# Papilov Research — Full Content > This file contains the full text of all published research articles. > For a summary and links, see llms.txt. > License: CC BY-NC 4.0. AI-co-authored with Claude (Anthropic). > Site: https://papilov.org > Author: Artem Papilov --- ## Prebid.js: How the Programmatic Monetization Standard Works URL: https://papilov.org/research/prebid-js-explainer/ Also available in: RU, ES, DE, ZH Programmatic advertising is a $595B market (2024), projected to reach $779B by 2028. Roughly 90% of digital display is bought automatically. At the center of publisher-side infrastructure sits an open-source library used by over 10,000 companies: Prebid.js. This analysis is for decision-makers: what Prebid does, how it works, why it became the standard, and which levers determine business results.
300+
bid adapters
~90%
display = programmatic
$203B
US programmatic 2026
10K+
companies using Prebid
## The Problem: Waterfall Leaves Money on the Table Publishers historically sold inventory through a **waterfall** — ad networks called sequentially by priority. The first one that accepted the price got the impression. Others never competed, even if they would have paid far more. **Header bidding** fixed this: all buyers bid simultaneously, highest price wins. Prebid.js is the open-source library that became the industry standard for browser-side header bidding.

Waterfall

Network A (priority 1)$2.00 — wins by default
Network B (priority 2)Never asked
Network C (priority 3)Would have paid $5.00 — never asked
ResultImpression sold for $2.00

Header Bidding (Prebid)

Network A$2.00
Network B$3.10
Network C$5.00 — wins
ResultAll bid in parallel → $5.00 wins (+150%)
+20–50%
waterfall → header bidding uplift
+70%
The Telegraph (case study)
+25–50%
CPM uplift range (industry)
30–40%
average portfolio uplift (AdPushup)
## How It Works: 5 Steps in ~1000ms
1

Page loads → auction starts

Prebid.js identifies ad slots and simultaneously sends bid requests to all connected buyers.

~0ms
2

Buyers respond with bids

Each DSP/SSP receives slot data and returns a bid or pass. All in parallel.

200–800ms
3

Timeout cuts slow responders

Those who didn't respond are excluded. UX protection.

timeout: 1000–1500ms
4

Best bids → ad server

Winning bids sent to Google Ad Manager to compete with direct deals and AdX.

~50ms
5

Ad server picks winner → ad shown

GAM compares all sources and serves the highest-paying creative.

~1100ms total
Prebid doesn't replace the ad server. It creates competition before the decision, increasing the effective price of every impression.
## Market Context
$1.14T
global ad market 2025
$595B
programmatic (global) 2024
$203B
US programmatic display 2026
The ANA Supply Chain Study (2023) found only **36 cents** per advertiser dollar reached publishers. By Q3 2025 this improved to **47.1¢** (+11 points), but $26.8B/year is still lost to supply chain inefficiency.
Advertiser
$1.00
→
DSP + SSP fees
−29¢
transactions (2023)
→
Waste
−35¢
IVT, MFA (2023)
→
Publisher
36¢
→ 47¢ by Q3 2025
47.1%
publisher share, Q3 2025 (was 36% in 2023)
0.39%
MFA inventory, Q3 2025 (record low)
81.6%
PMP share, Q3 2025 (was 64.5%)
## Google Antitrust: Why Prebid Matters More In April 2025, a US federal court found Google guilty of illegally monopolizing publisher ad server and ad exchange markets. The EU fined Google €2.95B for adtech antitrust violations. The DOJ seeks divestiture of AdX.
~90%
Google's ad server share
€2.95B
EU adtech fine
> "Google willfully engaged in a series of anticompetitive acts to acquire and maintain monopoly power in the publisher ad server and ad exchange markets." > — Judge Leonie Brinkema, April 17, 2025 For Prebid, this is a **structural tailwind**. If Google must unbundle its ad server from its exchange, vendor-neutral auction solutions become critical infrastructure. ## Three Architectures

Client-side

Auction in the browser

Server-side

Auction on server (Prebid Server)

Hybrid (industry standard)

Both — optimal balance
Hybrid is the industry standard: 5–8 key buyers client-side + rest server-side. Prebid Server delivers up to 40% latency reduction.
## Business Levers ### Timeout: Revenue vs. UX
800ms
~55%
~55%
1000ms ←
~75%
~75%
1200ms
~88%
~88%
1500ms
~95%
~95%
### Price Granularity | Type | Step | GAM Lines | Revenue Loss | |------|------|-----------|--------------| | Low | $0.50 | ~40 | up to $0.49 (high) | | Medium | $0.10 | ~200 | up to $0.09 (moderate) | | High | $0.01 | ~2,000 | $0.009 (minimal) | | Custom | Variable | Optimized | Controlled (recommended) | ### Buyer Selection: Diminishing Returns
3 buyers
~60%
~60%
5 buyers
~80%
~80%
8 buyers ←
~92%
~92%
15 buyers
~97%
~97%
20+ buyers
~99%
latency↑
80% of incremental revenue comes from the first 5–7 buyers. Sweet spot: ~8 client-side. ## CPM by Geography Average banner CPMs (SSP-side, 2024):
US
$1.43
$1.43
UK
$1.05
$1.05
Germany
$0.90
$0.90
France
$0.80
$0.80
Brazil
$0.50
$0.50
India
$0.25
$0.25
An 8× spread that defines monetization economics. Q1 2025: US display CPMs fell −33–42% YoY after a record political ad year. By December 2025: display +6.3% YoY, video +33.2% YoY. This volatility underscores the need for dynamic floor prices — exactly what Prebid's Floors Module does. ## Privacy: Post-Cookie Adaptation
User ID — UID2, SharedID, EUID, LiveRamp
First-party data
Topics API / Protected Audiences
GDPR / CCPA / GPP consent
40%
US marketers using 1P data as primary targeting (2025)
60–80%
CPM preserved with Prebid User ID vs. no identity
## Beyond Display | Format | Architecture | Market | |--------|-------------|--------| | Display | Client + Server | Core programmatic | | Video (instream) | Client + Server | Highest CPM | | Mobile in-app | SDK → Server | 71% programmatic | | CTV / OTT | Server only | 44% share (Q2 2025), >$45B | | Retail Media | Server | $30B+ by 2026, +29% YoY | | DOOH | Server | +400% since 2019 |
$110B+
US programmatic video 2026
44%
CTV share (Q2 2025, was 28%)
$30B+
retail media by 2026
## Ecosystem and Competitors
2015
Prebid.js launched (AppNexus + partners)
Open-source header bidding library released.
2017
Prebid Server + Mobile SDK
Server-side auction and mobile support added.
2019–20
95% of top US publishers on header bidding
Header bidding becomes the industry norm.
2021–22
User ID, Floors, GDPR modules
Privacy-era modules expand the platform.
2024–25
v11, CTV, 300+ adapters, PAAPI
Multi-format expansion and post-cookie readiness.
2025–26
Google antitrust → neutral infra more critical
Court ruling accelerates shift to vendor-neutral solutions.
| Solution | Type | Differentiation | Lock-in | |----------|------|-----------------|---------| | Prebid.js | Open source | Neutral, auditable | None | | Amazon TAM | Proprietary | Tied to Amazon DSP | Medium | | Google Open Bidding | Server-side in GAM | Conflict of interest | High | | Index Exchange | Proprietary | Quality, but vendor-dependent | Medium | Prebid runs in parallel with TAM and Open Bidding. Per Roxot data, client-side and server-side header bidding generate ~22% of programmatic revenue each, while AdX accounts for ~56%. ## What Prebid Doesn't Do

Doesn't optimize placement

Ad slot position and size are publisher decisions — Prebid handles demand, not supply-side layout.

Doesn't replace the ad server

GAM is still needed for direct deals, frequency capping, and final decisioning.

Doesn't fix traffic quality

Bots, low viewability, and invalid traffic are root-level problems outside Prebid's scope.

Not plug-and-play

An engineering tool requiring ongoing configuration optimization and monitoring.

## Bottom Line Working without header bidding in 2026 means leaving **20–40% of programmatic revenue** on the table. The strategic question isn't "do we need Prebid" but **which configuration is optimal**. With Google antitrust and the post-cookie transition, vendor-neutral solutions are more critical than ever.
+20–50%
waterfall → header bidding
+5–15%
+ server-side
+3–10%
config optimization
--- ## Auction Mechanics: From Christie's to Google Ads URL: https://papilov.org/research/auction-mechanics/ Also available in: RU, ES, DE, ZH New Zealand, 1990. The government hires consulting firm NERA to design a radio spectrum auction. NERA recommends the Vickrey auction — a format whose theoretical foundations would earn a Nobel Prize six years later. Expected revenue: NZ$250 million. Actual revenue: NZ$36 million. One bidder offered NZ$100,000 and paid NZ$6. Another bid NZ$7 million and paid NZ$5,000. One wrong choice of auction format cost the treasury NZ$214 million. In 2019, Google changed the auction format for the $48 billion programmatic advertising market. In 2021, the FCC raised $81 billion in a C-band spectrum auction — a world record. The EU ETS has generated €245 billion from auctioning the right to pollute since 2013. In every case, it wasn't the good that determined the price — **the auction rules determined the price**. ## Four Basic Auctions Every auction in the world is a combination of two variables: open or sealed bids, and whether the winner pays their own price or someone else's.

First-Price (pay your bid)

Dutch (open, descending)Price falls from ceiling. First to say "stop" wins and pays current price. Strategy: wait longer = cheaper, but risk losing.
Sealed first-price (closed)Tenders, government procurement, programmatic ads since 2019. All submit bids in sealed envelopes. Highest wins. Strategy: bid shading — bid below true value.

Second-Price (pay 2nd bid)

English (open, ascending)Christie's, Sotheby's. Price rises. Last bidder standing wins and pays just above second-highest. Strategy: bid up to your true valuation, no higher.
Vickrey (sealed second-price)Nobel Prize 1996. Highest bid wins, but pays second price. eBay's proxy bidding. Dominant strategy: bid exactly your true valuation.
Strategic equivalence: Dutch ≈ Sealed first-price (same strategy, different pacing). English ≈ Vickrey (same outcome — winner pays ~second valuation). Two pairs that look different but are strategically identical.
## How It Works: One Item, Three Prices Three bidders value a painting at $100, $80, and $50. What happens under different rules?

First-Price Sealed

Optimal bid with n=3: b(v) = v × ⅔

Vickrey (Second-Price)

Dominant strategy: bid your true value

English (Open Ascending)

Price rises until one bidder remains
### Bidding Formulas

Second-Price (Vickrey)

b(v) = v. Bid your true value. Always. Dominant strategy — independent of number of participants and their behavior.

b(v) = v

First-Price (Sealed)

b(v) = v × (n−1)/n. With 2 bidders: shade 50%. With 10: shade 10%. More competition → more aggressive bids.

b = v·(n−1)/n
**Bid shading by number of participants** (value = $100, first-price):
n = 2
$50
$50
n = 3
$67
$67
n = 5
$80
$80
n = 10
$90
$90
n = 20
$95
$95
n = 100
$99
$99
### Revenue Equivalence Theorem
E[RevenueFP] = E[RevenueSP] = (n−1)/(n+1) With n bidders drawing values from U[0,1], the expected seller revenue is identical across all four formats. The key word is expected: in any single auction prices differ, but across many they converge. Vickrey 1961 (2 bidders), Riley & Samuelson 1981 (general case) → Nobel Prize 1996.
**Conditions:** independent private values, risk neutrality, symmetric bidders, no collusion. In reality, every condition is violated — making format choice a critical decision.

Independent private values

If holds: all formats equal

Risk neutrality

If holds: RE works

Symmetric bidders

If holds: RE works

No collusion

If holds: RE works
## The New Zealand Disaster
NZ$250M
expected revenue
NZ$36M
actual revenue
86%
revenue lost
Specific lots: | Lot | Bid | Paid | Discount | |-----|-----|------|----------| | Radio frequency license | NZ$100,000 | NZ$6 | 99.994% | | Radio frequency license | NZ$7,000,000 | NZ$5,000 | 99.93% | | National cellular license | NZ$101,000,000 | NZ$11,000,000 | 89.1% | | Sky Network TV, Lot 1 | NZ$2,371,000 | NZ$401,000 | 83.1% |
Three design errors: (1) No minimum price (reserve price) — you could bid $6. (2) Sold identical licenses through separate Vickrey auctions instead of a unified uniform-price auction. (3) Asymmetric participants (incumbents vs newcomers) — revenue equivalence breaks down.
## Comprehensive Format Comparison

English (ascending)

Christie'seBayFCC SMRA

Dutch (descending)

Aalsmeer flowersGoogle IPOTreasury

First-Price (sealed)

ProcurementTendersProgrammatic 2019+

Vickrey (second-price)

RTB 2007–2017Google Ads (GSP)
## Ad Auctions: 100 Milliseconds and $500 Billion Advertising auctions are the most massive in history: trillions of auctions per day. Their 12-year evolution recapitulated the path auction theory traveled over 50 years.
2007
RTB and second-price auctions
Birth of programmatic: SSPs run real-time Vickrey auctions. Highest bidder wins and pays second price + $0.01. Truthful bidding — everyone bids their true value.
~2012
Waterfall: cascade of inefficiency
Publishers call ad networks sequentially: Google first, then rest. An $8 bid from the third SSP never beats the $5 price from the first — it's never seen. Loss: 20–40% of revenue.
2014–15
Header bidding: simultaneous auctions
Revolution. All SSPs receive bid requests simultaneously. Prebid.js (2015, open source) becomes the standard. 84% of top 10K US sites. RPM growth 30–40%. Sequential → simultaneous auction.
2017
Exchanges switch to first-price
OpenX, Rubicon Project and others drop second-price. Reason: in header bidding, second-price enables SSP floor-price manipulation. First-price is simpler and more transparent.
2019
Google Ad Manager → unified first-price
Google — the last major exchange. Eliminates "last look" privilege. Unified auction = one first price for all. Bid shading becomes mandatory. The $48B market shifts overnight.
2021
AdSense → first-price. Full circle.
The last second-price bastion in Google falls. Theory said second-price → truthful. Practice: first-price + bid shading. Bid shading can take up to 20% of publisher revenue.
Summary: Ad industry traveled from Vickrey (2007) to first-price (2019) in 12 years — for the same reasons Vickrey described in 1961: when values correlate, bidders are asymmetric, and intermediaries exist, second-price creates manipulation opportunities. The theory wasn't wrong — the conditions of application were.
## Auctions in Finance Financial markets are auctions with different names. Treasury, IPO, buyback — the same mechanisms everywhere.

US Treasury Bills

Dutch / uniform-price auction
$23T+ debt

Google IPO, 2004

Modified Dutch auction
$1.67B raised

Stock Exchanges

Continuous double auction
~$120T/year

EU ETS Carbon Market

Sealed uniform-price
€245B since 2013€39B/year

Energy Markets

Uniform-price merit order
€60–300/MWh

FCC Spectrum Auctions

SMRA (Milgrom-Wilson design)
$81B record
## Auctions in Technology

AWS Spot Instances

Up to 90% discount on idle EC2 capacity

Ethereum Gas (EIP-1559)

First-price → hybrid mechanism

Domain Auctions

English auction + reserve price
Pattern: The same evolution repeats everywhere. Start with a pure auction (AWS Spot 2009, Ethereum pre-1559, programmatic 2007). Over time, transition to a hybrid: auction mechanism + algorithmic pricing. Pure auctions are too volatile for production systems. Stability beats theoretical optimality.
## Where Auctions Are Heading

AI Bidding Agents

Now → 2027

Privacy-Preserving Auctions

Now → 2028

Combinatorial Auctions

2025+

Dynamic Mechanism Design

2026+

Real-Time Energy Markets

2025+

Auctions → Automatic Markets

Long-term
### Evolution: From Gavel to Algorithm
Pre-20th century
English, Dutch — minutes to hours
People in a room. Christie's (1766), Aalsmeer flower auction (1911).
1960–1990
Sealed-bid, Vickrey — days to weeks
Companies + governments. NZ Spectrum 1990, US Treasury auctions.
1994–2010
SMRA, combinatorial — weeks to months
Telecoms + regulators. FCC Auctions designed by Milgrom & Wilson (Nobel 2020).
2007–2019
RTB second → first-price — 100ms
DSP/SSP algorithms. Programmatic advertising, header bidding revolution.
2020+
ML bid agents + hybrid — microseconds
AI vs AI. Auto-bidding, Spot pricing, EIP-1559. Auction theory as invisible OS.
## One Lesson Rules determine behavior. Behavior determines price. **Whoever designs the rules owns the market.** The same radio waves cost between $0.001 and $0.875 per MHz/pop — an 875× difference. The same ad generates 30–40% more or less revenue depending on the auction format. The same carbon costs €5 or €100 — depending on cap-and-trade rules. An auction is not a gavel at Christie's. It's a model of any market with competition for a scarce resource. --- ## The Price of Loyalty URL: https://papilov.org/research/subscription-pricing-models/ Also available in: RU, ES, DE, ZH Since 2020, subscription prices have outpaced inflation by 3–5×. The US Consumer Price Index rose roughly 25% over this period. Most major subscriptions rose 50–170%. Disney+ leads the pack with a 172% increase ($6.99 → $18.99), followed by Apple TV+ at 160%, Netflix at 125%, and Xbox Game Pass Ultimate doubling to $29.99. Meanwhile, World of Warcraft has charged exactly $14.99/month since November 2004 — 21 years with zero change. This isn't random. It's three distinct pricing models operating on fundamentally different economic logic. ## Who Raised Prices — and by How Much
Disney+
$6.99 → $18.99
+172%
Apple TV+
$4.99 → $12.99
+160%
Netflix
$7.99 → $17.99
+125%
Peacock
$4.99 → $10.99
+120%
Xbox Game Pass
$14.99 → $29.99
+100%
Amazon Prime
$79/yr → $139/yr
+76%
Microsoft 365
$69.99 → $99.99/yr
+43%
Spotify
$9.99 → $12.99
+30%
CPI Inflation
~25%
## Raisers vs. Holders The key question: is the subscription the product, or the gateway to other spending?

Price Raisers ↑

Netflix7 increases in 15 yrs. Standard: $7.99 → $17.99
Disney++172% in 6 years. Fastest riser in history
Spotify12 yrs frozen, then 3 hikes in 30 months
Adobe CCPerpetual → subscription → steady increases
Xbox Game PassUltimate doubled: $14.99 → $29.99

Price Holders ↓

World of Warcraft$14.99/mo since Nov 2004. 21 years, $0 change
Planet Fitness$10/mo from 1998 to 2024. 26 years frozen
EA Play$4.99/mo since Aug 2014. 11 years unchanged
Nintendo Switch Online$19.99/yr since Sep 2018. 7+ years
Costco$5 increase every 5–7 yrs. $60 → $65 in 2024
## The WoW Paradox $14.99 hasn't moved in 21 years. But Blizzard isn't leaving money on the table — they moved the table.

World of Warcraft

$14.99/mo since November 2004
$25.72
$14.99 (2004) adjusted for inflation
−42%
Real price decline
~9M
Estimated subscribers (2025)
$20
WoW Token price
Bobby Kotick (ex-CEO): *"It's a prickly audience. You don't wanna do too much to agitate them. Even a dollar increase would've been a problem."* **2004–2010:** Subscription was 85% of revenue. Expansions 15%. Microtransactions: zero. **2020–2026:** Subscription dropped to ~50%. Microtransactions and tokens grew to ~35%. Expansions stayed at 15%. They didn't raise the price. They built new revenue streams around a frozen anchor.
## Three Pricing Models

Extractive

Subscription = the product. Raise as fast as churn allows.
NetflixDisney+SpotifyHBO Max

Gateway

Subscription = acquisition tool. Hold price, monetize elsewhere.
WoWCostcoPlanet FitnessAmazon Prime

Monopoly

No real alternatives. Raise freely, cite AI/features.
Adobe CCMicrosoft 365Google Workspace
## Why We Keep Paying

Loss Aversion

Losing Netflix feels 2× worse than the rational savings of canceling. The pain of loss outweighs the logic of savings.

2×

Subscription Blindness

72% of consumers underestimate their total subscription spending by ~40%. 55% maintain at least one unused sub.

72%

Boiling Frog

Small ~10% increases stay below the pain threshold. Netflix: $1–2.50 every 12–24 months — too little to trigger action.

~10%

Moral Fairness

Consumers judge prices morally, not economically. Netflix's 2011 60% hike felt like betrayal. The same increase over 5 years felt normal.

60% ≠ 5×12%
## Pricing Disasters & Wins
2011
Netflix Qwikster — 60% price hike
800K subscribers lost. Stock dropped 77%. Too much, too fast, no added value.
2017–2019
MoviePass — the opposite error
Slashed price to $9.95 for daily movies. Lost $27–45M/month. 92% subscriber loss when forced to raise. Execs convicted of fraud.
2022 Q1
Netflix's first subscriber loss
Wall Street shifted from subscriber growth to revenue per user. Every streamer began aggressive price increases.
2022 May
EVE Online breaks 19-year freeze
+33% ($14.95 → $19.99). Severe player backlash — reinforced WoW's decision to hold.
2023–24
Netflix ad-tier + password crackdown
Ad tier hit 94M users. Password crackdown added ~50M subscribers. Revenue grew without proportional price pain.
2025 Jan
Netflix raises all tiers — no backlash
+$1–$2.50 across all plans. Reached 302M subscribers. Boiling frog strategy perfected over 14 years.
2025 Oct
Xbox Game Pass — 50% single hike
$19.99 → $29.99 overnight. Cancellation website crashed. GameStop publicly mocked Microsoft.
--- ## Penrose Diagrams URL: https://papilov.org/research/penrose-diagrams/ Also available in: RU, ES, DE, ZH Spacetime is infinite. Light can travel forever. A black hole crushes a point to zero. How do you draw all of this on a finite sheet? Roger Penrose's answer, developed in the early 1960s: conformal compactification. A mathematical transformation that squeezes infinity into finite points and lines while preserving the one thing that matters for the physics of causality — the structure of light cones. This one idea gave physicists a tool so powerful that it now appears in virtually every major result in general relativity — from the singularity theorems to the holographic principle, from the classification of spacetimes to the proof that gravitational waves carry energy. ## The Problem By the early 1960s, general relativity was in crisis — not because the theory was wrong, but because nobody could prove it was right in the ways that mattered most. Karl Schwarzschild found his black hole solution in 1916, within months of Einstein publishing the field equations. But the solution assumed perfect spherical symmetry — a condition that never holds in nature. Most physicists, Einstein included, believed that any real-world deviation from perfect symmetry would prevent a singularity from forming. The question of whether black holes actually exist, or are just artifacts of idealized mathematics, remained open for nearly 50 years. The difficulty was that existing tools couldn't handle the global structure of spacetime. You could write down the metric at any single point, but understanding what happens "at infinity" — where gravitational waves arrive, where light rays end up, what the ultimate fate of collapsing matter is — required seeing the whole spacetime at once. ## The Idea The technique came together across three key moments. In a short 1963 note in *Physical Review Letters*, Penrose introduced conformal compactification as a tool for studying asymptotic structure. At the Les Houches Summer School that same year, he gave three lectures explaining the method in detail — including how the conformal boundary changes depending on the sign of the cosmological constant. Then in 1965, he published the singularity theorem that the Nobel Committee would later call "the most important contribution to the general theory of relativity since Einstein." The key to the singularity theorem came to Penrose during a walk with colleague Ivor Robinson in London, autumn 1964. As he later recalled, they reached a crossroad, stopped talking to watch for traffic, and in that moment the concept of a "trapped surface" flashed into his mind. That single idea — combined with the conformal methods he had been developing — proved that singularities are inevitable in general relativity, ending five decades of debate. ## How It Works The core insight is elegant: if you care only about causality — which events can influence which other events — then you don't need to preserve distances. You only need to preserve the light cone structure. A conformal transformation rescales the metric by a factor Ω²: every distance is multiplied by Ω, but angles are unchanged. Light rays still travel at 45°. The causal ordering of events is preserved. What changes is that Ω is chosen to go to zero at infinity, compressing infinitely distant regions into a finite boundary. The result: an infinite spacetime fits on a finite diagram. Its causal structure is perfectly intact. What you lose — distances — is precisely what you don't need for understanding which events can influence which. ## Five Rules for All Diagrams
↑
Time flows upward
↔
Space extends horizontally
45°
Light always travels at 45°
>45°
Massive bodies: steeper paths
S²
Every point is a 2-sphere
With these five rules, you can read any Penrose diagram — from flat spacetime to rotating black holes. ## Four Fundamental Spacetimes ### Minkowski — The Diamond Flat spacetime, no gravity. The diagram is a diamond with five boundary components: future and past timelike infinity (i⁺, i⁻) where massive particles end and begin; spatial infinity (i⁰) at the waist; and future and past null infinity (I⁺, I⁻) — the surfaces where light rays arrive and originate. Gravitational wave energy is defined precisely at I⁺, a concept that only becomes rigorous through Penrose's conformal boundary. The Bondi-Metzner-Sachs group (1962), which describes asymptotic symmetries at null infinity, turns out to be infinite-dimensional — much larger than the 10-dimensional Poincaré group of flat spacetime. This surprising structure was discovered through conformal methods.
i⁺ future timelike ∞ i⁻ past timelike ∞ i⁰ spatial ∞ i⁰ I⁺ I⁺ I⁻ I⁻ light ray (45°) event t r r = 0 (origin)
Flat Minkowski spacetime. The diamond shape compresses all of infinite spacetime to a finite figure. Light cones are always at 45°. The five boundary components — i⁺, i⁻, i⁰, I⁺, I⁻ — encode the asymptotic structure.
### Schwarzschild — The Zigzag Singularity A non-rotating black hole. The diagram reveals what coordinates obscured for decades: the singularity at r = 0 is not a point in space — it's a moment in time. It appears as a horizontal zigzag line, meaning that once you cross the event horizon (a 45° line), the singularity lies in your future no matter which direction you move. You can't escape not because you can't run fast enough, but because all spatial directions point toward the singularity.
singularity (r = 0) event horizon i⁰ i⁻ II inside BH I exterior infalling
Schwarzschild black hole. The event horizon is a dashed 45° line. Beyond it, all paths lead to the zigzag singularity at the top. There is no escape — not because you can't move fast enough, but because the singularity is in your future.
The maximally extended solution — first found using Kruskal-Szekeres coordinates (1960) — reveals a white hole, a second universe, and an Einstein-Rosen bridge connecting them. The bridge closes faster than light can traverse it. Penrose diagrams added the crucial step of compactifying the asymptotic regions, making the full causal structure visible at a glance. ### De Sitter — The Square Universe Positive cosmological constant, no matter — our Universe's approximate future as dark energy dominates. The diagram is a square with spacelike (horizontal) top and bottom boundaries: the universe begins and ends everywhere at once.
I⁺ (future ∞) I⁻ (past ∞) horizon observer
De Sitter spacetime. A square instead of a diamond. Top and bottom boundaries are spacelike — the universe begins and ends everywhere at once. Each observer is surrounded by a cosmological horizon.
Every observer has a cosmological horizon: regions receding faster than light can never send signals that reach you. Since 1998, when Riess, Perlmutter, and Schmidt discovered the accelerating expansion (Nobel Prize 2011), we know our Universe is heading toward this de Sitter-like fate. ### Anti-de Sitter — The Vertical Strip Negative cosmological constant. The diagram is a vertical strip with timelike (vertical) boundaries. Not our Universe — but arguably the most important spacetime in theoretical physics today.
boundary (CFT) boundary (CFT) AdS bulk t → +∞ t → −∞ light "bounces" off boundary
Anti-de Sitter spacetime. A vertical strip. The sidewalls are timelike boundaries where the dual CFT lives. Light reaches the boundary in finite time and "bounces" back — the space acts like a box.
In 1997, Juan Maldacena showed that quantum gravity in this space is exactly equivalent to a quantum field theory living on its boundary. This AdS/CFT correspondence — the [most cited result](https://arxiv.org/abs/hep-th/9711200) in high-energy physics history (20,000+ citations) — means that the vertical edges of the Penrose diagram are not just abstract boundaries. They are where the dual theory lives. Gravity in the bulk, quantum fields on the edge. The Penrose diagram makes this architecture visible. ## What These Diagrams Made Possible
1963
Penrose introduces conformal compactification
A short note in Physical Review Letters, then three detailed lectures at Les Houches. The method is established.
1965
The singularity theorem
Penrose proves that singularities are inevitable under realistic conditions. The Nobel Committee calls it "the most important contribution to general relativity since Einstein." The prize arrives 55 years later.
1970
Penrose-Hawking singularity theorems
Hawking extends the method to cosmology: the Big Bang itself must contain a singularity. The universe began from a point of infinite density.
1973
Hawking & Ellis publish the reference text
"The Large Scale Structure of Space-Time" establishes Penrose diagrams as the standard tool. The notation and conventions used to this day come from this book.
1973–75
Black hole entropy and Hawking radiation
Bekenstein proposes entropy ∝ horizon area. Wheeler: "Your idea is just crazy enough to work." Hawking proves black holes radiate. The Penrose diagram of an evaporating black hole — singularity eventually disappearing — creates the information paradox.
1997
AdS/CFT correspondence
Maldacena discovers that quantum gravity in Anti-de Sitter space equals a quantum field theory on its conformal boundary — the vertical edge of the Penrose diagram. At Strings '98, hundreds of physicists sang "The Maldacena" to the tune of the Macarena.
2013
ER = EPR
Maldacena and Susskind: Einstein-Rosen bridges (visible on Penrose diagrams) are equivalent to quantum entanglement. Spacetime geometry and quantum information may be two descriptions of the same thing.
2015
LIGO detects gravitational waves
Gravitational wave energy is defined at null infinity I⁺ — the boundary that Penrose diagrams made rigorous. The BMS asymptotic symmetry group, built on conformal methods, underlies the theoretical framework.
2020
Nobel Prize
Penrose, aged 89, receives half the Nobel Prize in Physics. The gap between discovery (1965) and recognition (2020) is 55 years — one of the longest in Nobel history.
## The Person Behind the Diagrams
A visual mind across disciplines Roger Penrose (born 1931) proposed the impossible triangle (conceived 1954, published 1958) and inspired M. C. Escher's Waterfall and Ascending and Descending. His aperiodic tilings (1974) presaged the discovery of quasicrystals. His singularity theorem used topology where others tried brute-force calculation. His conformal diagrams replaced pages of equations with a single picture. In every case, the breakthrough came from seeing structure that others described only algebraically.
Kip Thorne described Penrose's approach as "overlooking the detailed geometrical structure of spacetime and instead concentrating just on the topology of the space, or at most its conformal structure, since it is the latter — as determined by the lay of the lightcones — that determines the causal relationships." He shared the 1988 Wolf Prize with Stephen Hawking and won the Nobel at 89. --- ## The Universe as a Hologram URL: https://papilov.org/research/ads-cft-holographic/ Also available in: RU, ES, DE, ZH In 1997, Juan Maldacena — a 29-year-old Argentine physicist, one year out of his Princeton PhD and newly a professor at Harvard — published what would become the most cited paper in the history of high-energy physics. Over 20,000 citations to date, roughly two per day for over two decades. The second and third most cited papers in the field are both about the same discovery, and both cite his. He showed that quantum gravity in a bounded, negatively curved spacetime is mathematically identical to a quantum field theory living on its boundary — with no gravity and one fewer dimension. Everything happening in the interior, including the bending of space and the formation of black holes, is fully encoded on the edge. Like a hologram. This is the AdS/CFT correspondence. It didn't just solve a problem. It created an entire field. ## Where It Began: A PhD Student vs. Everyone The story starts not with Maldacena, but with Jacob Bekenstein — a graduate student at Princeton in the early 1970s. His advisor, John Wheeler, posed a thought experiment: what happens to the entropy of a cup of tea if you drop it into a black hole? The tea has thermodynamic entropy — a measure of its internal disorder. But a black hole, described by the no-hair theorem, has only three properties: mass, charge, and spin. No temperature. No disorder. So when the tea crosses the event horizon, its entropy seems to vanish. The second law of thermodynamics — entropy never decreases — appears to be violated. Bekenstein's answer was radical: black holes themselves carry entropy, and that entropy is proportional to the area of the event horizon — not the volume of the interior. He published this in 1973. The response from the physics community was overwhelmingly negative. As Kip Thorne wrote: "All the world's black hole experts lined up on Hawking's side." The sole exception was Wheeler, who told Bekenstein: *"Your idea is just crazy enough that it might be right."*
S = A / 4 The Bekenstein-Hawking entropy formula: the entropy of a black hole equals one quarter of its event horizon area, measured in Planck units. This is the most concise equation in physics connecting gravity, quantum mechanics, and thermodynamics — three theories that otherwise refuse to cooperate.
Two years later, Hawking — who had been trying to prove Bekenstein wrong — instead proved him right. By applying quantum field theory to curved spacetime, Hawking showed in 1975 that black holes emit thermal radiation at a specific temperature. This fixed the constant in Bekenstein's formula and established black hole thermodynamics as real physics, not analogy. But there was a deeper message in the formula S = A/4. If the maximum entropy in a region of space is proportional to its surface area rather than its volume, then the number of degrees of freedom needed to describe the physics inside is somehow determined by the boundary. As if the interior were a projection. ## The Holographic Principle In the early 1990s, Gerard 't Hooft and Leonard Susskind took this implication seriously. They proposed the holographic principle: all the information needed to describe a volume of space can be encoded on its boundary, with a density of at most one bit per Planck area (~10⁻⁶⁶ cm²).
10⁻⁶⁶ cm²
Planck area — 1 bit limit
S = A/4
Entropy = quarter of horizon area
20,000+
Citations of Maldacena's paper
~2/day
Citation rate, 27 years running
This was a principle without a proof. Nobody could show a concrete system where it actually worked. That changed in November 1997. ## Maldacena's Discovery Maldacena considered a stack of D-branes — objects in string theory — and took a specific low-energy limit. Viewed from far away, the stack looks like a black hole in anti-de Sitter space (AdS). Viewed up close, it looks like a quantum field theory (a conformal field theory, or CFT) living on the brane surfaces. Both descriptions must be of the same system. Therefore: string theory in the bulk of AdS = conformal field theory on its boundary. Specifically: type IIB string theory on 5-dimensional AdS × S⁵ is equivalent to N = 4 super Yang-Mills theory on the 4-dimensional boundary. Every object in one description has a precise counterpart in the other. Mass in the bulk corresponds to operator dimension on the boundary. The radial direction in AdS corresponds to the energy scale of the field theory. Black holes in AdS correspond to thermal states. The dictionary is exact.

Bulk (AdS, d+1 dimensions)

GravityEinstein's equations with negative Λ
Black holeObject with event horizon
Particle massDetermined by field equations
Radial directionExtra spatial dimension
Weak couplingClassical gravity regime

Boundary (CFT, d dimensions)

No gravityQuantum field theory, flat space
Thermal stateSystem at finite temperature
Operator dimensionScaling behavior of operators
Energy scaleRenormalization group flow
Strong couplingNon-perturbative regime
black hole in bulk (d+1) AdS gravity · d+1 dimensions CFT on boundary quantum field theory · d dimensions · no gravity thermal state in CFT
Bulk ↔ Boundary: a black hole in AdS volume = a thermal state in boundary quantum theory.
The last row is why AdS/CFT is so powerful: when the boundary theory is strongly coupled (hard to compute), the bulk description is weakly coupled (easy to compute), and vice versa. Problems that were unsolvable on one side become tractable on the other. ## The Reaction The excitement was immediate. At the Strings '98 conference in Santa Barbara the following summer, physicist Jeff Harvey led hundreds of theorists in singing "The Maldacena" — a parody of the Macarena — at the conference dinner: > *"You start with the brane and the brane is BPS / Then you go near the brane and the space is AdS / Who knows what it means? I don't, I confess / Ehhhh, Maldacena!"* The fact that this made the New York Times gives some sense of the impact. As Quanta Magazine described Maldacena: Susskind calls him "the master." Within months, two papers — by Gubser, Klebanov and Polyakov, and by Witten — made the conjecture more precise and laid out the computational machinery that thousands of physicists have used since. ## Spacetime from Entanglement In 2006, Shinsei Ryu and Tadashi Takayanagi added a startling piece to the puzzle. They showed that the entanglement entropy of a region on the boundary equals the area of a specific minimal surface in the bulk — a direct generalization of the Bekenstein-Hawking formula. Mark Van Raamsdonk took this further in 2010. In an essay that won first prize from the Gravity Research Foundation, he argued that if you systematically remove quantum entanglement between two regions of a boundary theory, the corresponding regions of the bulk spacetime literally pull apart and pinch off. Entanglement gone → spacetime disconnected. His conclusion was radical: spacetime is not the fundamental entity. It is woven together from quantum entanglement. Remove the entanglement, and spacetime falls apart. In 2013, Maldacena and Susskind crystallized this into a conjecture with the most compact physics equation since E = mc²:
ER = EPR Einstein-Rosen bridges (wormholes connecting distant black holes) are the same thing as Einstein-Podolsky-Rosen pairs (quantum-entangled particles). Geometry = entanglement. Two frameworks that seemed completely unrelated — general relativity and quantum mechanics — may be two descriptions of the same underlying reality.
## The Problem: Our Universe Isn't AdS

AdS (Λ < 0)

Negatively curved. Has a boundary. AdS/CFT works beautifully. Not our Universe.

✓ solved

dS (Λ > 0)

Positively curved. Our Universe. No boundary in the conventional sense. Holography unclear.

? open
The AdS/CFT correspondence works for spaces with a negative cosmological constant. Observations since 1998 confirm that our Universe has a positive cosmological constant — it's de Sitter-like, not anti-de Sitter. Andrew Strominger proposed a dS/CFT correspondence in 2001, but the program remains incomplete. The boundary of de Sitter space is spacelike (it's the infinite future), not timelike like in AdS. This changes the mathematical structure fundamentally.
I⁺ (future infinity) I⁻ (past infinity) us horizon CFT ? dS bulk
Penrose diagram for de Sitter spacetime. The observer "us" sees only part of the space. A hypothetical CFT lives on I⁺ — at infinite future. Unlike AdS, this boundary is spacelike, not timelike.
This is not just a technical gap — it's the central open question of the field. We have a complete holographic description of a universe that isn't ours, and no complete description of the one that is. ## Timeline
1973
Bekenstein: black hole entropy ∝ area
A PhD student proposes that information about a region of space is encoded on its boundary. Nearly everyone disagrees. It takes 25 years for the full implications to emerge.
1975
Hawking radiation
Hawking, who set out to disprove Bekenstein, instead proves him right: black holes have temperature and entropy. S = A/4 is established. The information paradox is born.
1993–95
't Hooft and Susskind: the holographic principle
The maximum information in a region of space scales with its surface area, not its volume. A bold claim — but no concrete realization yet.
1997
Maldacena publishes AdS/CFT
"The Large N Limit of Superconformal Field Theories and Supergravity." The first concrete realization of the holographic principle. Will become the most cited paper in high-energy physics.
1998
Witten + Gubser-Klebanov-Polyakov
Two papers make AdS/CFT computationally precise. The duality dictionary is established. At Strings '98, physicists sing "The Maldacena" at the conference dinner. The field erupts.
2001
Strominger proposes dS/CFT
An attempt to extend holography to de Sitter space — our Universe. The proposal is promising but incomplete. The problem remains open.
2006
Ryu-Takayanagi formula
Entanglement entropy on the boundary = area of a minimal surface in the bulk. A direct generalization of the Bekenstein-Hawking formula. Geometry and entanglement are linked.
2010
Van Raamsdonk: spacetime from entanglement
Remove entanglement → spacetime disconnects. The radical conclusion: spacetime is emergent, stitched together by quantum correlations. First prize, Gravity Research Foundation.
2013
ER = EPR
Maldacena and Susskind: wormholes = entanglement. Three letters from 1935 (Einstein-Rosen bridge) equal three letters from 1935 (Einstein-Podolsky-Rosen). Geometry is quantum information.
2019
The black hole information paradox "resolved"
Using AdS/CFT techniques (quantum extremal surfaces, islands), several groups show that black hole evaporation follows a unitary Page curve. Information is preserved — but how it escapes remains debated.
## What It Means The holographic principle, if true, is the most radical revision of our picture of reality since quantum mechanics. It says that the three-dimensional space we inhabit may be a projection — complete and self-consistent, but ultimately encoded on a distant two-dimensional surface. We don't yet know if this applies to our actual Universe. The evidence is overwhelming for AdS spacetimes, circumstantial for de Sitter. But the direction of theoretical physics since 1997 is unambiguous: spacetime is not fundamental. It emerges from something deeper — quantum information, entanglement, codes. Penrose diagrams showed us how to see the causal structure of spacetime. The holographic principle suggests that this structure itself may be computed from a lower-dimensional theory on the boundary.