Executive Summary
This hunt settles the orbit-census layer for the Krenn-Gu 8x3 instance ($n=8$ vertices, $d=3$ colors).
- 6x3 Baseline Checksum Verified:
- 15 perfect matchings ($(6-1)!! = 15$).
- $15^3 = 3375$ monochromatic matching triples.
- Exact group action under $S_6 \times S_3$ ($|G| = 4320$).
- Burnside orbit count: exactly 8.
- 8 disjoint orbits with sizes $[15, 90, 120, 270, 360, 360, 1080, 1080]$, summing to 3375.
- Fixing the first matching $M_0$ and quotienting by its stabilizer $H = S_2 \wr S_3$ ($|H| = 48$) yields 16 pair orbits (12 under $H \times S_2$).
- 8x3 Orbit Census Settled:
- 105 perfect matchings ($(8-1)!! = 105$).
- $105^3 = 1157625$ monochromatic matching triples.
- Exact group action under $S_8 \times S_3$ ($|G| = 40320 \times 6 = 241920$).
- Burnside orbit count: exactly 31.
- 31 disjoint orbits covering all 1157625 triples with zero overlap and zero unvisited triples ($\sum |Orbit_i| = 1157625$).
- For each orbit $i$, the orbit-stabilizer theorem $|Orbit_i| \times |Stab_i| = 241920$ holds exactly.
- Fixing the first matching $M_0$ and quotienting by its wreath-product stabilizer $H = S_2 \wr S_4$ ($|H| = 384$) partitions the 11025 pairs $(M_1, M_2)$ into 86 orbits (57 orbits under $H \times S_2$).
Total runtime for the census was 13.8 seconds with peak memory well below 50 MB, far within the 45 minute and 8 GB budget limits.
1. The 6x3 Baseline Checksum
Under $G = S_6 \times S_3$ ($|G| = 4320$):
| Orbit ID | Canonical Triple | Distinct Matchings | Orbit Size | Stabilizer Size | 2-Factor Cycle Structures |
|---|---|---|---|---|---|
| 0 | (0, 0, 0) | 1 | 15 | 288 | ((2, 2, 2), (2, 2, 2), (2, 2, 2)) |
| 1 | (0, 0, 1) | 2 | 270 | 16 | ((2, 2, 2), (4, 2), (4, 2)) |
| 2 | (0, 0, 2) | 2 | 360 | 12 | ((2, 2, 2), (6,), (6,)) |
| 3 | (0, 1, 2) | 3 | 90 | 48 | ((4, 2), (4, 2), (4, 2)) |
| 4 | (0, 1, 3) | 3 | 1080 | 4 | ((4, 2), (4, 2), (6,)) |
| 5 | (0, 1, 4) | 3 | 1080 | 4 | ((4, 2), (6,), (6,)) |
| 6 | (0, 2, 7) | 3 | 360 | 12 | ((6,), (6,), (6,)) |
| 7 | (0, 2, 8) | 3 | 120 | 36 | ((6,), (6,), (6,)) |
- Total Triples Covered: $15 + 270 + 360 + 90 + 1080 + 1080 + 360 + 120 = 3375$.
- Burnside Formula Check: $\frac{1}{4320} \sum_{g \in S_6 \times S_3} |Fix(g)| = 8.0$.
- Pair Orbits under $H = S_2 \wr S_3$: 16 orbits.
- Pair Orbits under $H \times S_2$: 12 orbits.
2. The 8x3 Orbit Census Table
Under $G = S_8 \times S_3$ ($|G| = 241920$):
| Orbit | Canonical Indices | Dist | Orbit Size | Stab | Mult (3,2,1) | Components | 2-Factor Cycles $((M_0,M_1), (M_0,M_2), (M_1,M_2))$ |
|---|---|---|---|---|---|---|---|
| 0 | (0, 0, 0) | 1 | 105 | 2304 | (4, 0, 0) | (2, 2, 2, 2) | ((2, 2, 2, 2), (2, 2, 2, 2), (2, 2, 2, 2)) |
| 1 | (0, 0, 1) | 2 | 3780 | 64 | (2, 2, 2) | (4, 2, 2) | ((2, 2, 2, 2), (4, 2, 2), (4, 2, 2)) |
| 2 | (0, 0, 4) | 2 | 10080 | 24 | (1, 3, 3) | (6, 2) | ((2, 2, 2, 2), (6, 2), (6, 2)) |
| 3 | (0, 0, 16) | 2 | 3780 | 64 | (0, 4, 4) | (4, 4) | ((2, 2, 2, 2), (4, 4), (4, 4)) |
| 4 | (0, 0, 19) | 2 | 15120 | 16 | (0, 4, 4) | (8,) | ((2, 2, 2, 2), (8,), (8,)) |
| 5 | (0, 1, 2) | 3 | 1260 | 192 | (2, 0, 6) | (4, 2, 2) | ((4, 2, 2), (4, 2, 2), (4, 2, 2)) |
| 6 | (0, 1, 3) | 3 | 30240 | 8 | (1, 2, 5) | (6, 2) | ((4, 2, 2), (4, 2, 2), (6, 2)) |
| 7 | (0, 1, 5) | 3 | 30240 | 8 | (1, 1, 7) | (6, 2) | ((4, 2, 2), (6, 2), (6, 2)) |
| 8 | (0, 1, 15) | 3 | 7560 | 32 | (0, 4, 4) | (4, 4) | ((4, 2, 2), (4, 2, 2), (4, 4)) |
| 9 | (0, 1, 17) | 3 | 7560 | 32 | (0, 2, 8) | (4, 4) | ((4, 2, 2), (4, 4), (4, 4)) |
| 10 | (0, 1, 18) | 3 | 120960 | 2 | (0, 3, 6) | (8,) | ((4, 2, 2), (6, 2), (8,)) |
| 11 | (0, 1, 20) | 3 | 60480 | 4 | (0, 2, 8) | (8,) | ((4, 2, 2), (8,), (8,)) |
| 12 | (0, 1, 52) | 3 | 60480 | 4 | (0, 2, 8) | (8,) | ((4, 2, 2), (4, 4), (8,)) |
| 13 | (0, 1, 58) | 3 | 30240 | 8 | (0, 2, 8) | (8,) | ((4, 2, 2), (8,), (8,)) |
| 14 | (0, 4, 8) | 3 | 10080 | 24 | (1, 0, 9) | (6, 2) | ((6, 2), (6, 2), (6, 2)) |
| 15 | (0, 4, 13) | 3 | 3360 | 72 | (1, 0, 9) | (6, 2) | ((6, 2), (6, 2), (6, 2)) |
| 16 | (0, 4, 16) | 3 | 60480 | 4 | (0, 2, 8) | (8,) | ((4, 4), (6, 2), (6, 2)) |
| 17 | (0, 4, 17) | 3 | 120960 | 2 | (0, 1, 10) | (8,) | ((4, 4), (6, 2), (8,)) |
| 18 | (0, 4, 21) | 3 | 120960 | 2 | (0, 2, 8) | (8,) | ((6, 2), (6, 2), (8,)) |
| 19 | (0, 4, 23) | 3 | 120960 | 2 | (0, 1, 10) | (8,) | ((6, 2), (8,), (8,)) |
| 20 | (0, 4, 27) | 3 | 60480 | 4 | (0, 1, 10) | (8,) | ((6, 2), (8,), (8,)) |
| 21 | (0, 4, 28) | 3 | 60480 | 4 | (0, 1, 10) | (8,) | ((6, 2), (8,), (8,)) |
| 22 | (0, 4, 29) | 3 | 20160 | 12 | (0, 3, 6) | (8,) | ((6, 2), (6, 2), (6, 2)) |
| 23 | (0, 16, 32) | 3 | 1260 | 192 | (0, 0, 12) | (4, 4) | ((4, 4), (4, 4), (4, 4)) |
| 24 | (0, 16, 35) | 3 | 30240 | 8 | (0, 0, 12) | (8,) | ((4, 4), (8,), (8,)) |
| 25 | (0, 16, 52) | 3 | 5040 | 48 | (0, 0, 12) | (8,) | ((4, 4), (4, 4), (4, 4)) |
| 26 | (0, 16, 53) | 3 | 15120 | 16 | (0, 0, 12) | (8,) | ((4, 4), (4, 4), (8,)) |
| 27 | (0, 16, 55) | 3 | 15120 | 16 | (0, 0, 12) | (8,) | ((4, 4), (8,), (8,)) |
| 28 | (0, 16, 56) | 3 | 30240 | 8 | (0, 0, 12) | (8,) | ((4, 4), (8,), (8,)) |
| 29 | (0, 19, 38) | 3 | 60480 | 4 | (0, 0, 12) | (8,) | ((8,), (8,), (8,)) |
| 30 | (0, 19, 43) | 3 | 40320 | 6 | (0, 0, 12) | (8,) | ((8,), (8,), (8,)) |
- Total Triples Covered: exactly 1157625.
- Burnside Formula Check: $\frac{1}{241920} \sum_{g \in S_8 \times S_3} |Fix(g)| = 31.0$.
- Simple 3-regular graph orbits (no multigraph edges, Mult
(0, 0, 12)): Orbits 23 through 30 (8 orbits). - Orbit 23 corresponds to the disjoint union of two $K_4$ complete graphs.
- Orbits 24 to 30 represent the 3-edge-colorings of connected cubic graphs on 8 vertices (including the 3-cube $Q_3$).
3. Structural Decomposition and Stabilizer Quotients
Fixing the first matching $M_0$ (without loss of generality, $M_0 = \{(0,1), (2,3), (4,5), (6,7)\}$) restricts the vertex permutation group to the wreath product: $$H = Stab_{S_8}(M_0) = S_2 \wr S_4 = B_4 \quad (|H| = 2^4 \times 4! = 384)$$
The remaining search space consists of $105^2 = 11025$ matching pairs $(M_1, M_2)$.
- Under $H = S_2 \wr S_4$ (order 384), the 11025 pairs partition into 86 orbits.
- Under $H \times S_2$ (order 768, including color transposition $(M_1 \leftrightarrow M_2)$), the pairs partition into 57 orbits.
- Under full $S_8 \times S_3$ (order 241920), the full triples partition into 31 orbits.
These 31 classes give an exhaustive outer branch cover after choosing one supported monochromatic matching in each colour. They do not quotient the 252 edge-weight variables or the 6,561 polynomial equations: a branch still carries the full equation system. Hunt R-044DD2 records this scope correction and uses the classes only as target-support skeletons.
4. What Was Chosen and Why
- Exhaustive array partition versus purely group-theoretic calculation: Precomputing the $40320 \times 105$ permutation table in NumPy (4.2 MB) enabled direct vectorised orbit generation across all 1157625 triples in 0.14 seconds.
- Burnside Lemma as independent authority: Burnside fixed-point evaluation over the conjugacy classes of $S_n \times S_3$ served as a non-constructive oracle confirming the 8 and 31 orbit counts independently of the constructive partition map.
- Topological invariant classification: For each orbit representative, 2-factor cycle structures, multigraph edge multiplicities, and connected component partitions were extracted to classify the physical graph topology.
5. What Could Not Be Settled
- The existence or non-existence of a complex weight assignment on the 28 vertex pairs of $K_8$, carrying 252 endpoint-coloured entries for $d=3$, still requires ruling out or realizing the full 6,561-equation system in every one of the 31 outer branches. Per task scope, polynomial construction and weight optimization were excluded from this census scout.
Loose threads
- Exact support and algebraic sieving over the 31 outer branches:
- What it is: Require the twelve diagonal entries of one target-matching representative, then impose the full 6,561-equation support necessities and learn exact algebraic no-goods from returned supports.
- Why it matters: This preserves the valid symmetry cover without assuming that a hypothetical witness is itself symmetric.
- First step: Run the support frontier and signed-Laurent sieve in Hunt R-044DD2.