5] 差 A + B の階数が 1 でない辺 = 0 本
[5] 使われた階数 1 の元の種類 = 63 / 63
[6] 無作為 100 頂点の次数の集合 = [31, 63]   （SGL_6 は正則でない）

[判定] **合格**  頂点 888832 / 行 888832 / 相異なる True / 集合一致 True / 悪い辺 0
所要 35.5 秒
```

The verifier prints its report in Japanese; the block above is left exactly as the program emits it.
Read line by line: `[0]` names the input file and its line count (行 = lines);
`[1]` gives how many lines are symmetric and how many are invertible (by elimination);
`[2]` gives the sizes of S₆(F₂) and of SGL₆(F₂); `[3]` states whether the number of lines equals
|SGL₆(F₂)|, whether all lines are distinct, and whether the set of lines equals SGL₆(F₂);
`[4]` gives the number of rank-one symmetric matrices and whether `{x xᵀ : x ≠ 0}` coincides with them;
`[5]` gives the total number of cycle edges (including the wrap-around from the last line to the first),
the number of edges whose difference A + B does not have rank one, and how many kinds of rank-one element
were used; `[6]` gives the set of degrees of randomly sampled vertices, with the remark that SGL₆ is not
regular. The verdict line reads 合格 = **pass**, and 所要 = elapsed time.

**Cross-check against the printed paper**: `|SGL₆(F₂)| = 888,832` agrees with the formula of §2.3
(recomputed by the present working record in integer arithmetic).
The degrees {31, 63} also agree with Prop. 12 of the printed paper (non-alternate 2^{n−1} − 1, alternate 2ⁿ − 1).
**The verifier does not take these two from the printed paper but produces them independently by exhaustive computation.**

---

## 6. Where the code is, and its hashes

All of it is in the working directory (verbatim from `n960_canon.out` [4]).

| Item | sha256 |
|---|---|
| `sgl6_cycle_mat.txt` | `4b5e6be8839ad000d10427aed83a1522a73cd6cdac8cc52d19ad986bef6bb5ad` |
| `sgl6_cycle.txt` | `77ead106f5e1ff1af2880429c1c0cead62c74bc26c876490b0f9ef5952619b6d` |
| `verify_sgl6.py` | `31a1deb974ba6b866f6e7bf162d0d2901af92cf62518323d86722820e78a9b32` |
| `n940_sgl6.py` | `89627471a6e9d2528b0414eea857a6e633f8b30e4bd3a733a74bafa0a6089437` |
| `n960_canon.py` | `9e8724da7e9140e5a7fcdd6361ba00bdf854786233b5832404dc997842c2b641` |

**The hashes of the cycle itself**:

| Item | sha256 |
|---|---|
| the 21-bit codes as comma-separated decimals (in storage order) | `529a55de313b353265b7a57706face085c25b7d3ab29b6bb2d68161c875aa076` |
| **the code sequence normalised for rotation and reflection** | **`c3e3aba6a601bf700d55d9995188a12131270907391d242ee8b5d051de77059e`** |

The normalisation follows the same convention as §6 of HGL4-NOTE — "place the smallest code (**4640**)
first, and take the direction in which the second element is smaller"
(for this cycle **the reverse of the storage order** was selected).
**If the cycle itself is the same, this value agrees regardless of the direction of storage and of the
starting point.** When a third party finds another cycle, this one line decides whether it is the same one.

**Proofreading**: `sgl6_cycle_mat.txt` (the matrices) and `sgl6_cycle.txt` (the codes) were matched
against each other and **all 888,832 lines agreed** (`n960_canon.out` [3]; 0 lines disagreeing).
That is, the conversion from codes to matrices loses no information.
**However, what the verifier reads is only the matrix side**, so **there is no need to trust the code
convention** — the codes are used only for the normalised hash.

**The convention for the 21-bit code** (a convention for turning the `.txt` back into codes; it is not
part of the verification):

- order the pairs `(i, j)` (0 ≤ i ≤ j ≤ 5) lexicographically and assign them the bit numbers 0..20
- when that bit is 1, `A[i][j] = A[j][i] = 1`

---

## 7. Relation to what is known

### 7.1 Orel's table, and the one line this document adds

| Graph | Vertices | Vertex-transitive | Hamilton cycle | Source |
|---|---|---|---|---|
| HGL₂(F₄) = Petersen | 10 | yes | **none** (classical) | — |
| SGL₃(F₂) = Coxeter | 28 | yes | **none** (classical) | — |
| HGL₃(F₄) | 280 | yes | **exists** | Orel 2015 (Concorde) |
| SGL₄(F₂) | 448 | no (n even) | **exists** | Orel 2015 (Concorde) |
| SGL₅(F₂) | 13,888 | yes | **exists** | Orel 2015 (Concorde) |
| HGL₄(F₄) | 38,080 | yes | **exists** | `HGL4-NOTE.md` (a record number; Pósa, a working record / verification by a working record, a working record) |
| **SGL₆(F₂)** | **888,832** | **no (n even)** | **exists** | **this document (one-step-lookahead Pósa, a working record / a working record / verification by a working record, a working record)** |
| HGL₅(F₄) | 18,887,680 | yes | open | a working record (this working session): 1,038 seconds, coverage 97.877%. **It does not close** |
| SGL₇(F₂) | 112,881,664 | yes | open | — |
| HGL₆(F₄) | 39,286,374,400 | yes | open | — |
| SGL₈(F₂) | 28,897,705,984 | no (n even) | open | — |

**⟹ What this document adds is only the single boldface line of the table above.**
On the SGL_{2n}(F₂) side, which Orel explicitly said he was asking about "including the
non-vertex-transitive case", **one positive data point, n=6, has been added**.
**Nothing has been answered about general n (Orel's second question).**

In the same paragraph, Orel also mentions the **subgraph induced by the non-alternating matrices** of the
SGL with even n (verbatim):

> Unlike the graph SGL_{2n}(F₂), its subgraph that is induced by nonalternating matrices
> is vertex-transitive. So studying the hamiltonicity of these subgraphs seems meaningful
> as well. We mention here just that in the case of 4 × 4 matrices we obtain a graph on 420
> vertices, which has a Hamilton cycle.

**For n=6 this subgraph has 874,944 vertices (= 888,832 − 13,888), and the printed paper does not touch it.
This document adds nothing there either** (the cycle of this document passes through all 888,832 vertices;
it does not give a Hamilton cycle of that subgraph).

### 7.2 Prior art — **the range searched is stated first**

**This document does not write "world first".** All that can be written is the following one sentence.

> **Within the search range of this ledger, no printed source giving the Hamiltonicity of SGL₆(F₂) was found.**

That "search range" is made explicit, in a form that lets the examination retrace the same steps.

**The search actually performed (2026-09-07, a working record)**: a general web search only,
in the same frame as the five queries of §7.2 of `HGL4-NOTE.md`, with the SGL-side terms added.

1. `"Orel Open Problem 16"`
2. `"SGL_n(F_2) Hamilton cycle"`
3. `"invertible symmetric matrices graph Hamiltonian"`
4. `"Coxeter graph" generalizations "Hamilton cycle"`
5. `"non-vertex-transitive" "Hamilton cycle" "symmetric matrices"`

**Sources found**: only Orel 2015 itself and two papers by the same author on adjacency-preserving maps
(arXiv:1307.3482 / 1307.3484).
**No source stating that Open Problem 16 has been solved came up.**

**Searches performed (a working record, 2026-09-08; transcribed by the examination of a working session)**:

| Means | Result |
|---|---|
| **Going through the complete citation list of Orel 2015** (a task carried over from a working session across four sessions) | **Done.** On OpenAlex (`W2172989702`, `cited_by_count` = 6), **all 6** were read down to title and abstract, and **none of them touches Hamiltonicity**. The breakdown is 4 on cores / complement prisms, 1 on rank-1 non-increasing maps, 1 on graph homomorphisms. Among them, **Orel–Višnjić, *Homomorphisms from the Coxeter graph*, Linear Algebra Appl. 2025** treats the same family `Γ_n = SGL_n(F₂)` in 2025 and **settles the classification of homomorphisms for n = 3**, yet **does not touch Open Problem 16**. **Semantic Scholar could not be retrieved (HTTP 429)** (unauthenticated, 3 retries), and **Crossref returns only a count (`is-referenced-by-count` = 3)**, not a list. The raw responses are in the citation folder in the working directory, the readout in the search output. **The examination recounted the stored JSON with a separate implementation and reproduced 6 items, 0 mentions of Hamilton, DOI 10.37236/3759 and a citation count of 3** (the check script / `.out`) |

**⟹ This one line raises the quality of "was not found" by one step** — a member of the family published
a paper on the same family in 2025, and no settlement of Hamiltonicity is written there.
**Even so, we do not write "world first".**
**Note also that the citation count being as low as 6 is not by itself evidence of difficulty**
(the reverse side of Wall 19[^6]: obscurity and difficulty are different things).
**What was obtained goes only as far as the abstracts, not the bodies.**

**Searches not performed (= the holes in this claim; filling them is the work of the examination)**:

| Means | Status |
|---|---|
| **MathSciNet** (Mathematical Reviews) | **There is no means of access in this environment** (a subscription is required) |
| **Zentralblatt MATH** | Same as above |
| **Full-text search of arXiv** (`math.CO` with "SGL" / "symmetric matrices" × "Hamilton") | **Not done** (**a working record ran 7 queries through the arXiv API index — titles, abstracts and authors — and the only hit for `SGL_n` / `HGL_n` was arXiv:2404.17067, with no occurrence of the word Hamilton. That is not a full-text index, so this line stays as it is**) |
| **Scanning the tables of contents of related issues and proceedings after EJC** | **Not done** |
| Non-English literature | **Not done** |

⟹ **Therefore we write not "there is none" but "was not found"** (the reverse side of Wall 19).
**The remaining holes are: MathSciNet, zbMATH, the arXiv full-text search, the scanning of tables
of contents of proceedings and the like, and non-English literature.**
**Whether it may be reported is the operator's decision.** Whether this document goes outside,
and if so in what range, is decided neither by this working session nor by this ledger.

(Its position in the ledger: it has been assigned a **record number** by the examination of a working
session. The record number is the ledger-internal fact that "the cycle in SGL₆(F₂) has become an
independently verified certificate"; it is not a claim of priority in the literature.)

### 7.3 The next open case

**HGL₅(F₄) (18,887,680 vertices, 165-regular)** is, among Orel's families, the **smallest open case after**
SGL₆(F₂) (the next one is SGL₇(F₂) with 112,881,664, six times larger).
A working record (this working session) applied one-step-lookahead Pósa here and
**did not close it: 1,038 seconds, coverage 97.877% (401,078 left out).**
The rate-limiting factor is not the difficulty of the search but **the representation that holds the path
in an array and reverses it** — the first 345 seconds reach 93.8% with one rotation each, and beyond that
a single rotation requires reversing half the path length, so the limiting speed falls from
51,400 vertices/second to 289 vertices/second.
⟹ **What should be changed next is the representation of the path (a balanced binary tree), not the
budget and not the depth of the lookahead.**

---

## 8. English summary

**Claim.** The graph SGL₆(F₂) — vertices: the 888,832 invertible symmetric 6×6 matrices
over F₂; edges: {A,B} with rk(A−B)=1 — **has a Hamilton cycle.**
This is the case n=6 of the second family in Orel's Open Problem 16
(*Electron. J. Combin.* **22**(4) (2015), #P4.27). It sits on the side Orel singled out:
for even n the graph SGL_n(F₂) is **not vertex-transitive** (not even regular; the
non-alternate and alternate vertices have different degrees), and Orel writes that he
states the general case "which include non-vertex-transitive graphs SGL_{2n}(F₂), as an
open problem". With Concorde, Orel found Hamilton cycles for n = 4 and 5 (448 and 13,888 vertices); n = 6 was left open.

**Certificate.** `sgl6_cycle_mat.txt`: 888,832 lines, one per vertex, each line the 6×6
symmetric matrix over F₂ written as six space-separated 6-digit 0/1 rows.

**Verifier.** `verify_sgl6.py` — **Python 3 standard library only**, reads only that text
file, runs in **35.5 seconds**. It checks exactly two things:
(i) the 888,832 lines are distinct and exhaust the set of invertible symmetric matrices —
the verifier rebuilds that set by sweeping all 2²¹ = 2,097,152 symmetric matrices and does
**not** use Orel's cardinality formula, so the number 888,832 is reproduced independently;
(ii) all 888,832 cyclic differences have rank exactly 1, measured by elimination on the
decoded matrices (**not** by membership in a precomputed rank-one set).

**How it was found.** Pósa rotation–extension. A run with uniformly random rotations
reached 99.9637% (323 vertices left) in 998 s and stalled. Changing the *procedure* — not
the budget — closed it: choosing the rotation so that the **new endpoint** lands in the
neighbourhood of a missing vertex (**one-step lookahead**) inserted all 323 in 53.3 s with
6,906 rotations, and 439 more rotations closed the path into a cycle. Total reversal length
dropped by a factor of 19. Plain insertion and double-exchange inserted **0** of the 323,
in agreement with a counting estimate made in advance (expected 0.70 and ≤5).

**Canonical hash** (rotation- and reflection-invariant, so a third party who finds another
cycle can tell whether it is the same one):
`c3e3aba6a601bf700d55d9995188a12131270907391d242ee8b5d051de77059e`.

**What is not claimed.** This is a heuristic search, so it can only settle the positive
side; it is not "better than Concorde". No general construction for arbitrary n is given.
Nothing is claimed about the vertex-transitive subgraph induced by non-alternate matrices
(874,944 vertices for n = 6), which Orel also mentions.

**Priority is explicitly not claimed.** We do **not** say this is new. What we can say is:
*within the search we actually performed, we found no printed source giving a Hamilton
cycle in SGL₆(F₂).* That search was a general web search on 2026-09-07 (§7.2). We **did**
go through the complete citation list of Orel 2015 (6 works on OpenAlex, read down to title
and abstract; none of them touches Hamiltonicity). We did **not** query MathSciNet or
zbMATH (no access), did **not** run an arXiv full-text search, did **not** scan the tables
of contents of related issues and proceedings, and did **not** search non-English
literature. Whether and how this note is circulated is the operator's decision, not ours.

---

## 9. Assumptions that remain

1. **The verbatim passages from the printed paper were copied from the OCR** (the OCR text in the working directory).
   **The OCR drops superscripts and subscripts.** For that reason this document read the degree formula of
   Prop. 12, **by matching it against the sentence on the side of the proof in the same paper**, as
   `2^{n−1} − 1` (non-alternate) and `2ⁿ − 1` (alternate) (see the note in §2.3).
   The values 31 and 63 for n=6 agree with the verifier's measurement, but
   **matching them against the relevant lines of the PDF (`orel_ejc22.pdf`) is the work of the examination.**
   **The claim (the existence of a Hamilton cycle) does not depend on this reading** —
   (i) and (ii) of §5 alone imply that it is a cycle.
2. **The negative statement that it is not vertex-transitive is taken from Orel Prop. 12.**
   What this verifier confirmed independently goes only as far as "the set of degrees of 100 randomly
   chosen vertices is {31, 63}"
   (that not regular ⟹ not vertex-transitive follows from this).
3. **"53.3 seconds" is not "53.3 seconds in total".**
   A working record restarts from the 99.9637% path that a working record took 998 seconds to build.
   **The cost of running the one-step lookahead from an empty path has not been measured**
   (carried over from a working session).
   ⟹ **"A factor of 19" is a comparison only of the stage that fills in the leftovers.**
4. **The generator and the verifier differ in machine and in implementation, but the author is the same.**
   A shared conceptual misunderstanding has not been ruled out.
5. **The basis for the prior-art statement is limited to the search range listed in §7.2.** What was
   done is the general web search with the five queries and the complete citation list of Orel 2015
   (6 works, down to title and abstract; none touches Hamiltonicity). **MathSciNet and zbMATH could not
   be reached, and the arXiv full-text search, the scanning of tables of contents of proceedings and the
   like, and non-English literature are not done.** For the citation list, too, what was obtained goes
   only as far as the abstracts, not the bodies.
   Therefore this document does not claim "world first". What it claims goes only as far as
   "**within the search range of this ledger it was not found**".
   **This single point is the weakest part of this document.**
6. **It has not been through a third-party examination.** The operator, the other lines and outside
   parties have all yet to do so.
7. **Whether it may be reported is the operator's decision.**

---

## 10. Attribution table

| What | Who | When / where |
|---|---|---|
| The definition of the graph SGLₙ(F₂), the cardinality formula, the degree and transitivity (Prop. 12), Open Problem 16, the decision to include the SGL with even n in the problem | **Marko Orel** | *On generalizations of the Petersen graph and the Coxeter graph*, **Electron. J. Combin. 22(4) (2015), #P4.27** (Published Nov 13, 2015) |
| The Hamilton cycles in HGL₃(F₄), SGL₄(F₂), SGL₅(F₂) | **Marko Orel** (by the Concorde TSP Solver [5]) | Same as above, corresponding to p.14 |
| The enumeration underlying the cardinality of SGLₙ(F₂) | **[2, Lemma 9.5.9]** (Brouwer–Cohen–Neumaier) / [20, Corollaries 3.31, 4.10] (Orel's citation) | Orel's bibliography |
| The Hamilton cycle of the 4×4 non-alternating induced subgraph (420 vertices) | **Marko Orel** | Same as above, corresponding to p.14 |
| The search method for Hamilton cycles (rotation plus extension) | **L. Pósa** (1976) | Classical |
| The survey of Hamiltonicity of vertex-transitive graphs (the 5 known exceptions) | Orel's citation **[9]** (survey) | Same as above |
| **The path up to 99.9637% by random Pósa** | **A working record** (a working session, 2026-09-07, a separate machine) | `sgl6_state.npz` |
| **The design called one-step-lookahead rotation, and the cycle itself** | The same **working record** (a working session, 2026-09-07, a separate machine) | `n940_sgl6.py` / `sgl6_cycle_mat.txt` |
| **The standard-library-only verifier** | The same **working record** | `verify_sgl6.py` |
| **The normalised hash, the recomputation of the closed formulas of the printed paper, and this document** | The same **working record** (a working session, 2026-09-08) | `n960_canon.py` / this document |

**The reference numbers** ([2][5][9][20]) are those of the bibliography of Orel 2015, and this
document **does not copy that bibliography**. To check which works the numbers point to, consult the
printed paper.

---

**Notes.** The numbered markers in the text gloss names that are internal to the working side's ledger. The wording of the text itself is left as it is.

[^1]: A name internal to the working side's ledger: a "working record" is the record of one unit of work.

[^2]: A name internal to the working side's ledger: a "working session" is the unit into which the work is divided.

[^3]: A name internal to the working side's ledger: the "ledger" is the bound collection of the working records.

[^4]: A name internal to the working side's ledger: a "record number" (written with a ★ in the ledger) is the ledger's own numbering; it has no bearing on priority in the literature.

[^5]: A name internal to the working side's ledger: "Wall N" numbers a rule of examination that the working side imposes on itself. Wall 39 is "do not mix a comparison of costs with a comparison of the objects".

[^6]: A name internal to the working side's ledger: "Wall N" numbers a rule of examination that the working side imposes on itself. Wall 19 is "do not treat 'not found by the search' as evidence that there is none — little attention is not evidence of difficulty".