Stable patterns which are not glider-constructible

[HartmutHolzwart]

**Subject: Exploring the Preimage and Loop Landscape of Skew N x 1 Tori**

Hi everyone,

Following the landmark work on unconstructible still lifes and agar-based limitations by Ilkka Törmä and Ville Salo (https://arxiv.org), I would like to propose a systematic look into a specific 1D projection of Conway's Game of Life: **collapsing periodic agars into an N × 1 Skew Torus.**

Many space-filling, periodic 2D agars naturally collapse into highly compact 1D structures via a coordinate lattice projection (essentially a structural Hadamard-type matrix reduction). By focusing specifically on these N × 1 Skew Tori with a horizontal skew factor s, the 8-cell Moore neighborhood maps onto a 1D ring of length N with the following coordinate offsets:

[-1, +1] U [s-1, s, s+1] U [-s-1, -s, -s+1] (mod N)

While this strips away the standard 2D visual plane, it provides a highly structured, lower-dimensional environment for analyzing global transition graphs. The system can be modeled using the group ring F_2[t]/(t^N - 1), and the state-transition graph collapses cleanly into a quotient graph under the Dihedral group D_N.

### Observations from Small Sizes
Brute-force checks on small universes show that varying N and the skew s radically alters the topology of the quotient graph:

* **N=17, s=3:** The state space is dominated by the All-Dead (0) basin (~96%). The remaining states collapse into pure, static 1D Still Lifes (Solitons).
* **N=17, s=4:** The All-Dead basin expands to over 97%, but the system unlocks a true **Period-16 cycle** in the quotient graph (Period-32 in raw space) that deforms and breathes through different cell counts (3 -> 5 -> 4 -> 9 ...).
* **N=14, s=3:** Even dimensions yield true 1-loop classes that are pure **Phoenix Agars** (cell overlap between consecutive generations is exactly zero).

### SAT Solving Advantages
Given that searching for specific preimage structures or long grandfather chains is generally PSPACE-hard, the linear 1D chain topology of these skew tori seems uniquely suited for SAT solvers.

Because the All-Dead basin is overwhelmingly dominant, random bit assignments trigger immediate conflicts during Unit Propagation. A CDCL solver should be able to prune the 2^N search space exceptionally fast. Additionally, symmetry-breaking clauses can be strictly derived from the D_N orbits.

### Questions for Discussion
I am interested in using this framework to study:
1. **Quantitative Entropies:** Can we find a formal bound for the exponential decay of non-Garden-of-Eden states as N scales?
2. **Isolated Cycles:** What is the distribution of true, strict d-loops (isolated cycles with an In-Degree of exactly 1)?
3. **Asymptotic Behavior:** What kind of hidden structural properties or unexpected attractor behaviors emerge as N grows significantly larger, particularly when N is prime?

I would love to get your thoughts on this, or collaborate on optimizing the CNF clause generation for these asymmetric 1D neighborhoods to test larger rings (N >= 31).

Cheers,
Hartmut