Sites, Their Sizes, and the Site Lift

The BPT law

BPT Part 3.14 (Proto Wells: Where Wells Will Form) makes each Proto Well a site: "Formed in the data before the physical layer, it is the deepest place of the Data Matter, the seed of a halo (Part 3.13), and at the Physical Nova it becomes a site: where the gas will gather, where the first stars will light, and where a well will be born. It is the data's prediction of where the wells of matter will form, as the peaks of the early density field are where halos collapse in standard cosmology (Bardeen, Bond, Kaiser & Szalay, 1986)." A well is born at a site by direct collapse early, or by its first dying star sinking to its heart. Then "the prediction scored. If the Proto Wells are where wells will form, wells are born inside sites far more often than chance":

Site Lift               Λ_s = N_in / (N_w · A_s)

"How much more often wells are born inside sites than by chance", with Λ_s "the lift: 1 is chance", N_in "wells born within a site's core, the radius holding half its matter", N_w "all wells born", and A_s "the share of the surface within the sites' cores"; "Read in every universe the game plays, with the share of sites holding their well over its life." The Game's Reading adds: "Every well born is tested for a site's core, and the Site Lift and the share of sites holding their well over the life go on the universe's record with each site's conditions and how and when its well was born." BPT Part 3.14 (Testable Predictions) lists it among the theory's predictions: "wells, and the galaxies round them, should form where the Proto Wells stood, far more often than chance (Site Lift)."

The established physics

That halos collapse at the peaks of the early density field is standard cosmology (Bardeen, Bond, Kaiser & Szalay 1986, as BPT Part 3.14 cites it). The lift itself is an enrichment ratio, a test of the theory's prediction against chance; no physics source gives it. A game method.

The game's math

A site's sizes (haloRadius(w, g, share) app.js:1409, set at app.js:3107 every reading). Over the galaxy's own bodies (Part 6.3), sorted by periodic distance from the well, the radius at which the running mass first reaches share × total, never under SITE_R_MIN = 1 cell (1 cell if it has no bodies):

siteR    = haloRadius(w, g, 0.9)      its halo
siteCore = haloRadius(w, g, 0.5)      its core

siteNear(r, c) app.js:1425 returns the waiting site nearest (r, c) whose siteR contains it, the site a dying star or a collapsing clump there fills (the filling itself is in the wells chapter).

Scoring every birth (scoreBirth(r, c) app.js:1483, called for every physical well born after the Physical Nova, app.js:1530). If any sites exist: N_w += 1; and if the birth lies within the siteCore of any well that was ever a site (waiting or filled), N_in += 1. A direct-collapse birth adds to both N_w and N_in in fillSite() app.js:1446, since it is born in its site by definition.

The surface share (sampleSites() app.js:1489, at each life sample, app.js:926). The sites S are every well with siteBorn set. A 96 × 96 grid of points is laid on the field at ((i + 0.5)·rows/96, (j + 0.5)·cols/96); the sites are binned by the widest core, Rm = max(1, min(min(rows, cols)/2, max siteCore)); a point is a hit if it lies inside any site's core. Then

area sample = hits / 9216
filled curve point = (life share, filled sites / all sites)     the last 200 kept

The record (siteRecord() app.js:1502):

A_s = mean of the area samples
Λ_s = (N_in / N_w) / A_s

with A_s to 4 decimals, Λ_s to 3, and the first 400 sites' entries.

Example: if the cores cover 461 of the 9216 points (A_s = 0.050) and 30 of 60 wells were born inside a core, Λ_s = 0.5 / 0.05 = 10: wells are born inside the sites ten times as often as chance.

In the game

The sites themselves are not drawn: BPT Part 3.14 (The Game's Reading) says a site "is not counted, drawn, ranked or shining, and it swallows nothing", and the code keeps that. What the player sees of them is their work: the gas and the first stars gather round a site from the big bang on, and a galaxy is drawn round it (Part 6.2) before it has any well. When the site fills, by direct collapse or by its first stellar well sinking to its heart, a heart appears at the galaxy's center, drawn as the small mark of a galaxy's heart, and blazing if it feeds as a Seyfert or quasar (wells chapter).

The lift is not shown on screen. It is a number of the run record: runRecord() puts siteRecord() under sites (app.js:2213), with the sites' total, how many filled by direct collapse, by a remnant or by a clump, how many still wait, the births scored and born inside, the area share, the lift, the filled-share curve over the life and each site's entry. The record is saved at each nova and at the final well, and it travels in the exported packet ("Export .ud": "Download this universe as a .ud packet: its genesis, Pulsecore, record and lineage", index.php:260). So every universe a player runs scores the theory's prediction, whether or not they look.

The scale model

Exact: no scale model in the counts and the lift, which are ratios of counts and of areas. The area is sampled, not integrated: the grid points are rows/96 cells apart, 2.67 cells on a desktop and 0.67 on a phone.

Where the game and the theory differ

• Two features of the count shift the lift upward and should be read with it: every direct-collapse birth is inside by construction, and the cores of sites already filled still count as targets for later births. BPT's N_in counts "wells born within a site's core" and says neither.
• On a desktop a core under about one and a half cells across may fall between the sample points and read as no area, which would raise the lift.
• A site's radius is never under SITE_R_MIN = 1 cell, and a site with no bodies is a point a cell across; BPT's core is "the radius holding half its matter" with no floor.

Fixed numbers