The Dark Sector: Data Matter, the Null Vacuum, and Vacuum Pressure

What is the dark side of a universe made of? Our universe is about 5 % ordinary matter, 27 % dark matter and 68 % dark energy (Planck Collaboration, 2020): almost everything in it is something no telescope sees directly. Binary Pulse Theory does not need two new substances for the dark 95 %. Both are already in the book, made by the novas that make every universe. The data that never becomes physical is the dark matter, here named Data Matter. The Pulsecore that the novas empty is a vacuum, here named the Null Vacuum (the Hollow of earlier drafts), and its outward push is the dark energy, here named Vacuum Pressure. They are one story, from the Data Nova to the bifurcation: Data Matter builds the web, returns through the wells, and feeds the Null Vacuum, whose Vacuum Pressure spreads everything apart until the universe empties into it and the next one begins.

Data Matter: the Data That Never Became Physical

At the Data Nova the Pulsecore pours its data onto the scaffold (Part 3.6). Before the physical layer that data has no mass: it is radiation, moving freely at the local light speed (Part 3.7). At the Physical Nova, the Causal Nova of Part 3.6, mass appears, and part of the data condenses into matter, the protons and nuclei whose masses the strong force sets (Part 2.10). The rest stays data. From that moment it has mass, by the Data–Matter Equivalence of Part 2.9, but it was never resolved into charged particles, so it has nothing to emit, absorb or scatter light with. That is what dark matter is observed to be.

Everything known about dark matter fits it. Dark matter has mass and gravitates: it was found by its pull, in the speeds of galaxies in clusters (Zwicky, 1933) and of stars in the outskirts of galaxies (Rubin & Ford, 1970). It never touches light. It passes through itself without colliding: in the Bullet Cluster the gas of two colliding clusters was stopped while their dark matter sailed on (Clowe et al., 2006); data flows through matter and through itself. It is cold: it moves slowly next to light and so clumps on every scale (Blumenthal, Faber, Primack & Rees, 1984). And it comes first. In cosmology the dark matter builds the cosmic web, and ordinary matter falls into that web later; dark matter can hardly clump while radiation rules, and once matter rules it gathers first, because ordinary matter is still held back by its tie to light. Binary Pulse Theory has that order built into its novas: the data is laid on the scaffold at the Data Nova, the Proto Wells form at its densest places, and only at the Physical Nova does matter condense and fall into a web already there. The Proto Wells are the first wells of Data Matter, the seeds of the halos that every galaxy sits in (Navarro, Frenk & White, 1996), and of their central wells.

Data Matter Mass G

M◍ = N◍ · k_B · T↯ · ln(2) / c²

Dimensional analysis:
Where (from the book): M◍ [𝕄], N◍ [∅], T↯ [Θ], c [𝕃·𝕋⁻¹]
[𝕄]
[∅] × [𝕄·𝕃²·𝕋⁻²·Θ⁻¹] × [Θ] × [∅] ÷ [𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²·Θ⁻¹] × [Θ] × [∅] ÷ [𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] ÷ [𝕃²·𝕋⁻²] = [𝕄]
[𝕄] = [𝕄] ✓ Valid PulseCore Verified ✓

The mass of the data that stays data, frozen at the Physical Nova

Where:

• M◍ [𝕄] — the mass of a body of Data Matter
• N◍ [∅] — the bits of data in it that never resolved into matter
• k_B [𝕄·𝕃²·𝕋⁻²·Θ⁻¹] — the Boltzmann constant
• T↯ [Θ] — the temperature of the universe at the Physical Nova, the pin head (Part 3.11)
• c [𝕃·𝕋⁻¹] — light speed

➢ The Data–Matter Equivalence (Part 2.9) gives a bit the mass k_B·T·ln 2 / c². For the data that stays data it is set once, at the Physical Nova, when mass appears, and does not change after: so Data Matter thins only as the universe grows, as a⁻³, as cold dark matter does, and does not cool away as radiation would. Data Matter is matter in the sense that counts for gravity, and data in the sense that counts for light.

Data Matter Ratio G

ρ◍ / ρ_b = N◍ / N_b

Dimensional analysis:
Where (from the book): ρ◍ [𝕄·𝕃⁻³], ρ_b [𝕄·𝕃⁻³], N◍ [∅], N_b [∅]
[𝕄·𝕃⁻³] ÷ [𝕄·𝕃⁻³] = [∅]
[∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

How much more Data Matter than ordinary matter: the share of the data that never condensed

Where:

• ρ◍ [𝕄·𝕃⁻³] — the density of Data Matter
• ρ_b [𝕄·𝕃⁻³] — the density of ordinary matter
• N◍ [∅] — the data left unresolved at the Physical Nova
• N_b [∅] — the data condensed into matter there, counted by the same equivalence

➢ In our universe the ratio is 5.36: Ω_c h² = 0.120 against Ω_b h² = 0.0224 (Planck Collaboration, 2020). If Data Matter is the dark matter, this number is the Physical Nova's own split of the Pulsecore's data, and it should follow from the Pulsecore's counts at our level, not be fitted to them. It is the test this reading has to pass: a ratio near 5.4 reached from the counts would be a result; one tuned to reach it would not. The count follows.

The book fixes both shares. The Data Nova fires when the Pulsecore's data, f(n) = (n + 1)², fills the Bifold's χ_∂ = 16 · 4^L places (Part 3.6), at n + 1 = 4 · 2^L, and that whole skin turns out onto the scaffold: χ_∂ bits laid before the physical layer, which stay data, the Data Matter. The Pulsecore goes on filling, and the Physical Nova is the third layer, which comes 2^L · (1 − 2/3) = 2^L / 3 Tempos after the Data Nova, at n + 1 = (13/3) · 2^L. What the core gathered between the two novas, f − χ_∂ = (169/9 − 16) · 4^L = (25/9) · 4^L, is the first data born at the physical layer, the data the Physical Nova sends out as the big bang: the ordinary matter.

Data Matter Count G

N◍ / N_b = 16 · 4^L / ((13 / 3)² · 4^L − 16 · 4^L)

Dimensional analysis:
Where (from the book): N◍ [∅], N_b [∅], L [∅]
[∅] ÷ [∅] = [∅]
[∅] × [∅]^([∅]) ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] × [∅] ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ ([∅]² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] × [∅] - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] - [∅] × [∅]) = [∅] ÷ ([∅] - [∅]) = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The skin laid before the physical layer, against the data the core gathers until it

Where:

• N◍ [∅] — the data on the scaffold at the Physical Nova, the Bifold's χ_∂ = 16 · 4^L turned out at the Data Nova
• N_b [∅] — the data the Pulsecore gathered between the Data Nova and the Physical Nova, f(n↯) − χ_∂
• L [∅] — the level

➢ The level cancels: N◍ / N_b = 16 / (25/9) = 144/25 = (12/5)² = 5.76 at every level, the square of the ratio of the two radii of data, 4 against 5/3. Measured, the ratio is 5.36 ± 0.06 (Ω_c h² = 0.1200 ± 0.0012 against Ω_b h² = 0.02237 ± 0.00015; Planck Collaboration, 2020). The count lands 7 % high, from the book's own laws with nothing tuned, but well outside the measurement, so it is not yet a match. How sensitive it is says where to look: the measured value would need the Physical Nova at 0.357 of the life rather than the third the layer law gives, or a seventh part of the scaffold's data to have condensed as well, for instance the data that fell into the Proto Wells and their matter. Which of these the book's own rules give is taken up next; neither is assumed.

Failed Proto Wells

The book's answer is the second, and it is already written. Proto Wells are collapses: they form at the Data Nova where the data is densest, as every Null Well forms where the overflow overwhelms the Zinf point. Every collapse meets the Reproductive Outcome Distribution of Part 3.9, and in a fraction P⦰ ≈ 0.06 of them it fails: no child is seeded, and the data returns to the Source without continuing. 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. Which data a well holds is its basin, every bit nearest to it, the rule by which every bit returns through a well (the Return to Bifurcation), and with 2^(L/3) wells the failed basins are the fraction P⦰ of the scaffold. The ordinary matter is untouched: it is the core's new data, not on the scaffold.

Data Matter Count with Failed Proto Wells G

N◍ / N_b = (1 − P⦰) · 16 · 4^L / ((13 / 3)² · 4^L − 16 · 4^L)

Dimensional analysis:
Where (from the book): N◍ [∅], N_b [∅], P⦰ [∅], L [∅]
[∅] ÷ [∅] = [∅]
([∅] - [∅]) × [∅] × [∅]^([∅]) ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] × [∅] × [∅]^([∅]) ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] × [∅]^([∅]) ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] × [∅] ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ (([∅] ÷ [∅])² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ ([∅]² × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] × [∅]^([∅]) - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] × [∅] - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] - [∅] × [∅]^([∅])) = [∅] ÷ ([∅] - [∅] × [∅]) = [∅] ÷ ([∅] - [∅]) = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The skin laid before the physical layer, less what its failed Proto Wells return, against the data the core gathers until it

Where:

• N◍ [∅] — the Data Matter at the Physical Nova
• N_b [∅] — the data the Pulsecore gathered between the Data Nova and the Physical Nova, the ordinary matter
• P⦰ [∅] — the probability that a collapse fails, ≈ 0.06 (Reproductive Outcome Distribution, Part 3.9)
• L [∅] — the level

➢ With the Dimensional Failure Law of Part 3.9, P⦰ = 2^(−D) = 1/16 in a universe of four actual dimensions, the count is N◍ / N_b = (15/16) × 144/25 = 27/5 = 5.40, against the measured 5.36 ± 0.06 (Planck Collaboration, 2020): within the measurement, 0.55 standard deviations away. The level cancels; only the number of actual dimensions remains. Three laws of this book written for other reasons, the Data Nova's containment, the layer law that times the Physical Nova, and the failure of a collapse whose every dimension returns null, give our universe's ratio of dark matter to ordinary matter as a pure number, twenty-seven to five, with nothing fitted.

Read backwards, the count measures a universe's dimensions. The dark-matter ratio a sky shows fixes 2^(−D), and so D.

Dimension Count from the Dark Sector G

D = −ln(1 − (25 / 144) · ρ◍ / ρ_b) / ln(2)

Dimensional analysis:
Where (from the book): D [∅], ρ◍ [𝕄·𝕃⁻³], ρ_b [𝕄·𝕃⁻³]
[∅]
ln([∅] - ([∅] ÷ [∅]) × [𝕄·𝕃⁻³] ÷ [𝕄·𝕃⁻³]) ÷ [∅] = ln([∅] - [∅] × [𝕄·𝕃⁻³] ÷ [𝕄·𝕃⁻³]) ÷ [∅] = ln([∅] - [𝕄·𝕃⁻³] ÷ [𝕄·𝕃⁻³]) ÷ [∅] = ln([∅] - [∅]) ÷ [∅] = ln([∅]) ÷ [∅] = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

A universe's dark matter counts its actual dimensions

Where:

• D [∅] — the number of actual dimensions
• ρ◍ [𝕄·𝕃⁻³] — the density of dark matter
• ρ_b [𝕄·𝕃⁻³] — the density of ordinary matter, so that ρ◍ / ρ_b is the measured ratio

➢ Our sky's 5.36 ± 0.06 gives 2^(−D) = 0.069 ± 0.011 and D = 3.9 (+0.3 / −0.2): four, the three of space and one of time that every experiment finds. Three is ruled out at five standard deviations (it would give 5.04) and five at more than three (5.58). Other universes, with other numbers of actual dimensions, would show other ratios: 5.04 with three, 5.40 with four, 5.58 with five, 5.67 with six.

The Return of Data Matter

Data Matter is not permanent. Every bit returns through a well on its loop home (Part 3.10): the Data Loop carries data through the Null Wells and the Zinf back to the UniSpheral Data Source. So over the life of a universe its Data Matter drains away, some of it falling into the wells, and the rest returning as the space it leaves. Decaying dark matter is a studied possibility in cosmology, and the sky limits it: if dark matter decayed into radiation, the microwave background and the growth of structure require its lifetime to exceed about 160 billion years, more than ten times our universe's age (Poulin, Serpico & Lesgourgues, 2016). The return is therefore slow through the life, and the great emptying belongs to the close, where Part 3.10 places the dissipation.

Data Matter Return G

ρ◍ = ρ◍⊚ · a^(−3) · exp(−Γ◍ · t)

Dimensional analysis:
Where (from the book): ρ◍ [𝕄·𝕃⁻³], ρ◍⊚ [𝕄·𝕃⁻³], a [∅], Γ◍ [𝕋⁻¹], t [𝕋]
[𝕄·𝕃⁻³]
[𝕄·𝕃⁻³] × [∅]^([∅]) × exp([𝕋⁻¹] × [𝕋]) = [𝕄·𝕃⁻³] × [∅]^([∅]) × exp([𝕋⁻¹] × [𝕋]) = [𝕄·𝕃⁻³] × [∅] × exp([𝕋⁻¹] × [𝕋]) = [𝕄·𝕃⁻³] × exp([𝕋⁻¹] × [𝕋]) = [𝕄·𝕃⁻³] × exp([∅]) = [𝕄·𝕃⁻³] × [∅] = [𝕄·𝕃⁻³]
[𝕄·𝕃⁻³] = [𝕄·𝕃⁻³] ✓ Valid PulseCore Verified ✓

Data Matter thins with the expansion and slowly returns

Where:

• ρ◍ [𝕄·𝕃⁻³] — the density of Data Matter
• ρ◍⊚ [𝕄·𝕃⁻³] — its density when the scale factor is 1
• a [∅] — the scale factor (Part 3.11)
• Γ◍ [𝕋⁻¹] — the return rate, the share of Data Matter going home each unit of time
• t [𝕋] — the time since the Physical Nova

➢ With Γ◍ = 0 this is the cold dark matter of standard cosmology. In Binary Pulse Theory Γ◍ is not zero, but it is small while a universe lives, held by the limits above, and it is tied to the universe's own life of 2^L Tempos (Part 3.10). How it grows toward the close is the next thing to derive.

The Null Vacuum: One Vacuum, Seen From Both Sides

The Pulsecore, emptied first of its null at the Null Nova and then of its data at the Data Nova, holds neither (Part 3.6). It is a vacuum, and it is not Nothing. Part 1.1 keeps Nothing, ∅, as absolute absence, with no properties at all, and that ruling stands. The emptied core is the Pre-Pulse Field brought back: potential with no content, the state a collapse returns to rather than to ∅ itself (Part 1.1). Physics calls such a state a vacuum, a real physical emptiness that can carry energy.

The Null Vacuum is also the bottom of the well. A universe begins across three parts (Part 3.7): the parent's Nullcore, the singularity at the bottom of a null well where the parent's data sinks to 0; the Bifold, the fold where one universe becomes two; and the child's Pulsecore. The data sinks to 0 at the Nullcore, crosses the Bifold, fills the Pulsecore, and erupts out of it at the novas, leaving the Pulsecore empty. That emptiness is the Nullcore again, the bottom of the parent's well, now seen from inside the child. The Nullcore and the Null Vacuum are one vacuum, seen from its two sides, and the loop of a universe closes on it.

Nor is the Null Vacuum off at a centre. Before the Physical Nova the whole scaffold is drawn back to the Pulsecore's own size, the pin head (Part 3.11), so every place in the universe was inside the core. The vacuum the core became is the space everything sits in, everywhere at once, as the Big Bang happened everywhere and not at a point.

Vacuum Pressure: Why Emptiness Pushes

The energy of a vacuum does not behave like matter. However space grows, a vacuum stays the same vacuum, so its energy density stays the same; for the energy in a growing volume to obey energy conservation, its pressure must be the negative of its density. In general relativity pressure gravitates too, and negative pressure gravitates outward. Einstein put such a term into his equations as the cosmological constant (Einstein, 1917); Zel'dovich showed that the energy of the vacuum is one (Zel'dovich, 1968); and the expansion of our universe was found to be speeding up, driven by it (Riess et al., 1998; Perlmutter et al., 1999).

Vacuum Pressure G

p⦵ = −ρ⦵ · c²

Dimensional analysis:
Where (from the book): p⦵ [𝕄·𝕃⁻¹·𝕋⁻²], ρ⦵ [𝕄·𝕃⁻³], c [𝕃·𝕋⁻¹]
[𝕄·𝕃⁻¹·𝕋⁻²]
[𝕄·𝕃⁻³] × [𝕃²·𝕋⁻²] = [𝕄·𝕃⁻¹·𝕋⁻²]
[𝕄·𝕃⁻¹·𝕋⁻²] = [𝕄·𝕃⁻¹·𝕋⁻²] ✓ Valid PulseCore Verified ✓

The pressure of the Null Vacuum is the negative of its energy density

Where:

• p⦵ [𝕄·𝕃⁻¹·𝕋⁻²] — the pressure of the Null Vacuum
• ρ⦵ [𝕄·𝕃⁻³] — its density, its energy density divided by c²
• c [𝕃·𝕋⁻¹] — light speed

➢ The equation of state w = p / (ρc²) = −1, the vacuum's own: the same energy in every cubic metre however the universe grows (Carroll, 2001).

Acceleration Equation G

ä / a = −(4 · π · G / 3) · (ρ + 3 · p / c²)

Dimensional analysis:
Where (from the book): ä [𝕋⁻²], a [∅], G [𝕄⁻¹·𝕃³·𝕋⁻²], ρ [𝕄·𝕃⁻³], p [𝕄·𝕃⁻¹·𝕋⁻²], c [𝕃·𝕋⁻¹]
[𝕋⁻²] ÷ [∅] = [𝕋⁻²]
([∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] ÷ [∅]) × ([𝕄·𝕃⁻³] + [∅] × [𝕄·𝕃⁻¹·𝕋⁻²] ÷ [𝕃²·𝕋⁻²]) = ([∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] ÷ [∅]) × ([𝕄·𝕃⁻³] + [∅] × [𝕄·𝕃⁻¹·𝕋⁻²] ÷ [𝕃²·𝕋⁻²]) = ([𝕄⁻¹·𝕃³·𝕋⁻²] ÷ [∅]) × ([𝕄·𝕃⁻³] + [∅] × [𝕄·𝕃⁻¹·𝕋⁻²] ÷ [𝕃²·𝕋⁻²]) = [𝕄⁻¹·𝕃³·𝕋⁻²] × ([𝕄·𝕃⁻³] + [∅] × [𝕄·𝕃⁻¹·𝕋⁻²] ÷ [𝕃²·𝕋⁻²]) = [𝕄⁻¹·𝕃³·𝕋⁻²] × ([𝕄·𝕃⁻³] + [𝕄·𝕃⁻¹·𝕋⁻²] ÷ [𝕃²·𝕋⁻²]) = [𝕄⁻¹·𝕃³·𝕋⁻²] × ([𝕄·𝕃⁻³] + [𝕄·𝕃⁻³]) = [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄·𝕃⁻³] = [𝕋⁻²]
[𝕋⁻²] = [𝕋⁻²] ✓ Valid PulseCore Verified ✓

Pressure gravitates: a negative enough pressure turns the pull into a push

Where:

• ä [𝕋⁻²] — the acceleration of the scale factor
• a [∅] — the scale factor
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the universe's gravitational constant (Part 3.11)
• ρ [𝕄·𝕃⁻³] — the density of what fills the universe
• p [𝕄·𝕃⁻¹·𝕋⁻²] — its pressure
• c [𝕃·𝕋⁻¹] — light speed

➢ The second Friedmann equation. Matter, with no pressure, slows the expansion. The Null Vacuum, with p⦵ = −ρ⦵c², gives ρ + 3p/c² = −2ρ⦵, and the expansion speeds up: ä / a = +8πGρ⦵ / 3. That is Vacuum Pressure, and it is why the late universe accelerates.

Cosmological Constant of the Null Vacuum G

Λ = 8 · π · G · ρ⦵ / c²

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

The cosmological constant is the Null Vacuum's density in geometric units

Where:

• Λ [𝕃⁻²] — the cosmological constant, 1.1 × 10⁻⁵² m⁻² in our universe (Planck Collaboration, 2020)
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the gravitational constant
• ρ⦵ [𝕄·𝕃⁻³] — the density of the Null Vacuum, about 5.8 × 10⁻²⁷ kg m⁻³ in ours
• c [𝕃·𝕋⁻¹] — light speed

➢ Einstein's Λ and the vacuum's energy are one quantity in two units.

How Much the Vacuum Holds: Counted on Its Boundary

Summing the zero-point energy of every mode of space down to the Zinf, as Part 5.2's vacuum expectation does, gives about the Planck density, some 10¹²³ times what is measured: the cosmological constant problem, the largest mismatch in physics (Weinberg, 1989; Carroll, 2001). The way out that the holographic principle points to is that a vacuum's energy is not counted through its volume. A region cannot hold more energy than would make it a black hole, which ties its vacuum energy to its boundary and gives a density of order one over the square of its size (Cohen, Kaplan & Nelson, 1999); with the horizon as the size it is the density of dark energy measured today (Li, 2004).

Holographic Vacuum Density G

ρ⦵ = 3 · c² / (8 · π · G · R◎²)

Dimensional analysis:
Where (from the book): ρ⦵ [𝕄·𝕃⁻³], c [𝕃·𝕋⁻¹], G [𝕄⁻¹·𝕃³·𝕋⁻²], R◎ [𝕃]
[𝕄·𝕃⁻³]
[∅] × [𝕃²·𝕋⁻²] ÷ ([∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕃²]) = [𝕃²·𝕋⁻²] ÷ ([∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕃²]) = [𝕃²·𝕋⁻²] ÷ ([∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕃²]) = [𝕃²·𝕋⁻²] ÷ ([𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕃²]) = [𝕃²·𝕋⁻²] ÷ [𝕄⁻¹·𝕃⁵·𝕋⁻²] = [𝕄·𝕃⁻³]
[𝕄·𝕃⁻³] = [𝕄·𝕃⁻³] ✓ Valid PulseCore Verified ✓

The vacuum a horizon of radius R◎ can hold

Where:

• ρ⦵ [𝕄·𝕃⁻³] — the density of the Null Vacuum
• c [𝕃·𝕋⁻¹] — light speed
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the gravitational constant
• R◎ [𝕃] — the radius of the cosmological horizon

➢ With the Cosmological Constant of the Null Vacuum this is Λ = 3 / R◎², the de Sitter horizon, whose entropy is S◎ = 3π / (Λ ℓ_P²) (Gibbons & Hawking, 1977), the horizon count of Part 1.9.

Binary Pulse Theory already counts that boundary. The Holographic Pixel Bound of Part 1.9 gives the view at level L as 16 × 2^(2L) pixels, each worth one bit, and finds it equal, at our level, to the horizon's capacity S◎ / ln 2, with no Λ in its derivation. Read the other way, the count fixes the vacuum, and so the cosmological constant, of every level.

Level Λ Law G

Λ · ℓ_P² = 3 · π / (16 · ln(2) · 2^(2 · L))

Dimensional analysis:
Where (from the book): Λ [𝕃⁻²], ℓ_P [𝕃], L [∅]
[𝕃⁻²] × [𝕃²] = [∅]
[∅] × [∅] ÷ ([∅] × [∅] × [∅]^([∅] × [∅])) = [∅] ÷ ([∅] × [∅] × [∅]^([∅] × [∅])) = [∅] ÷ ([∅] × [∅]^([∅] × [∅])) = [∅] ÷ ([∅] × [∅]^([∅])) = [∅] ÷ ([∅] × [∅]) = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

Each level's cosmological constant, from its own pixel count

Where:

• Λ [𝕃⁻²] — the cosmological constant of a universe at level L
• ℓ_P [𝕃] — the Planck length of that universe
• L [∅] — the universe's level, 202 for ours (Part 1.6)

➢ At L = 202 the law gives Λ ℓ_P² = 2.1 × 10⁻¹²²; the measured value is 2.9 × 10⁻¹²² (Planck Collaboration, 2020): the same factor of 1.4 that Part 1.9 leaves to the base multiplier 16 and the conversion from nats to bits, and the two close together when it is accounted for. Two cautions belong with it. Λ in Planck units is close to one over the square of our universe's age in Planck times, a coincidence cosmology has long noted (Carroll, 2001), and a universe's life in this book is 2^L Tempos; part of the match is that coincidence again, and the law earns its place by the derivation of its factor, not by the power of two alone. And it predicts what no fit can: a shallower universe has a far stronger Vacuum Pressure, Λ ℓ_P² ≈ 1.8 × 10⁻²² at level 36, in step with its short life.

Vacuum Pressure Fed by Returning Data

Part 9.1 finds the dark energy in the tension of uncollapsed states, and uncollapsed, unresolved data is what Data Matter is. The two dark components are one ledger. As Data Matter returns, the space it leaves is the Null Vacuum, so the vacuum is fed by the returning data, and Vacuum Pressure is not a fixed constant across a universe's life: it is smallest early and grows as the data goes home. An energy flow between dark matter and dark energy is the interacting dark sector of cosmology (Wetterich, 1995; Amendola, 2000).

Vacuum Feeding G

ρ⦵′ = Γ◍ · ρ◍

Dimensional analysis:
Where (from the book): ρ⦵′ [𝕄·𝕃⁻³·𝕋⁻¹], Γ◍ [𝕋⁻¹], ρ◍ [𝕄·𝕃⁻³]
[𝕄·𝕃⁻³·𝕋⁻¹]
[𝕋⁻¹] × [𝕄·𝕃⁻³] = [𝕄·𝕃⁻³·𝕋⁻¹]
[𝕄·𝕃⁻³·𝕋⁻¹] = [𝕄·𝕃⁻³·𝕋⁻¹] ✓ Valid PulseCore Verified ✓

The Null Vacuum gains what Data Matter returns

Where:

• ρ⦵′ [𝕄·𝕃⁻³·𝕋⁻¹] — the rate at which the Null Vacuum's density grows
• Γ◍ [𝕋⁻¹] — the return rate of Data Matter
• ρ◍ [𝕄·𝕃⁻³] — the density of Data Matter

➢ The balance of the two dark components: what leaves the one enters the other, so their sum keeps the universe's books. A vacuum that changes over time no longer has w = −1 exactly, and Part 9.1's prediction of an evolving equation of state, w(z), follows from it. Observation has begun to test exactly this: the DESI survey's 2025 measurements, joined with the microwave background and supernovae, favour a dark energy that changes over time at between 2.8σ and 4.2σ, depending on the supernova sample (DESI Collaboration, 2025). These are hints, not proof, and the shape matters: DESI's fits have the dark energy weakening today after a stronger past, while steady feeding makes it grow. Whether the return rate as Binary Pulse Theory fixes it, small through the life and rising to the close, gives the shape the sky shows is a calculation still to do.

The Bottom of the Well Radiates

Where the vacuum meets a horizon it shows itself. Near a black hole's horizon the vacuum turns into real particles that stream away, so every black hole glows and slowly evaporates (Hawking, 1975); an observer accelerating through empty space sees the same vacuum as a warm bath (Unruh, 1976). In Binary Pulse Theory the horizon of a Null Well is a computational threshold (Part 2.4) and its bottom is the Nullcore, the same vacuum as the Null Vacuum of the child it seeds. So the bottom of every well radiates: a well that has no more to feed on returns what it holds by this glow, and with it a universe's last wells empty at the close.

The Close: Everything Returns to the Null Vacuum

The life of a universe ends in its own vacuum. Ordinary matter gathers into stars and falls into wells; Data Matter returns, slowly through the life and fully at the close; the wells evaporate from their horizons; and what is left is the Null Vacuum, its pressure spreading what remains apart. When every bit has gone home, the field dissipates into pure nothingness (Part 3.10), and the null left behind is the Pre-Pulse Field where the next pulse begins: Return to Bifurcation, ∅_L → (0 ⥂ 1)_(L+1). The vacuum a universe ends in and the vacuum at the bottom of the well its children are born from are the same vacuum.

Links to Binary Pulse Theory

• Part 1.1: Nothing stays absolute absence; the Null Vacuum is the Pre-Pulse Field returned, a vacuum, not ∅.
• Part 1.9: the Holographic Pixel Bound becomes the Level Λ Law when it is read toward the vacuum.
• Parts 2.4 and 2.7: the Null Well's horizon and its Nullcore; the bottom of the well is the Null Vacuum of the child, and it radiates.
• Part 2.9: the Data–Matter Equivalence gives Data Matter its mass, frozen at the Physical Nova.
• Part 3.9: the Reproductive Outcome Distribution fails Proto Wells as it fails every collapse, and their data goes home before the Physical Nova.
• Parts 3.6 and 3.7: the Data Nova lays Data Matter on the scaffold; the emptied Pulsecore, once called the Hollow, is the Null Vacuum.
• Parts 3.10 and 3.11: Data Matter returns along the Data Loop; Vacuum Pressure accelerates the expansion late in the life, and the close empties everything into the Null Vacuum.
• Part 9.1: unresolved computation is Data Matter, and its tension is Vacuum Pressure.