Stellar Formalism: The Gas, the Stellar Mass Unit, and the Stellar Null Well

The physical picture of gas, stars and Stellar Null Wells is set out in Part 2.9; here we give its formal structure. Every law below is established by observation and standard astrophysics; what Binary Pulse Theory adds is where each constant comes from: the Planck values of the level (Part 6.5), inherited from the well a universe was born in, and the gas the Physical Nova leaves (Part 3.6). Masses are in kilograms or in Suns (M☉ = 1.989 × 10³⁰ kg), times in seconds or years.

The Collapse of the Gas

Jeans Length G

λ_J = σ · √(π / (G · ρ))

Dimensional analysis:
Where (from the book): λ_J [𝕃], σ [𝕃·𝕋⁻¹], G [𝕄⁻¹·𝕃³·𝕋⁻²], ρ [𝕄·𝕃⁻³]
[𝕃]
[𝕃·𝕋⁻¹] × √([∅] ÷ ([𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄·𝕃⁻³])) = [𝕃·𝕋⁻¹] × √([∅] ÷ [𝕋⁻²]) = [𝕃·𝕋⁻¹] × √[𝕋²] = [𝕃·𝕋⁻¹] × [𝕋] = [𝕃]
[𝕃] = [𝕃] ✓ Valid PulseCore Verified ✓

The smallest size at which a region of gas collapses under its own gravity

Where:

• λ_J [𝕃] — the Jeans length
• σ [𝕃·𝕋⁻¹] — the speed of the gas's random motion, its sound speed
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling
• ρ [𝕄·𝕃⁻³] — the density of the gas

➢ Below λ_J pressure waves cross the region faster than it can fall in, and it rings instead of collapsing; above it, gravity wins. The Jeans mass of Part 2.9 is the mass of a sphere of diameter λ_J, M_J = (π/6)·ρ·λ_J³, which gives its form (π^(5/2)/6)·σ³/(G^(3/2)·ρ^(1/2)).

Free-Fall Time G

t⇓ = √(3·π / (32 · G · ρ))

Dimensional analysis:
Where (from the book): t⇓ [𝕋], G [𝕄⁻¹·𝕃³·𝕋⁻²], ρ [𝕄·𝕃⁻³]
[𝕋]
√([∅] × [∅] ÷ ([∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄·𝕃⁻³])) = √([∅] ÷ ([∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄·𝕃⁻³])) = √([∅] ÷ ([𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄·𝕃⁻³])) = √([∅] ÷ [𝕋⁻²]) = √[𝕋²] = [𝕋]
[𝕋] = [𝕋] ✓ Valid PulseCore Verified ✓

How long a cloud takes to collapse with nothing holding it up

Where:

• t⇓ [𝕋] — the free-fall time
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling
• ρ [𝕄·𝕃⁻³] — the density of the cloud

➢ It depends on the density alone: a molecular cloud core of 10⁴ molecules a cubic centimetre falls in about 3 × 10⁵ years. This is the clock of star formation, and in this book it is also the clock the game's G is set by, the collapse time of the universe's own matter (Part 3.10, the Rupture Rule).

Star Formation Rate G

Ṁ⊛ = ε⇓ · M_gas / t⇓

Dimensional analysis:
Where (from the book): Ṁ⊛ [𝕄·𝕋⁻¹], ε⇓ [∅], M_gas [𝕄], t⇓ [𝕋]
[𝕄·𝕋⁻¹]
[∅] × [𝕄] ÷ [𝕋] = [𝕄] ÷ [𝕋] = [𝕄·𝕋⁻¹]
[𝕄·𝕋⁻¹] = [𝕄·𝕋⁻¹] ✓ Valid PulseCore Verified ✓

Gas turns into stars at about one percent of its mass per free-fall time

Where:

• Ṁ⊛ [𝕄·𝕋⁻¹] — the rate gas becomes stars
• ε⇓ [∅] — the efficiency per free-fall time, ≈ 0.01 (Krumholz & McKee, 2005)
• M_gas [𝕄] — the mass of dense gas
• t⇓ [𝕋] — the free-fall time

➢ The efficiency is low because stars push back: their light, winds and supernovae heat and disperse the gas that would form more of them. A universe therefore turns its gas into stars slowly, over many free-fall times, and the gas left over is what its later generations, and its Null Wells, feed on.

The Stellar Mass Unit

Planck Mass of a Level G

m_P = √(ħ · c / G)

Dimensional analysis:
Where (from the book): m_P [𝕄], ħ [𝕄·𝕃²·𝕋⁻¹], c [𝕃·𝕋⁻¹], G [𝕄⁻¹·𝕃³·𝕋⁻²]
[𝕄]
√([𝕄·𝕃²·𝕋⁻¹] × [𝕃·𝕋⁻¹] ÷ [𝕄⁻¹·𝕃³·𝕋⁻²]) = √([𝕄·𝕃³·𝕋⁻²] ÷ [𝕄⁻¹·𝕃³·𝕋⁻²]) = √[𝕄²] = [𝕄]
[𝕄] = [𝕄] ✓ Valid PulseCore Verified ✓

The mass of one Pulse at the level, built from its Planck constant, light speed and gravity

Where:

• m_P [𝕄] — the Planck mass, 2.176 × 10⁻⁸ kg in our universe
• ħ [𝕄·𝕃²·𝕋⁻¹] — the reduced Planck constant of the level (Part 2.5)
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling

➢ In this book ħ, and with it the Planck mass, is a constant of the level, set at genesis by the well (Part 6.5, Density and the Relative Planck Constant). Every stellar quantity below carries m_P, and so carries the lineage.

Stellar Mass Unit, Formal G

M⋆ = (m_P / m_H)² · m_P

Dimensional analysis:
Where (from the book): M⋆ [𝕄], m_P [𝕄], m_H [𝕄], ħ [𝕄·𝕃²·𝕋⁻¹], c [𝕃·𝕋⁻¹], G [𝕄⁻¹·𝕃³·𝕋⁻²]
[𝕄]
([𝕄] ÷ [𝕄])² × [𝕄] = [∅]² × [𝕄] = [∅] × [𝕄] = [𝕄]
[𝕄] = [𝕄] ✓ Valid PulseCore Verified ✓

The one mass every star, limit and remnant is a multiple of

Where:

• M⋆ [𝕄] — the Stellar Mass Unit: 3.68 × 10³⁰ kg = 1.85 M☉ in our universe
• m_P [𝕄] — the Planck mass of the level
• m_H [𝕄] — the hydrogen mass, 1.6735 × 10⁻²⁷ kg
• ħ [𝕄·𝕃²·𝕋⁻¹] — the reduced Planck constant of the level (Part 2.5)
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling

➢ A star in balance has its gravity held off by pressure; its central temperature must reach fusion, its electrons must not yet be crushed together, and its own light must not blow it apart. Each of those conditions sets its mass within a factor of order unity of M⋆ (Carr & Rees, 1979): the hydrogen-burning limit ≈ 0.04 M⋆, the Chandrasekhar limit ≈ 0.77 M⋆, the most massive stable stars ≈ 50 – 100 M⋆. The ratio m_P/m_H ≈ 1.3 × 10¹⁹ is why a star holds about 10⁵⁷ atoms.

Solar Mass of a Universe G

M☉(u) ≈ 0.54 · M⋆(u)

Dimensional analysis:
Where (from the book): M☉(u) [𝕄], M⋆(u) [𝕄]
[𝕄] ≈ [∅] × [𝕄] = [𝕄] ≈ [𝕄] ✓ Valid PulseCore Verified ✓

Each universe's own Sun, a fixed share of its Stellar Mass Unit

Where:

• M☉(u) [𝕄] — the mass of a Sun-like star in universe u
• M⋆(u) [𝕄] — that universe's Stellar Mass Unit, from its own Planck values

➢ Our Sun is 0.54 of our Stellar Mass Unit. A universe born of a different well, with a different ħ and so a different m_P, has a Sun of a different mass, and its whole stellar sequence (its red dwarfs, its supernova threshold, its Stellar Null Wells) moves with it, in the same proportions.

Stars in Balance

Mass–Luminosity Relation G

L⊛ = L☉ · (M/M☉)^(3.5)

Dimensional analysis:
Where (from the book): L⊛ [𝕄·𝕃²·𝕋⁻³], L☉ [𝕄·𝕃²·𝕋⁻³], M [𝕄], M☉ [𝕄]
[𝕄·𝕃²·𝕋⁻³]
[𝕄·𝕃²·𝕋⁻³] × ([𝕄] ÷ [𝕄])^([∅]) = [𝕄·𝕃²·𝕋⁻³] × ([𝕄] ÷ [𝕄])^([∅]) = [𝕄·𝕃²·𝕋⁻³] × [∅]^([∅]) = [𝕄·𝕃²·𝕋⁻³] × [∅] = [𝕄·𝕃²·𝕋⁻³]
[𝕄·𝕃²·𝕋⁻³] = [𝕄·𝕃²·𝕋⁻³] ✓ Valid PulseCore Verified ✓

A star's brightness rises steeply with its mass

Where:

• L⊛ [𝕄·𝕃²·𝕋⁻³] — the luminosity of a main-sequence star
• L☉ [𝕄·𝕃²·𝕋⁻³] — the Sun's, 3.828 × 10²⁶ W
• M [𝕄] — the star's mass
• M☉ [𝕄] — the mass of the Sun

➢ The exponent runs from about 2.3 for red dwarfs to about 4 near a Sun, and falls toward 1 for the heaviest stars, which shine close to their Eddington limit; 3.5 is the average over the stars that matter most for the light of a galaxy.

Main-Sequence Lifetime, Formal G

τ⊛ = ε⊛ · f⊛ · M · c² / L⊛

Dimensional analysis:
Where (from the book): τ⊛ [𝕋], ε⊛ [∅], f⊛ [∅], M [𝕄], c [𝕃·𝕋⁻¹], L⊛ [𝕄·𝕃²·𝕋⁻³]
[𝕋]
[∅] × [∅] × [𝕄] × [𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻³] = [∅] × [𝕄] × [𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻³] = [𝕄] × [𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻³] = [𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻³] = [𝕋]
[𝕋] = [𝕋] ✓ Valid PulseCore Verified ✓

A star lives as long as the fuel in its core lasts at the rate it shines

Where:

• τ⊛ [𝕋] — the main-sequence lifetime
• ε⊛ [∅] — the share of rest mass hydrogen fusion releases, 0.007
• f⊛ [∅] — the share of the star's hydrogen its core burns, ≈ 0.1
• M [𝕄] — the star's mass
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)
• L⊛ [𝕄·𝕃²·𝕋⁻³] — its luminosity

➢ For the Sun this gives 1.04 × 10¹⁰ years. With L⊛ ∝ M^3.5 it gives τ⊛ ∝ M^(−2.5), the law of Part 2.9: the fuel grows as the mass and the burning as its 3.5th power.

Eddington Luminosity G

L⇑ = 4 · π · G · M · m_H · c / σ⊤

Dimensional analysis:
Where (from the book): L⇑ [𝕄·𝕃²·𝕋⁻³], G [𝕄⁻¹·𝕃³·𝕋⁻²], M [𝕄], m_H [𝕄], c [𝕃·𝕋⁻¹], σ⊤ [𝕃²]
[𝕄·𝕃²·𝕋⁻³]
[∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕃³·𝕋⁻²] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕄·𝕃³·𝕋⁻²] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕄·𝕃⁴·𝕋⁻³] ÷ [𝕃²] = [𝕄·𝕃²·𝕋⁻³]
[𝕄·𝕃²·𝕋⁻³] = [𝕄·𝕃²·𝕋⁻³] ✓ Valid PulseCore Verified ✓

The brightest a body of mass M can shine before its light pushes its own matter away

Where:

• L⇑ [𝕄·𝕃²·𝕋⁻³] — the Eddington luminosity, 1.26 × 10³¹ W per solar mass
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling
• M [𝕄] — the mass of the star or Null Well
• m_H [𝕄] — the hydrogen mass
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)
• σ⊤ [𝕃²] — the Thomson cross-section of the electron, 6.652 × 10⁻²⁹ m²

➢ It caps the mass of stars, since the heaviest would outshine their own gravity, and it caps how fast a Null Well can feed, since the matter falling in shines as it falls.

The Ends of Stars

Chandrasekhar Limit, Formal G

M⊘ = 3.098 · m_P³ / (μₑ · m_H)²

Dimensional analysis:
Where (from the book): M⊘ [𝕄], m_P [𝕄], μₑ [∅], ħ [𝕄·𝕃²·𝕋⁻¹], c [𝕃·𝕋⁻¹], G [𝕄⁻¹·𝕃³·𝕋⁻²], m_H [𝕄]
[𝕄]
[∅] × [𝕄³] ÷ ([∅] × [𝕄])² = [𝕄³] ÷ ([∅] × [𝕄])² = [𝕄³] ÷ [𝕄]² = [𝕄³] ÷ [𝕄²] = [𝕄]
[𝕄] = [𝕄] ✓ Valid PulseCore Verified ✓

The heaviest a white dwarf can be: electron pressure holds no more

Where:

• M⊘ [𝕄] — the Chandrasekhar mass, 1.43 M☉ for a carbon–oxygen white dwarf
• m_P [𝕄] — the Planck mass of the level, √(ħ·c/G)
• μₑ [∅] — the mass per electron in units of m_H, 2 for carbon and oxygen
• ħ [𝕄·𝕃²·𝕋⁻¹] — the reduced Planck constant of the level (Part 2.5)
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling
• m_H [𝕄] — the hydrogen mass

➢ It is M⋆ times 3.098/μₑ², about 0.77 M⋆: a pure ratio of the Pulse to the matter quantum. A dead core below it settles as a white dwarf; one above it collapses on, to a neutron star or a Stellar Null Well.

White Dwarf Initial–Final Mass Relation G

M⊘f ≈ 0.109 · M + 0.394 · M☉

Dimensional analysis:
Where (from the book): M⊘f [𝕄], M [𝕄], M☉ [𝕄]
[𝕄] ≈ [∅] × [𝕄] + [∅] × [𝕄] = [𝕄] ≈ [𝕄] + [∅] × [𝕄] = [𝕄] ≈ [𝕄] + [𝕄] = [𝕄] ≈ [𝕄] ✓ Valid PulseCore Verified ✓

The white dwarf a star leaves, from its birth mass

Where:

• M⊘f [𝕄] — the white dwarf's mass
• M [𝕄] — the star's birth mass, 0.8 – 8 M☉
• M☉ [𝕄] — the mass of the Sun

➢ From white dwarfs in star clusters of known age (Kalirai et al., 2008). A Sun leaves about 0.5 M☉; a star of 8 M☉ leaves about 1.3, just under the limit. The rest of the star is returned to the gas, enriched.

Stellar Null Well Threshold G

M▣ > M_TOV

Dimensional analysis:
Where (from the book): M▣ [𝕄], M_TOV [𝕄]
[𝕄] > [𝕄] ✓ Valid PulseCore Verified ✓

A dead core heavier than any neutron star can be, M_TOV ≈ 2.3 M☉, becomes a Null Well

Where:

• M▣ [𝕄] — the mass of the collapsed core
• M_TOV [𝕄] — the Tolman–Oppenheimer–Volkoff limit, 2.1 – 2.3 M☉ by the heaviest neutron stars observed

➢ With the neutron-star equation of state still uncertain, the limit is taken from observation. Cores above it meet the Null Well Formation Condition (Part 6.2): recursion at the core passes the Zinf-scaled critical threshold and the region goes to physical silence behind a horizon r▣ = 2GM▣/c².

Stellar Null Well Fraction G

f▣ = N▣ / N⊛

Dimensional analysis:
Where (from the book): f▣ [∅], N⊛ [∅], M▣⁰ [𝕄], M⇓ [𝕄], M⇑ [𝕄], dN/dM [𝕄⁻¹]
[∅]
[∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The share of all stars that end as Stellar Null Wells

Where:

• f▣ [∅] — the fraction of stars born that end as black holes, ≈ 1.3 – 1.8 × 10⁻³ for the Kroupa Initial Mass Function
• N▣ [∅] — the stars born between M▣⁰ and M⇑, the integral of dN/dM over those masses
• N⊛ [∅] — all the stars born, the integral of dN/dM from M⇓ to M⇑
• M▣⁰ [𝕄] — the lowest birth mass that leaves a black hole, ≈ 20 – 25 M☉
• M⇓ [𝕄] — the hydrogen-burning limit, 0.075 M☉
• M⇑ [𝕄] — the most massive stars, ≈ 150 M☉
• dN/dM [𝕄⁻¹] — the Initial Mass Function (Part 2.9)

➢ About one star in six hundred. By the same integral, 76 percent of stars are red dwarfs and 0.6 percent end in core-collapse supernovae.

The Stellar Null Well After Birth

Eddington Growth Time G

t⇑ = ε⇑ · σ⊤ · c / (4 · π · G · m_H · (1 − ε⇑))

Dimensional analysis:
Where (from the book): t⇑ [𝕋], ε⇑ [∅], σ⊤ [𝕃²], c [𝕃·𝕋⁻¹], G [𝕄⁻¹·𝕃³·𝕋⁻²], m_H [𝕄]
[𝕋]
[∅] × [𝕃²] × [𝕃·𝕋⁻¹] ÷ ([∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × ([∅] - [∅])) = [𝕃²] × [𝕃·𝕋⁻¹] ÷ ([∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × ([∅] - [∅])) = [𝕃³·𝕋⁻¹] ÷ ([∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × ([∅] - [∅])) = [𝕃³·𝕋⁻¹] ÷ ([∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × ([∅] - [∅])) = [𝕃³·𝕋⁻¹] ÷ ([𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × ([∅] - [∅])) = [𝕃³·𝕋⁻¹] ÷ ([𝕃³·𝕋⁻²] × ([∅] - [∅])) = [𝕃³·𝕋⁻¹] ÷ ([𝕃³·𝕋⁻²] × [∅]) = [𝕃³·𝕋⁻¹] ÷ [𝕃³·𝕋⁻²] = [𝕋]
[𝕋] = [𝕋] ✓ Valid PulseCore Verified ✓

The fastest a Null Well can grow by feeding: e-folding its mass in this time

Where:

• t⇑ [𝕋] — the Salpeter time, ≈ 5 × 10⁷ years
• ε⇑ [∅] — the share of infalling rest mass radiated away, ≈ 0.1
• σ⊤ [𝕃²] — the Thomson cross-section
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling
• m_H [𝕄] — the hydrogen mass

➢ A well feeding at its Eddington limit grows as e^(t/t⇑), independent of its mass. A 100 M☉ Population III seed reaches 10⁹ M☉ in about 0.8 × 10⁹ years only if it feeds at the limit without pause, which is why the giant Null Wells already present in the first billion years require early, heavy seeds, mergers, or both.

Hawking Lifetime G

t⇣ = 5120 · π · G² · M³ / (ħ · c⁴)

Dimensional analysis:
Where (from the book): t⇣ [𝕋], G [𝕄⁻¹·𝕃³·𝕋⁻²], M [𝕄], ħ [𝕄·𝕃²·𝕋⁻¹], c [𝕃·𝕋⁻¹]
[𝕋]
[∅] × [∅] × [𝕄⁻²·𝕃⁶·𝕋⁻⁴] × [𝕄³] ÷ ([𝕄·𝕃²·𝕋⁻¹] × [𝕃⁴·𝕋⁻⁴]) = [∅] × [𝕄⁻²·𝕃⁶·𝕋⁻⁴] × [𝕄³] ÷ ([𝕄·𝕃²·𝕋⁻¹] × [𝕃⁴·𝕋⁻⁴]) = [𝕄⁻²·𝕃⁶·𝕋⁻⁴] × [𝕄³] ÷ ([𝕄·𝕃²·𝕋⁻¹] × [𝕃⁴·𝕋⁻⁴]) = [𝕄·𝕃⁶·𝕋⁻⁴] ÷ ([𝕄·𝕃²·𝕋⁻¹] × [𝕃⁴·𝕋⁻⁴]) = [𝕄·𝕃⁶·𝕋⁻⁴] ÷ [𝕄·𝕃⁶·𝕋⁻⁵] = [𝕋]
[𝕋] = [𝕋] ✓ Valid PulseCore Verified ✓

How long a Null Well with nothing to eat takes to evaporate

Where:

• t⇣ [𝕋] — the evaporation time
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational coupling
• M [𝕄] — the mass of the Null Well
• ħ [𝕄·𝕃²·𝕋⁻¹] — the reduced Planck constant of the level (Part 2.5)
• c [𝕃·𝕋⁻¹] — light speed, one Pulse Length a Tempo (ch 1)

➢ For one Sun, 2.1 × 10⁶⁷ years; it grows as M³, so the lightest evaporate first and each quickens at its end. Evaporation is the last way a Null Well returns its data to the Source (Part 3.10); for a well in our universe it begins in earnest only when the universe around it is colder than the well, long after the last stars.

The Game's Reading, Formally

The simulation cannot reach the scale at which m_P and m_H are separate numbers: its cells are Pulse Lengths and its matter is counted in bits, so its universes are ratios of ours. It keeps every law above in its proportions. The gas is the matter the Physical Nova condenses; ρ is its mass per cell, σ the spread of its velocities about their local mean. At the game's scale the Pulse and the matter quantum are one cell each, m_P = m_H, so M⋆ = 1 cell and the game's Sun is M☉(game) = 0.54 cells by the Solar Mass of a Universe. Star-forming gas sits at its radiative floor, where the Jeans mass of dense cores is about a Sun (Part 2.9); the game's bodies move as whole clumps, whose motion does not cool, so a body holding more than a Sun of gas forms stars at Ṁ⊛ = ε⇓ · M_gas / t⇓. Every mass then reads in those Suns, and the type, colour, luminosity (M^3.5), lifetime (M^(−2.5), the game's Sun living a set share of its universe's life), end and remnant follow the tables of Part 2.9. Only a core past the neutron-star limit becomes a well: a Stellar Null Well, born small, growing at most at its Eddington rate, merging, and evaporating by Hawking radiation (ħ = 1 in the game's units) when starved.

Chapter 6 Review

This review gathers what each part of the chapter formalizes. The physical picture behind it, and the predictions it shares with Chapter 2, are set out in the Chapter 2 Review.

Key Developments

Information-Energy Equivalence Framework

The Key Equations open the chapter with E = ℏ × I × ω and its differential form ΔE = ℏ × ΔI × ω, with entropy counted as S = k_B × ln(W), or S = log₂(W) in bits.

Temporal Quantization

Part 6.2 writes the Planck Pulse as the complete cycle 0 → 1 → 0 with period ①⥂⧗⌂, giving the Planck frequency ν_P = 1/①⥂⧗⌂, the energy quantum E_P = ℏ/t_P and the discrete interval Δt = n·①⥂⧗⌂.

Null Well Formation Dynamics

Parts 6.2–6.4 set the formation condition ℜ⌂(x,τ) ≥ ℜρ⨶ = k × ρ_P, the collapse evolution dℜ/dτ = -γℜ² + σ∇²ℜ, and the horizon condition lim[r→r_s] A(r,τ) = 0 with A(r,τ) = A₀ · exp(-(r-r_s)/λ). Part 6.7 adds the collapse trajectory and its time scale τ⦰.

Genesis Reactivation Mechanisms

Reactivation is the threshold T⫷ ≥ T⇌ (Part 6.2), with T⇌ = k⇌ · ρ_P · V∅ · l_P² at the boundary (Part 6.3). Part 6.6 gives the genesis sequence and its bifurcation condition ∂²S/∂τ² |_τ=0 = δ(M_n - M⨶), and Part 6.7 the genesis function G[0∅].

Density-Dependent Constants

Part 6.5 makes the Pulse Tempo local, ①⥂⧗'⌂ = ①⥂⧗⌂ · fρ(ρ⦰) with fρ(ρ) = (ρ_P/ρ⦰)^(1/2), and binds the modified constants through g₁(ρ) · g₂(ρ) = g₃(ρ)⁵ and α + β = 5γ. Part 6.6 writes the same scaling through Null Mass, ①⥂⧗'⌂ = ①⥂⧗⌂ · (M_n/M_P)^(1/2).

Information Conservation Across Transitions

Part 6.4 balances information at the horizon, I_in = I◎ + I⟹, with storage I◎ = A◎/(4l_P²) · ln(2). Part 6.6 carries it through genesis as I⊚ = I∅⚛ + Iℜ▢, and Part 6.7 maps volume information onto the Null Well boundary through the projection Π[I_3D].

Merger-Origin Universe Classification

Part 6.8 defines the Origin Compression Factor C(⇌) = F(M₁, M₂, a₁, a₂, θ⋎) and places our Universe at about 0.92–0.94, a high-spin binary Kerr–Kerr merger. Part 6.9 derives the spin contribution in closed form, with a Kerr floor near 0.932 and a single inherited preferred axis.

Stars and the Stellar Null Well

Part 6.10 formalizes the stellar road of Part 2.9: the Jeans length, free-fall time and star formation rate of the gas; the Stellar Mass Unit M⋆ = m_P³/m_H² with m_P = √(ħc/G) the Planck mass of the level, and each universe's Sun at 0.54 M⋆; the mass–luminosity relation, the lifetime τ⊛ = ε⊛·f⊛·M·c²/L⊛, and the Eddington limit; the Chandrasekhar limit 3.098·(ħc/G)^(3/2)/(μₑm_H)², the white dwarf initial–final mass relation, the neutron-star limit and the Stellar Null Well fraction f▣ ≈ 1.5 × 10⁻³; and the Stellar Null Well's Eddington growth time and Hawking lifetime.

Theoretical Integration

Substrate Architecture Connection

Part 6.1 states the Zero Substrate formally: Quintuple Nullity ∅▱ = {∅☐, ∅⚕, ∅ℹ, ∅⧗, ∅◉}, the activation A(∅) = ∅ → (0 ↔ 1), and the irreversibility ∅₀ ≠ ∅▷.

Recursive State Evolution Extension

Part 6.7 writes collapse as binary state evolution, P(τ+Δτ) = F[P(τ), ∂P/∂τ, ℜ(τ)], ending at the critical collapse condition where ∂P/∂τ and P(τ) go to zero and ℜ(τ) reaches ℜ⥣.

Empirical Predictions

Each part closes with its own testable predictions; those this chapter shares with Chapter 2 are gathered by observational domain in the Chapter 2 Review.