The Matter of a Level

The BPT law

BPT Part 2.10 (The Strong Force and the Mass of Matter) reads the hierarchy between the Pulse and matter as the level itself. Its Matter Scale of a Level is Λ⊗ = E_P · 2^(−L/3): "the strong scale of a universe is its Planck energy over the cube root of its life in Pulses". Its Proton Mass is m_p·c² = κ⊗·Λ⊗, "a proton is a fixed multiple of the strong scale", κ⊗ ≈ 14.3 against the level's E_P·2^(−202/3). Together they give m_P/m_p = 2^(L/3)/κ⊗, κ⊗ = 14.3: at our level 202, 1.30 × 10¹⁹.

From that one ratio the Part derives the stars of every level. How Big a Star Can Be: a star holds about (m_P/m_p)³ = 2^L/κ³ protons, "at level 12 it is about one: a star would be a single proton. Below about level 12 there is not even one". Protons in a Star: N_p★ = 0.227·(m_P/m_p)³, where 0.227 = 0.42/1.85 is the mean birth mass of the Initial Mass Function over the Stellar Mass Unit. Stars of a Universe: N★ = 925·(m_P/m_p), with 925 = 0.06 × 3,500 / 0.227: Eddington's 3,500·(m_P/m_p)⁴ protons, 6 % of them in stars; "about 65 stars to a well at every level". Star Existence Condition: m_P/m_p > α^(−1/2), "stars can stand only where gravity between two protons is weaker than their electric force"; with κ = 14.3 the condition is a level, 2^(L/3) > 14.3·√137 ≈ 167, L > 3·log₂ 167 ≈ 22.2. "Below it gravity outweighs the electric force between protons and the universe holds matter but no stars." The book states its one assumption plainly: α is the same at every level.

The established physics

• The fine-structure constant α ≈ 1/137.036.
• Stars require α_G = (m_p/m_P)² < α (Carr & Rees 1979; Barrow & Tipler 1986; cited in BPT Part 2.10).
• 6 % of ordinary matter in stars (Fukugita & Peebles 2004, cited at app.js:4255).
• The strong scale and lattice QCD for κ (Dürr et al. 2008, cited in BPT Part 2.10).

The game's math

matterLaw(L, deficitBits = 0) app.js:4261:

Ld      = max(0, L + deficitBits/2)
mPmp    = 2^(Ld/3) / 14.3                    m_P / m_p
stable  = mPmp > 1
perStar = 0.227 · mPmp³                      protons in a typical star
stars   = 925 · mPmp                         stars the universe forms in its life
wells   = 2^(L/3)

The task notes said the stellar mass per Pulsecore is m_P/m_p = 2^(L/3)/14.3: confirmed, with L read as the universe's depth (scaleModel() app.js:4245 and app.js:5262 pass uni.depth when it is finite, else uni.L). deficitBits is never passed in 8.3, so Ld = L; the Pulsecore's measured shortfall pulsecoreInfo() is kept on the record only (app.js:5262).

STAR_LINE = √137.036 app.js:4260. In sendOutMatter() app.js:1252:

matter.unstable = !stable                    no stable matter
matter.noStars  = stable and mPmp < STAR_LINE

• No stable matter (L below about 12): each sub-step every body vanishes with probability 0.35·dt, flashing out as radiation (app.js:663); stepStars() is skipped; collapsing clumps are not registered as wells (app.js:691).
• Matter but no stars (about 12 to 22.2): stepStars() is skipped.
• Stars (above 22.2): everything in this chapter runs.

Worked values:

At every level the law gives 925/14.3 ≈ 65 stars a well, as BPT Part 2.10 says.

In the game

The player chooses the level before Start with the Level L control. Its tooltip (index.php:143) states the law: "stars stand only where gravity between two protons is weaker than their electric force, from about level 23 (the Star Existence Condition, Part 2.10). Shallower universes hold matter and wells but no stars; deeper ones have more and larger stars." The default level is 25, "a couple of levels past where stars begin".

The law is read at the Physical Nova, the third nova (PHYSICAL_LAYER = 3), when burstPulsecore() app.js:1259 sends the Pulsecore's data out as matter. Until then the HUD's Stars line (index.php:86) reads "none before the Physical Nova". From then it shows one of three universes (app.js:2145):

• Below level 12: "none: no stable matter at this level, a universe of light and wells". The matter the nova sends out does not stay: each body has a chance of 0.35 a flip to wink out as a warm-white flash on the torus (the "merge" flash, scene3d.js:1061), so within a few flips the field is light and wells.
• From 12 to 22.2: "none: gravity outweighs the electric force between protons at this level". The matter gathers into filaments and clumps as in Chapter 4, but stays gas haze; no point of starlight appears.
• Above 22.2: the stars of this chapter, starting with "none yet · the gas is still too warm to collapse" (stars.js:255) until the first stars form.

The run record keeps the law (law: { info_depth, mp_ratio, stable, protons_per_star, stars, wells }) and matter_state as 'unstable', 'no-stars' or 'stars' (app.js:2207–2208); the universe's language block lists the stars routine as "skipped: matter unstable at this level" or "skipped: m_P/m_p … ≤ α^−½ = 11.7" (app.js:2234).

The scale model

The law is exact at the true level, computed from the depth, not from the drawn field. What it sets is a target: how many stars the drawn ones must stand for (Part 5.3). The thresholds are exact.

Where the game and the theory differ

• BPT Part 2.10 says that below about level 12 a star would hold less than one proton, so no star can form; it does not say the matter itself is unstable. The game reads the same line as no stable matter at all: every body leaves at 0.35 per flip as radiation, and collapsing clumps there are not counted as wells (app.js:691). The rate 0.35 is the game's, not the book's.

Fixed numbers