The series — The Simulated Universe

The diamond

Everything you could ever see or influence, drawn as one region of space and time.

DefinitionPart I · Foundations

Everything an observer can ever measure sits inside one region. Its past tip $p$ is the first event. Its future tip $q$ is the cosmic horizon. Where its two light-sheets meet sits the screen $\sigma$ , a sphere of maximal area.

The Robertson–Walker metric is one way to draw it, and nothing depends on that choice. The predictions depend only on the algebra of what can be observed inside.

Edition: 24 editions · from $V_4 = \tfrac{\pi}{24}\,\tau^4$
In the master: §2 · eq. (1) · p. 2
Built on: Gary Gibbons Sergey Solodukhin

00 · The diamondeq. (1)

\mathcal{D}(p,q) = J^{+}(p)\,\cap\,J^{-}(q)

(1)

Modular flow

The universe's own clock: time generated by its state, not supplied from outside.

DerivedPart I · Foundations

Give a region its state and the state supplies the dynamics. Tomita and Takesaki proved it: the modular flow $\theta_t$ is generated by the state and the region alone. No external clock. No Hamiltonian put in by hand.

Bisognano and Wichmann showed what the flow looks like: a boost, normalized by a $2\pi$ that can't be adjusted. On the diamond it's the vector field $\xi$ — timelike inside, null on the sheets, zero on the screen.

Edition: 628 editions · from $2\pi$
In the master: §3 · eq. (7), (11) · p. 3
Built on: Minoru Tomita Masamichi Takesaki Joseph Bisognano Eyvind Wichmann Rudolf Haag Nicolaas Hugenholtz Marinus Winnink Peter Hislop Roberto Longo Horacio Casini Marina Huerta Robert Myers John von Neumann

01 · Modular floweq. (7), (11)

\theta_t(A) = \Delta^{it}\,A\,\Delta^{-it}

(7), (11)

K = \textcolor{#FFB000}{2\pi}\!\int_\Sigma T^{\rm info}_{\mu\nu}\,\xi^\mu\,d\Sigma^\nu

The area law

How much information the observable universe can hold, set by the area of its boundary.

DerivedPart I · Foundations

The entropy of the modular state is its expected modular energy. On the diamond, that's an area.

Today: $S_H = 2.268\times10^{122}$ degrees of freedom on $N_P = 9.07\times10^{122}$ Planck tiles — one degree of freedom per four tiles. Two counts, kept apart. Swap one for the other and $\alpha^{-1}$ moves by 0.69.

The holographic principle comes out of the state. It isn't put in.

Edition: 4 editions · from $S = \mathcal{A}/4$
In the master: §3 · eq. (15) · p. 5
Built on: Jacob Bekenstein Stephen Hawking Gerard 't Hooft Leonard Susskind Raphael Bousso

02 · The area laweq. (15)

S_H = \frac{\mathcal{A}}{4G\hbar} = \frac{\pi c^5}{G\hbar H^2}

(15)

The obit

The single assumption: making anything definite costs exactly one natural unit of irreversibility.

PostulatePart I · Foundations

One definite record costs one nat. Not $\ln 2$ . One.

The content of the postulate isn't the 1. It's the nat: irreversibility comes in natural units, not bits. This is the only number the framework doesn't derive. It stands where $\hbar$ once stood — postulated, then tested by everything that follows from it.

What kills itA bench measurement that saturates at

k_BT\ln 2

however hard the bit is watched.

Edition: 1 · reserved for the author · the one postulate
In the master: §4 · eq. (22) · p. 6
Built on: Bryce Weiner

03 · The obiteq. (22)

S_{\rm obit} = 1\ \text{nat}

(22)

The partition

Where that cost goes: part to the record, the remainder to the boundary. Nothing is lost.

DerivedPart I · Foundations

An entangled pair carries $\ln 2$ nats of coherent capacity. A record costs 1. The shortfall, $\ln 2 - 1 = -0.307$ nats, goes to the boundary as negentropy. Nothing is destroyed. It's re-partitioned.

The ratio $\eta$ doesn't care about scale. It turns up at recombination, today, near black holes, and on a lab bench.

Edition: 226 editions · from $\eta = 2.2589$
In the master: §4 · eq. (24), (25) · p. 6
Built on: John Bell Wojciech Zurek John von Neumann

04 · The partitioneq. (24), (25)

Data

The partition
Entry	Value (nat)
S_coh = ln 2	0.6931
S_obit	1
S_decoh = ln 2 − 1	−0.3069
η = ln 2/(1 − ln 2)	2.2589

\ln 2 = 1 + (\ln 2 - 1)

(24), (25)

\eta = \frac{\ln 2}{1-\ln 2} = 2.2589

The clock rate

How fast the universe updates itself, and why that rate barely changes over time.

DerivedPart I · Foundations

Two bounds fix the shape. Margolus and Levitin cap how fast a state can change; the area law caps how much there is to change. One normalization — every degree of freedom on the screen updated once per Hubble time — sets the rest.

Today $\gamma_0 = 7.75\times10^{-21}\ \mathrm{s^{-1}}$ . Because the logarithm of $10^{122}$ barely moves, neither does $\gamma/H$ : $1/262$ at recombination, $1/281.7$ now.

Edition: 282 editions · from $\ln S_H = 281.7$
In the master: §5 · eq. (32) · p. 8
Built on: Norman Margolus Lev Levitin Seth Lloyd

05 · The clock rateeq. (32)

Data

γ/H at three epochs
Epoch	log₁₀ H_P/H	ln S_H	γ/H
the first tick	0.00	1.14	0.8736
recombination	56.62	261.88	0.003819
today	60.93	281.73	0.003549

\gamma = \frac{H}{\ln S_H}

(32)

Time as ticks

Time as a count of updates, and why we can't notice the rate changing from inside.

DefinitionPart I · Foundations

If spacetime is the projection of processing, proper time is processing, accumulated.

Every clock is built from the same ticks it counts. So the drift of $\gamma$ is invisible from inside any epoch: hydrogen reads 1420 MHz at recombination and 1420 MHz today, and both readings are right.

What kills itA confirmed change in atomic

\alpha

or

m_p/m_e

at the

10^{-5}

level.

Edition: 41 editions · from $1/\gamma_0 = 4.1\times10^{12}$ yr
In the master: §6 · eq. (35) · p. 8

06 · Time as tickseq. (35)

d\lambda = \gamma\,d\tau

(35)

Gravity

Gravity as the cost of existence.

DerivedPart II · What the framework provides

Preparing a state costs relative entropy against the vacuum. Gravity is that cost: the cost of existence, of being distinguishable from nothing.

Require the ledger to be writable on the boundary and Einstein's equations follow. The local argument leaves $\Lambda$ free. The modular vacuum of the whole diamond fixes it.

Edition: 8 editions · from $8\pi G$
In the master: §9 · eq. (50) · p. 14
Built on: Ted Jacobson Amal Kumar Raychaudhuri Thomas Faulkner Mark Van Raamsdonk Robert Myers William Unruh Albert Einstein

07 · Gravityeq. (50)

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}

(50)

The first event

How the first event could happen with nothing before it.

BoundaryPart III · The boundary

At the past tip there's no geometry, no clock, no record. A purely coherent state has no definable partition, and that instability fires the first event — the second law, applied to the ledger itself.

With no past to evict into, the first event ejects forward. That forward cone becomes every later observer's past. Only the first event ejects forward.

In Planck units the first diamond holds $S_H = \pi$ , ticks at $\gamma = 1/\ln\pi$ , and lands at Planck density without being told to.

Edition: 1 · reserved for the author · only the first ejects forward
In the master: §10 · eq. (55) · p. 15
Built on: Norman Margolus Lev Levitin Max Planck

08 · The first eventeq. (55)

\ln 2 \;\longrightarrow\; \ln 2 + (\ln 2 - 1) + 1 = 2\ln 2

(55)

The line

Nature's handedness: the theory's one data input, a known unknown left to solve.

BoundaryPart III · The boundary

The dynamics are parity-even. They can't originate a handedness, and the world has one. So chirality enters as data, a known unknown written at the first event. So does the generation count, by the rigidity of the index.

The screen carries an $E_8\times E_8$ edge: two light-cone sectors, one chirality, $c = 8 + 8 = 16$ . The boundary writes the index. The modular flow reads it and projects it. It does not write.

What kills itChromatic cosmic birefringence at

3\sigma

, or a generation count that varies independently of the net chirality.

Edition: 248 editions · from $\dim E_8 = 248$
In the master: §13 · eq. (65) · p. 22
Built on: Michael Atiyah Isadore Singer David Gross Jeffrey Harvey Emil Martinec Ryan Rohm Edward Witten Alain Connes Tsung-Dao Lee Chen-Ning Yang Chien-Shiung Wu

09 · The lineeq. (65)

\text{boundary data} \;\to\; E_8\times E_8\ \text{edge} \;\to\; \theta_t \;\to\; \text{bulk records}

(65)

The cascade

The far future isn't heat death. It's the seed of the next cycle.

BoundaryPart III · The boundary

As $H \to 0$ the horizon grows without bound and $\gamma$ falls toward zero. That's not heat death. It's maximum coherence, and maximum coherence is unstable.

One decoherence seed raises $\gamma$ around it, which speeds the next event, which raises $\gamma$ again. Seen from inside, the leading edge of the cascade is a hot, dense early universe.

From any internal frame, the transition takes no time at all.

Edition: Open edition · open from the asymptote to the close
In the master: §14 · eq. §14 · p. 24
Built on: Roger Penrose

10 · The cascadeeq. §14

H \to 0 \;\Rightarrow\; \gamma = \frac{H}{\ln S_H} \to 0

§14

The ruler

A ruler written into the sky, stretched 2.18% because the early plasma was watched so intensely.

ProjectionPart IV · The projections

At recombination every baryon was hit by photons about $10^9$ times per Hubble time. That plasma wasn't lightly watched. It sat nine orders of magnitude inside the quantum Zeno regime.

Watching that hard flips the sign of the damping. The acoustic ruler stretches by 2.18%, with no new field and nothing fitted to the BAO data.

What kills itBAO or CMB data excluding

\alpha = -5.7

at high significance.

Edition: 151 editions · from $r_d = 150.71$ Mpc
In the master: §15 · eq. (88) · p. 30
Built on: Baidyanath Misra George Sudarshan Daniel Eisenstein Göran Lindblad Joseph Silk

11 · The rulereq. (88)

r_d = r_{s,\Lambda{\rm CDM}}\bigl[1 - \alpha\,(\gamma/H)\bigr] = 150.71\ \text{Mpc}

(88)

The 10⁻¹²³

The biggest mismatch in physics, read as a count instead of a fine-tuning.

DerivedPart IV · The projections

Symmetry forces the vacuum's form, $-\rho_\Lambda g_{\mu\nu}$ . Its total is the critical density, at every epoch. Divide by the Planck density and the famous $10^{-123}$ is exact: three-halves of the vacuum fraction over the number of Planck tiles on the horizon.

It's a count, not a fine-tuning. And it pins $w = -1$ to eleven decimal places.

What kills itA confirmed

w_0 \neq -1

from DESI, Euclid or Roman.

Edition: 123 editions · from $10^{-123}$
In the master: §17 · eq. (100) · p. 33
Built on: Steven Weinberg

12 · The 10⁻¹²³eq. (100)

Data

The hierarchy as a count
Quantity	Value
N_P (Planck tiles)	9.07 × 10¹²²
S_H = N_P / 4	2.268 × 10¹²²
Ω_Λ	0.6847
(3/2) Ω_Λ / N_P	1.13 × 10⁻¹²³

\frac{\rho_\Lambda}{\rho_{\rm Pl}} = \frac{3}{2}\,\frac{\Omega_\Lambda}{N_P} = 1.13\times10^{-123}

(100)

Fine structure

The strength of electromagnetism, computed from the size of the universe's information budget.

ProjectionPart IV · The projections

Three terms, all geometry: half the log of the horizon entropy, minus the phase space of a boundary photon, minus one obit per modular cycle. $140.867 - 3.676 - 0.159 = 137.032$ .

Measured: 137.035999. A 0.003% gap, from a formula with nothing to adjust.

What kills itQED contradicted at any precision where the energy–Hubble correspondence can be tested.

Edition: 137 editions · from $\alpha^{-1} = 137.032$
In the master: §18 · eq. (110) · p. 36
Built on: Arnold Sommerfeld

13 · Fine structureeq. (110)

Data

The fine-structure constant
Term	Value
½ ln S_H	140.8672
− ln(4π²)	-3.6758
− 1/2π	-0.1592
α⁻¹ predicted	137.0323
α⁻¹ measured (CODATA 2018)	137.035999

\alpha^{-1} = \tfrac12\ln S_H - \ln(4\pi^2) - \frac{1}{2\pi} = 137.032

(110)

H₀ from a bench

How fast the universe is expanding, worked out from two laboratory measurements.

DerivedPart IV · The projections

The $\alpha$ relation ties three measured numbers together. Solve it for $H$ , give it $\alpha$ from atomic spectroscopy and $G$ from a torsion balance, and no cosmology enters at all.

67.11 ± 0.25 km/s/Mpc. Planck agrees at $0.4\sigma$ . The Cepheid ladder's 73.04 is excluded at $5.5\sigma$ .

What kills itA cosmology-free

H_0

that disagrees, or TRGB distances converging on 73.

Edition: 67 editions · from $H_0 = 67.11$
In the master: §19 · eq. (126) · p. 40
Built on: Henry Cavendish

14 · H₀ from a bencheq. (126)

Data

Determinations of H₀
Determination	H₀	Tension with laboratory value
Laboratory · α and G	67.11 ± 0.25	—
Planck 2018 · CMB	67.36 ± 0.54	0.4σ
Projection · NGC 4258	69.15 ± 0.56	3.3σ
Projection · LMC	70.36 ± 0.56	5.3σ
SH0ES · Cepheids	73.04 ± 1.04	5.5σ

H_0 = \Bigl(\frac{\pi c^5}{\hbar G\,S_H}\Bigr)^{1/2} = 67.11 \pm 0.25

(126)

Dead pixels

Dark matter as frozen pixels of the universe's screen, not an undiscovered particle.

ProjectionPart IV · The projections

A tile on the screen precipitates by paying one nat. Its own entanglement supplies $\ln 2$ . A tile that can't fund the 0.307-nat gap freezes: still gravitating, never shining, never decaying. A dead pixel.

A baryon's cut carries three conserved charges. Covering the cost drains two shells of a framed lattice: $q^2 = 16$ dead pixels for every three charges. 5.333, against Planck's 5.36 ± 0.07.

What kills itA detected dark-matter particle.

Edition: 16 editions · from $q^2 = 16$
In the master: §20 · eq. (132) · p. 43
Built on: Fritz Zwicky Vera Rubin Kent Ford John Bell Brian Swingle

15 · Dead pixelseq. (132)

\frac{\Omega_{\rm dm}}{\Omega_{\rm b}} = \frac{q^2}{\nu} = \frac{16}{3}

(132)

The Little Bang

What black holes do with information: store it until they're full, then make room.

ProjectionPart IV · The projections

A black hole isn't an eraser. It's the extreme organizer of coherent entropy, filling its screen toward saturation.

At saturation it can't copy — no-cloning — and it can't compress — the holographic bound. So it makes room: the region expands, keeping its information by growing the space that holds it.

Edition: 693 editions · from steps at $n\ln 2$
In the master: §21 · eq. (139) · p. 48
Built on: William Wootters Wojciech Zurek

16 · The Little Bangeq. (139)

\frac{dI}{dt} = \gamma I\Bigl(1 - \frac{I}{I_{\max}}\Bigr)

(139)

The bench test

The experiment that can settle the theory's one assumption, on a lab bench.

ProjectionPart IV · The projections

Landauer's $k_BT\ln 2$ is the slow limit of the ledger, recovered exactly. Watch a bit faster than it can hand its residual to the boundary, and the residual lands in the heat bath instead. The minimum cost climbs to $k_BT$ .

The ratio of the two is $\eta$ — the same constant, measured on a bench. A transmon, an ion trap or a lattice of cold atoms can settle it.

What kills itSaturation at

k_BT\ln 2

, or no crossover at all.

Edition: 307 editions · from $\Delta W_{\rm sat} = 0.307\,k_BT$
In the master: §22 · eq. (149), (151) · p. 50
Built on: Rolf Landauer Charles Bennett

17 · The bench testeq. (149), (151)

Data

The Landauer crossover
μ	χ(μ) = e^(−1/μ)	W / k_BT
0.001	0.0000	0.6931
0.1	0.0000	0.6932
1	0.3679	0.8060
10	0.9048	0.9708
1000	0.9990	0.9997
∞	1	1

W(\mu) = k_BT\bigl[\ln 2 + \chi(\mu)\,(1-\ln 2)\bigr]

(149), (151)

\frac{W(0)}{\Delta W_{\rm sat}} = \eta = 2.2589

Eighteen tokens, in the master's order.

The diamond

Modular flow

The area law

The obit

The partition

The clock rate

Time as ticks

Gravity

The first event

The line

The cascade

The ruler

The 10−123

Fine structure

H0 from a bench

Dead pixels

The Little Bang

The bench test

The 10⁻¹²³

H₀ from a bench