Domain Walls and Quasicrystals

The BPT law

BPT Part 4.5 (True Expansion: Nova Containment and Dimensional Folds) speaks of boundaries between domains. Folds made at rupture "allow the UniSphere to expand through topological rewriting. This creates the computational analogue of cosmic strings or domain walls, while ensuring conservation laws remain intact and new universes retain independence." It adds that "regions of concentrated negative curvature resemble core geometry of cosmic strings and domain walls described by Vilenkin (1985), albeit instantiated in computational rather than field-theoretic form." BPT Part 3.9 (Fractal Progeny) describes Null Well Collapse, the event that seeds child universes, and the code maps the domain wall to "Null Well Collapse boundaries (ch 3 Part 3.9)" (states.js:55).

For the quasicrystal, BPT Part 3.6 (Data Novas and Dimensional Evolution) says that after the Dimensional Saturation Threshold, further novas are Harmonic Novas: "discharges of recursive density that no longer create new axes but instead reinforce alignment within the existing four-dimensional lattice". The code maps the quasicrystal to "Harmonic Novas as inner layers (ch 3 Part 3.6)" (states.js:54): order that reinforces without a new axis of repetition. BPT does not state either test. Both are game methods.

The established physics

A domain wall is the boundary where two domains of the same order meet out of step (Kibble 1976, as states.js:30 cites it; Vilenkin 1985, Cosmic strings and domain walls, in the BPT sources). A quasicrystal is ordered but never periodic, and it is built from a finite set of local patches, a property called finite local complexity (Shechtman 1984, as states.js:28 cites it).

The game's math

Domain walls (scan step 3, states.js:385–413). A cell is a wall cell if all of these hold:

1. It is a pulser of period exactly 2, outside any lattice region.
2. Looking along its row or its column, up to 8 cells each way, the nearest lattice cell on both sides belongs to the same lattice id.
3. The raw, uneroded agreement under one of that lattice's two basis shifts fails at this very cell. The order's own rule breaks here.

The 8-connected components of wall cells with at least 12 cells are kept if cells ≤ 3 · span, where span = max(h, w), so the shape is a line and not a patch. The signature is FNV of wall | round(log₂ cells), so walls are one species per size octave. Rarity is tier 2 at 96 cells or more, otherwise tier 1.

Quasicrystals (scan step 4, states.js:415–452). The candidates are pulsers of period 2 that belong to no lattice. A 4-connected component of them is a quasicrystal if it passes every test below:

1. It has at least 256 cells and a box of at least 16 × 16.
2. Its cells show at most 32 distinct 3 × 3 patches of the current frame (finite local complexity).
3. At least 16 interior cells have their whole 17 × 17 neighborhood, sampled every second cell, inside the region. A bulk, not a seam.
4. At no shift with 0 ≤ dr ≤ 12 and −12 ≤ dc ≤ 12 does the interior agree with itself in 98% of cells for two non-parallel shifts. If it does, the region is a big-tiled crystal: it is kept as a lattice with the reduced basis of those two shifts, and it is not a quasicrystal.

A quasicrystal's signature is FNV of its sorted patch set, and it is always tier 5 (Legendary).

Worked case. A 20 × 20 ordered region has 400 cells, which passes 256 and 16 × 16. Say it shows 14 distinct patches, has 16 or more cells whose 17 × 17 neighborhood lies inside it, and finds no two non-parallel 98% repeats up to 12 cells. It is reported as a quasicrystal of 400 cells, tier 5.

In the game

Walls and quasicrystals appear in the universe epoch where lattices have formed: a wall needs a lattice on both sides of it, and a quasicrystal is what is left ordered after the lattices are taken out. Neither is drawn on the field. Each lands in the Collection under "states of the field", worded "Domain wall · N cells" or "Quasicrystal · N cells" (finds.js:49). Their notice icons would be ⫽ and ✷ (states.js:54–55). A quasicrystal is always Legendary, the gold hue 48, so with the notices on it would always pop (Part 7.7). The run summary names them in its "◇ N states:" chip.

The scale model

Exact on the drawn field: no scale model.

Where the game and the theory differ

• BPT Part 4.5 places domain walls at the folds rupture makes, boundaries between regions that become separate universes. The game's domain wall is a line inside one universe's field where a lattice's own repeat breaks. The two share the name and the idea of two domains of one order meeting out of step. The game's wall has no link to rupture or to Null Well Collapse beyond the label in STATE_FAMILIES.
• The comment at states.js:26–27 and states.js:427 counts "32 of the 512 possible 3×3 patches". 512 = 2⁹ is the count for two states. The patch string is built from the three-state codes (0, 1, 2), and a patch at the region's edge can include null cells, so the possible patches number up to 3⁹ = 19,683. The test (≤ 32) is unaffected. Only the comment's denominator is wrong.

Fixed numbers