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

[glider_rider]

Are any of these infinite patches also self forcing? I assume they are all self forcing if my conjecture is true. These are all transformations of known patches, with a few exceptions.
EDIT:
All of these are self forcing. I did a quick check.

The pond agar is not self-forcing, it has the following predecessor:

Code: Select all

x = 6, y = 6, rule = B3/S23:T6,6
obo$o2b2o$o$3bobo$2obo$3bo!

I'm guessing what you meant is that it doesn't have any predecessors on a 3x3 torus.

[Anivec]

Are any of these infinite patches also self forcing? I assume they are all self forcing if my conjecture is true. These are all transformations of known patches, with a few exceptions.
EDIT:
All of these are self forcing. I did a quick check.

The pond agar is not self-forcing, it has the following predecessor:

Code: Select all

x = 6, y = 6, rule = B3/S23:T6,6
obo$o2b2o$o$3bobo$2obo$3bo!

Ah, I didn’t think it was truly self forcing.

[400spartans]

153-cell unsynthesizable still life:

Code: Select all

x = 22, y = 23, rule = B3/S23
13b2o$5bo6bobo$4bobo5bo6b2o$4bo2bo2b2obo2b2o2bo$2b2ob2obo2bo2bo2bobo$
2bo2bo2b2o2bob2o2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo3b2o$2bo2bo2b2o2b
ob2o2bo2bo$bobo2bo2bob2o2bob2obo$bob2ob2o2bo2bo2bo2bo$2bo2bo2b2o2bob2o
2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo$2bo2bo2b2o2bob2o2bo$bobo2bo2bob
2o2bob2o$o2b2ob2o2bo2bobo$obo2bo2b2o2bo2bo$bobo2bo2bob2obo$3b2ob2o2bo
2bo$2bo2bo5b2o$3bobo$4bo!

and a LifeHistory view of a 245-cell self-forcing patch for this still life:

Code: Select all

x = 22, y = 23, rule = LifeHistory
13.2A$5.A6.A.A$4.A.A4.DCD5.2A$4.A2.C2DACDC2D2C2.A$2.2A.2CDC2DC2DC2DCD
A$2.A2.C2D2C2DCD2C2DC$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D.2A$2.A2DC
2D2C2DCD2C2DA2.A$.A.C2DC2DCD2C2DCD2CDA$.A.ACD2C2DC2DC2DC2DC$2.A.DC2D
2C2DCD2C2DCD$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D$2.A2DC2D2C2DCD2C2D
C$.A.C2DC2DCD2C2DC.2C$A.D2CD2C2DC2DC.A$A.C2DC2D2C2DC2.A$.ADC2DC2DCD2C
.A$3.ACD2C2DCD.A$2.A.DCD4.2A$3.A.A$4.A!

[I6_I6]

153-cell unsynthesizable still life:

Code: Select all

x = 22, y = 23, rule = B3/S23
13b2o$5bo6bobo$4bobo5bo6b2o$4bo2bo2b2obo2b2o2bo$2b2ob2obo2bo2bo2bobo$
2bo2bo2b2o2bob2o2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo3b2o$2bo2bo2b2o2b
ob2o2bo2bo$bobo2bo2bob2o2bob2obo$bob2ob2o2bo2bo2bo2bo$2bo2bo2b2o2bob2o
2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo$2bo2bo2b2o2bob2o2bo$bobo2bo2bob
2o2bob2o$o2b2ob2o2bo2bobo$obo2bo2b2o2bo2bo$bobo2bo2bob2obo$3b2ob2o2bo
2bo$2bo2bo5b2o$3bobo$4bo!

and a LifeHistory view of a 245-cell self-forcing patch for this still life:

Code: Select all

x = 22, y = 23, rule = LifeHistory
13.2A$5.A6.A.A$4.A.A4.DCD5.2A$4.A2.C2DACDC2D2C2.A$2.2A.2CDC2DC2DC2DCD
A$2.A2.C2D2C2DCD2C2DC$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D.2A$2.A2DC
2D2C2DCD2C2DA2.A$.A.C2DC2DCD2C2DCD2CDA$.A.ACD2C2DC2DC2DC2DC$2.A.DC2D
2C2DCD2C2DCD$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D$2.A2DC2D2C2DCD2C2D
C$.A.C2DC2DCD2C2DC.2C$A.D2CD2C2DC2DC.A$A.C2DC2D2C2DC2.A$.ADC2DC2DCD2C
.A$3.ACD2C2DCD.A$2.A.DCD4.2A$3.A.A$4.A!

Brilliant! That's a new record by 1 cell. How did you find that?

[NNlk05]

Brilliant! That's a new record by 1 cell. How did you find that?

They said on Discord:

i decided to fire up the ol' unsynth life search program, no new population improvements, but found a few unsynthesizable still lifes that i thought were interesting:
...
153-cell unsynthesizable still life
...