Failed Proto Wells: failProtoWells()

The BPT law

BPT Part 3.9 (Fractal Progeny) gives the Reproductive Outcome Distribution, P▢ = 0.67, P≀ = 0.18, P⑂ = 0.09, P⦰ = 0.06: most collapses seed a viable universe, and in a fraction P⦰ "no child is seeded, and the data returns to the source without continuing back"; "the same law fails Proto Wells before the physical layer." Under "Why One in Sixteen Fails" it derives that share as the Dimensional Failure Law, P⦰ = 2^(−D): "A collapse fails only when every actual dimension of its universe returns null." D is the count of actual, fundamental dimensions, the prime harmonics of its pulse (Part 4.8); for each one "the returning data either carries it, 1, or holds its null, 0: the two states of a pulse, equally weighted and independent." Ours has four, so P⦰ = 1/16; the Zinf, whose harmonics hold the primes 2 and 3 alone, has D = 2 and "loses one collapse in four." BPT Part 3.13 (Failed Proto Wells) follows the data: "A Proto Well that fails never reaches the physical layer. Its data does not stay on the scaffold to become Data Matter, and it is not sent out as matter at the big bang: it goes home along the Data Loop before the Physical Nova." That sets the Data Matter Count with Failed Proto Wells, N◍ / N_b = (1 − P⦰) · 16 · 4^L / ((13/3)² · 4^L − 16 · 4^L).

The established physics

None: a BPT law. Each dimension returns 1 or 0 with equal weight, independently, so the all-null return has probability 2^(−D).

The game's math

failProtoWells(frame) app.js:1357 runs first thing in promoteDataWells() app.js:1378, at the Physical Nova.

D = actualDimensions(L) = 2 if L ≤ 0, 3 if L = 1, else 4      app.js:1329
P = 2^(−D)

For each Proto Well i, a 32-bit hash h(i, frame) is mixed from i + 1 and frame + 7. The well fails if h/2³² < P. The draw is deterministic: the same universe at the same frame fails the same wells.

Each pulsing cell of the field goes to its nearest Proto Well (wrapped distance, squared), BPT's basin, "every bit nearest to it." Cells whose nearest well failed are cleared: their data "went home to the Source and will be neither dark matter nor matter." Each failed well's record ends. wellsStand loses gone · wellWeight(). The ledger keeps failedWells and failedReturn (cells returned).

Worked case: at level 28, D = 4 and P = 1/16. Of 633 drawn Proto Wells about 40 fail, each standing for 1.019 true wells.

In the game

It happens at Harmonic 3, the moment the banner "Physical Nova · the big bang" comes up. On the torus about one magenta ring in sixteen goes out together with the patch of data skin round it, its whole basin, so dark gaps open in the scaffold just as the rest of the data condenses into matter. The notice the code writes, "40 of 633 Proto Wells failed (one in 16: every dimension of theirs returned null) · 2,315 cells of their data went home to the Source and will be neither dark matter nor matter" (app.js:1375), is held while notices are quiet; the counts go on the universe's record (failedWells, failedReturn) and the failed wells' life lines end there. The HUD's counted wells drop by the failed share.

The scale model

The failure share is exact. The drawn count of failures is a sample of P · N. The cells returned are drawn cells.

Fixed numbers