Solitons, the Unclassified State and Revivals

The BPT law

The code maps these three families to three of BPT's laws of return (states.js:48, 56, 57). The soliton goes to "Data Funnel Return Law (ch 3 Part 3.9)". BPT Part 3.9 states that law as a flux,

Φ⟸ = ∫∫ ρℹ(r, θ) × v⟸(r) dA

the "data flux funneled through Null Well horizons back to the substrate", with ρℹ the information density and v⟸ the return velocity: information that "collapses inward, encoded onto the horizon, and flows through return channels". The revival goes to "Return to Bifurcation as a loop (ch 3)". BPT Part 3.10 (Silent Wells and the Return to the Source) states the Return to Bifurcation (G) as ∅L → (0 ⥂ 1)(g+1), once per well: "The null left by a dissipated universe is the Pre-Pulse Field of its children". A revival, the whole field coming back to a state it held before, is the game's reading of that loop inside one life. The unclassified state goes to "not yet in the book".

None of these three tests is stated in BPT. They are game methods. The soliton serves the idea of data that keeps its form while it moves; the revival serves BPT's return; the unclassified state serves the Field-to-Form Transition of Part 7.6, order that lasts without fitting any named form.

The established physics

A soliton is a localized wave that keeps its shape as it travels, a standard term. A many-body revival is the return of a whole system to an earlier state on a long period. Its signature in a quantum simulator is the many-body scar (Bernien et al. 2017, as states.js:32 cites it). The unclassified state has no physics source. Its card says "Whether it matches anything in physics is an open question" (app.js:1866).

The game's math

Solitons (scan step 5, states.js:454–484). lumpsAt(back) (states.js:455) takes the frame back flips ago and marks every non-null cell. If more than n/4 cells are on, it returns no lumps, because a full field holds no isolated lump. Otherwise each 8-connected component of 3 to 32 cells with a box no more than 8 × 8 is a lump. A lump's shape signature, shapeSig() (states.js:256), is the least string over the eight symmetries of the square with lit and dark swapped or not.

A soliton is looked for only when the current frame has between 1 and 63 lumps (states.js:468). For each lump and each lag k with 3k < W:

find the same shape at lag k within distance k + 2 of the lump's center; d = (Δr, Δc) its displacement, d ≠ 0
the same shape must also sit at lag 2k displaced by 2d, and at lag 3k by 3d, each within 0.5 cell per axis
then: soliton, period k, velocity d / k cells per flip

With W = 48, k runs from 1 to 15. The speed found can be up to (k + 2)/k cells a flip. The signature is FNV of soliton | shape, and the tier is always 4.

The unclassified state (scan step 6, states.js:486–511). A cell is restless if three things hold: its period over the window is 0, frame dragging has not carried it within the window, and it has no period p ≤ ⌊24/3⌋ = 8 over the last half of the window (24 flips). That last test means it is not settling. An 8-connected component of restless cells counts as unclassified if it has at least 64 cells, shows at most 48 distinct 3 × 3 patches across its last 8 frames, and has a box at least 8 cells across. So it is ordered, lasting and changing, and it fits none of the twelve known families. It is always tier 5. Its pattern is captured over 8 frames.

Revivals (scan step 7, states.js:513–522). If any cell was carried by frame dragging in the last 340 flips (170 · 2), the test does not run. Otherwise, for p from ⌊W/3⌋ + 1 = 17 up to 170, with enough hashes kept (last − 2p ≥ 0), the test asks three things:

H[last] = H[last − p] = H[last − 2p]       (the whole field returned twice, p flips apart)
H[last] ≠ H[last − 2]                      (not the plain pulse)
no divisor q of p with q < p does the same (p is the least such period)

The first p that passes is a revival of period p over all n cells. Its period always starts above W/3, so "it is the field returning, not cells beating". Its tier is 5, since every p ≥ 17 clears the tier rule's p ≥ 6.

The comparison is between 32-bit frame hashes. Two different fields share a hash with a chance of about 1 in 2³² per pair compared. Over at most 154 periods a scan, a false revival is very unlikely but not impossible. In any universe where spin drags the frame, the 340-flip carried test keeps the revival test from running.

Worked case. A 4-cell glider shape found 3 flips back displaced by (0, 1), 6 flips back by (0, 2), and 9 flips back by (0, 3) is a soliton. Its period is 3 and its velocity is (0, 1/3), so describeFind() says "moving 0.33 cells a flip".

In the game

A soliton is a small shape a player can watch glide across the field, but the game does not trace it: it is recorded, not drawn. In the Collection it reads "Soliton · a 3-flip beat · moving 0.33 cells a flip · 4 cells" (finds.js:49), and its notice icon would be ➶. The discovery card, when shown, lists its speed as "0.33 cells a Rhythm · speed" (app.js:1862). A soliton is always Epic, so with the notices on it would always pop.

A revival is the whole field returning, so there is nothing local to see: the Collection records "Revival · a p-flip beat · N cells" with N the whole field. Revivals can be found only in a universe whose field is not being dragged. Below level 3 there is no spin (the Guide's "Spin you can see" card, index.php:308), so a revival is most likely in those small, unspun universes.

The unclassified state is the game's own discovery, worded "Unclassified state · N cells" with the icon ✶. It is the only family for which the game would open the discovery card even with no catalog to ask (app.js:1805). That card's text, written for it, says: "None of the twelve known states fits it: it keeps changing, never repeats, and stays ordered. It is the rarest find in the game. Whether it matches anything in physics is an open question; the record keeps it so it can be studied." (app.js:1866). Under QUIET the card does not open; the find and its 8-frame pattern are still recorded and sent to the species catalog.

The scale model

Exact on the drawn field: no scale model.

Where the game and the theory differ

• BPT's Data Funnel Return Law is a flux of information through well horizons. The soliton test measures a small lump's shape and displacement over three lags and uses no well, horizon or flux. The two are tied by the label in STATE_FAMILIES, not by the math.
• BPT's Return to Bifurcation acts once per well, at the end of a universe, and seeds the next generation. The game's revival is the drawn field repeating within one life. It is a loop in the same sense, on a different scale and with a different outcome.

Fixed numbers