Building a Computer from Dirt and Rocks: A Journey from First Principles & Binary Energy Dynamics: Resolving the Information Loss Paradox in Black Holes

 

When people talk about computers, they sound like priests reciting a code no one else can read. Everything is hidden under layers—kernels, stacks, caches, quantum foam if you let them talk long enough. It’s impressive, sure, but it builds the wrong mythology. The machine isn’t that mysterious. It’s just the modern mask of a very old impulse: to trap thought in matter.

A computer is a physical argument. Someone once said, “if I push this and not that, I can make a pattern appear.” That’s all this has ever been. A chain of “if” and “then” baked into stone, then copper, then silicon. The words have gotten fancier, the voltages smaller, the loops tighter. But the logic hasn’t aged a day.

The strange thing about learning computing now is that it’s usually taught upside down. You start with syntax before you ever learn what a state is. You’re handed a glowing screen and told to “write a loop,” as if you already know what repetition feels like to a machine. The field buries the elegance under ceremony. You’re memorizing incantations before you even know what they summon.

It’s like walking into a cathedral to learn how to stack stones. All that beauty, but you never get to touch the foundation. You’re told about abstraction, about high-level languages and distributed architectures, but you’re never told the simplest truth: everything the machine does is just a dance of differences. One thing is not another. That’s the whole show.

The real wonder is that difference can remember itself. That’s what we call state. The fact that one moment can leave a mark that shapes the next. Once that was done—once we learned to make “change” stay still—everything else followed. You could count. You could predict. You could build a memory that didn’t die when you blinked.

Modern computing tries to overwhelm you with scale. Billions of transistors, trillions of operations per second, floating-point precision down to the breath of an electron. It’s meant to sound ungraspable. But you could strip all of it down and still have the same skeleton: a pattern that listens to itself. Cavemen did that when they drew constellations, or stacked rocks by a river to mark a season. They made maps of time. That’s all data ever is—memory stretched across a landscape.

So if computers feel difficult, it’s because we meet them at the top of their complexity, not their beginning. The screen hides the simplicity behind glass. But the same mind that could start a fire or carve a spear could build one, given enough boredom and curiosity. Because logic isn’t a product of industry—it’s a side effect of noticing.

Every new generation of students relearns this. They wrestle with syntax until it hurts, then one day they realize they aren’t fighting the machine—they’re fighting the way it’s been explained. Once you stop expecting it to be mystical, it becomes almost childlike. Flip, wait, remember. The machine doesn’t think; it echoes the smallest truths we ever found in the dirt.

And when that finally sinks in—when you see that every computer, no matter how fast or small, is just a sculpted sequence of “if this, then that”—the tension leaves. The awe stays, but the fear goes. You realize we’ve been building computers for as long as we’ve been aware of patterns. The only real invention was persistence: keeping the thought long enough to share it.

So, no, computing was never meant to be hard. We just let the explanations grow taller than the idea. Strip them away, and what’s left is ancient: the simplest distinction in the world, made permanent. A whisper of logic written in matter. Something even a caveman could have done—if he’d had a reason to remember.

Part 1: The Foundation - What is Computation?

Before we dig our first hole or place our first rock, we need to understand what we're actually building. A computer, at its most fundamental level, is a machine that manipulates information according to rules. The information doesn't care what physical form it takes—it could be voltages in silicon, beads on an abacus, or rocks in holes. What matters is that we can distinguish between states and transform those states predictably.

Let's start with the absolute simplest possible computer: a system that can store one bit of information.

Part 2: The Bit - Our First Hole

Dig a hole in the dirt. Make it about the size of your fist, just deep enough that you can clearly see whether there's a rock in it or not. This hole represents one bit of storage.

The rules are simple:

• Empty hole = 0
• Rock in hole = 1

Congratulations. You've just created one bit of memory. You can store exactly two possible states: rock or no rock, yes or no, true or false, 1 or 0.

This seems trivial, but it's profound. This single hole can answer one yes/no question. Is the gate open? Is the king alive? Did the scout find water? Place a rock for yes, leave it empty for no.

Now dig seven more holes in a row. You now have eight bits—one byte of storage. With eight holes, you can represent any number from 0 to 255.

How? Each hole represents a power of 2:

Hole position:  [7] [6] [5] [4] [3] [2] [1] [0]
Value if rock:  128  64  32  16   8   4   2   1

Want to store the number 5? That's 4 + 1, so put rocks in holes 2 and 0:

[ ] [ ] [ ] [ ] [ ] [●] [ ] [●]  = 5

Want to store 200? That's 128 + 64 + 8:

[●] [●] [ ] [ ] [●] [ ] [ ] [ ]  = 200

You've just invented positional notation in base-2. This is how all digital computers store numbers, whether they're made of dirt or silicon.

Part 3: Memory - A Grid of Holes

Now dig a grid: 8 rows by 8 columns = 64 holes. Each row is one byte. You now have 64 bits of storage—enough to store eight numbers from 0-255, or 64 yes/no answers, or eight letters of text (using ASCII encoding).

But there's a problem: how do you find a specific hole quickly? If someone says "give me the value in byte 5," you need a system.

Solution: Addressing

Number your rows 0 through 7. When someone asks for "byte 5," you go to row 5, read the rocks left-to-right, and convert to a number.

This is Random Access Memory (RAM). "Random access" means you can jump directly to any address without reading through all the previous ones. Your grid of holes is RAM.

Part 4: The Problem of Permanence

There's an issue with our dirt computer: wind, rain, animals, or mischievous children can disturb the rocks. We need a way to make information more permanent and to process it without destroying it.

The Solution: Reading Without Disturbing

When you "read" a byte, you don't remove the rocks—you just look at them and write down what you see on a piece of bark or scratch it in the dirt beside you. This temporary workspace is like a register in a real CPU—fast, temporary storage for the number you're currently working with.

The Solution: Backup Storage

For permanent storage, you might carve notches in stones: one notch = 0, two notches = 1. These are your "hard drive"—slower to read/write than the dirt holes, but permanent. When you need to use the data, you copy it from carved stones into your dirt-hole RAM.

Part 5: The Arithmetic Logic Unit (ALU) - Actually Computing

So far, we've just built storage. Now let's build the part that actually computes: the ALU. We'll start with the most basic operation: addition.

Adding Two Numbers: The Procedure

Dig two new 8-hole rows labeled "INPUT A" and "INPUT B", and one row labeled "OUTPUT".

The Addition Algorithm:

We'll add bit by bit, right to left, keeping track of the carry.

Example: Add 5 + 3

Set up your input rows:

INPUT A:  [ ] [ ] [ ] [ ] [ ] [●] [ ] [●]  = 5
INPUT B:  [ ] [ ] [ ] [ ] [ ] [ ] [●] [●]  = 3
OUTPUT:   [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]  = ?

Start at the rightmost hole (position 0):

Position 0:

• A has rock (1)
• B has rock (1)
• 1 + 1 = 10 in binary (that's 0 with a carry of 1)
• OUTPUT position 0: leave empty (0)
• CARRY: 1 (remember this)

Position 1:

• A has no rock (0)
• B has rock (1)
• 0 + 1 + carry(1) = 10 in binary
• OUTPUT position 1: leave empty (0)
• CARRY: 1

Position 2:

• A has rock (1)
• B has no rock (0)
• 1 + 0 + carry(1) = 10 in binary
• OUTPUT position 2: leave empty (0)
• CARRY: 1

Position 3:

• A has no rock (0)
• B has no rock (0)
• 0 + 0 + carry(1) = 1
• OUTPUT position 3: place rock (1)
• CARRY: 0

Positions 4-7: All zeros, no carries, so leave empty.

Result:

OUTPUT: [ ] [ ] [ ] [ ] [●] [ ] [ ] [ ]  = 8 ✓

5 + 3 = 8. It works!

The Addition Rules (The Logic)

You've just executed an algorithm. Let's formalize the rules you were following:

For each bit position (starting from the right):

1. Look at bit from A
2. Look at bit from B
3. Look at carry from previous position
4. Apply these rules:
   • 0 + 0 + 0 = 0, carry 0
   • 0 + 0 + 1 = 1, carry 0
   • 0 + 1 + 0 = 1, carry 0
   • 0 + 1 + 1 = 0, carry 1
   • 1 + 0 + 0 = 1, carry 0
   • 1 + 0 + 1 = 0, carry 1
   • 1 + 1 + 0 = 0, carry 1
   • 1 + 1 + 1 = 1, carry 1

These rules are hardwired into every computer's ALU. In silicon, they're implemented with transistor gates. In your dirt computer, they're implemented by you following the procedure.

Part 6: Automation - The Dream of the Mechanical Computer

Right now, you are the control unit. You're reading the instructions, looking at the holes, placing and removing rocks, following the rules. This works, but it's slow and error-prone.

The dream—realized in Charles Babbage's designs and eventually in electronic computers—is to make the rules themselves physical.

Mechanical Logic: The Rolling Rock Adder

Imagine this mechanical system:

Build a sloped channel system in the dirt:

1. Three input channels (A, B, Carry-in) feed into a junction
2. Each channel has a gate: if there's a rock in the corresponding input hole, the gate opens
3. Rocks roll down open channels and meet at a junction
4. The junction has a scale:
   • 0-1 rocks: they roll out the "SUM" channel (output 1)
   • 2-3 rocks: they're too heavy, trigger a mechanism that:
     • Blocks the SUM channel (output 0)
     • Opens the CARRY channel (output 1)

This is a mechanical full adder. The physical behavior of rolling rocks implements the addition rules automatically.

In practice, building this with dirt would be extremely difficult—you'd need precise slopes, channels, gates, triggers. This is why Babbage's Analytical Engine, though theoretically sound, was nearly impossible to build with 19th-century technology.

But the principle is clear: The logical rules can be embodied in physical mechanisms. In modern computers, transistors are the gates, voltage is the rolling rocks.

Part 7: XOR - The Different Detector

Let's implement another crucial operation: XOR (exclusive or).

XOR rules:

• 0 XOR 0 = 0
• 0 XOR 1 = 1
• 1 XOR 0 = 1
• 1 XOR 1 = 0

In words: Output 1 only if inputs are different.

Dirt implementation:

Create two input holes (A and B) and one output hole. Follow this procedure:

1. Look at A and B
2. If one has a rock and the other doesn't: place rock in OUTPUT
3. If both have rocks or both are empty: leave OUTPUT empty

Why is this useful? XOR is the "sum without carry" part of addition. It's also used in hashing, encryption, error detection—anywhere you need to detect differences or mix information.

Example: Detecting Changes

Store a message in 8 holes (one byte). Store a copy in another 8 holes. Later, to check if the message was altered, XOR the two:

• If all outputs are empty (all 0s): the message is unchanged
• If any output has a rock: something changed

This is the basis of checksums and error detection.

Part 8: Multiplication - Shift and Add

Now let's multiply. Remember: multiplication is just repeated addition with shifts.

Example: 5 × 3

5 in binary: 101
3 in binary: 011

The algorithm:

1. Look at rightmost bit of multiplier (3): it's 1
   • Add 101 (shifted 0 positions) to result
   • Result so far: 101
2. Look at next bit of multiplier: it's 1
   • Add 101 (shifted 1 position left = 1010) to result
   • Result: 101 + 1010 = 1111
3. Look at next bit of multiplier: it's 0
   • Add nothing
   • Final result: 1111 = 15 ✓

Dirt implementation:

You need:

• Input A holes (multiplicand)
• Input B holes (multiplier)
• Multiple rows of working holes (for shifted versions)
• Output holes (accumulator)

The procedure:

1. For each bit in B (right to left):
   • If bit is 1: copy A into a working row, shifted left by the current position
   • If bit is 0: skip
2. Add all the working rows together (using the addition procedure repeatedly)
3. Final sum is the answer

This is tedious with rocks and dirt, which is exactly why multiplication in real CPUs requires hundreds or thousands of transistors arranged in complex trees to do it fast. But the algorithm is the same.

Part 9: The Control Unit - The Instruction Set

So far, you've been the brain—reading instructions, deciding what to do. Let's formalize this.

Create an instruction system:

Dig a special row called the INSTRUCTION register (8 holes). The pattern of rocks in this row tells you what operation to perform.

Instruction encoding:

00000001 = LOAD from address X to register A
00000010 = LOAD from address Y to register B
00000011 = ADD registers A and B, store in C
00000100 = STORE register C to address Z
00000101 = MULTIPLY A and B, store in C
... (more operations)

Dig another row called the PROGRAM COUNTER - this stores the address of the current instruction.

Dig a large grid: the PROGRAM MEMORY. Each row is one instruction.

Example program: Add two numbers

Row 0: 00000001  (LOAD from address 64 to register A)
Row 1: 00000010  (LOAD from address 65 to register B)
Row 2: 00000011  (ADD A and B, result in C)
Row 3: 00000100  (STORE C to address 66)
Row 4: 00000000  (HALT)

The execution cycle:

1. Read the PROGRAM COUNTER (starts at 0)
2. Go to that row in PROGRAM MEMORY
3. Look at the rocks—that's your instruction
4. Decode it: "00000001 means LOAD"
5. Execute: perform the load operation
6. Increment PROGRAM COUNTER (move to next row)
7. Repeat

You're now executing a stored program. The instructions are data, stored the same way as the numbers they manipulate. This is the von Neumann architecture—the foundation of nearly all modern computers.

Part 10: Evolution - From Manual to Mechanical to Electronic

Let's trace the evolution:

Stage 1: Manual Dirt Computer (What We've Built)

• Storage: holes with rocks
• Processing: human following rules
• Speed: one operation per minute
• Reliability: terrible (wind, rain, mistakes)

Advantages:

• Easy to understand
• Easy to debug (just look at the rocks)
• Cheap (dirt is free)

Limitations:

• Slow
• Error-prone
• Doesn't scale

Stage 2: Mechanical Computer (Babbage's Vision)

Replace human with mechanisms:

• Storage: gear positions (gear at position 0-9 represents digit)
• Processing: gears, levers, cams physically embody the rules
• Speed: one operation per second
• Reliability: better, but gears wear out, need lubrication

Example: The Difference Engine

Babbage's Difference Engine calculated polynomial tables:

• Input: turn cranks to set initial values
• Process: turn main crank, gears rotate, addition happens mechanically
• Output: read numbers from gear positions

It worked, but was enormous (thousands of parts) and expensive.

Stage 3: Electromechanical Computer (Relays)

Replace gears with electromagnetic switches (relays):

• Storage: relay positions (on/off)
• Processing: relays wired to implement logic
• Speed: 10-100 operations per second
• Reliability: much better

Example: Harvard Mark I (1944)

• 765,000 components
• 3,000 electromechanical relays
• 500 miles of wire
• Could multiply in 6 seconds

How relays work:

• Coil of wire with iron core
• When current flows through coil, iron becomes magnetic
• Magnetism pulls a metal switch closed
• Switch closing/opening controls another circuit

Relay as logic gate:

• Wire two relays in series: AND gate (both must close)
• Wire two relays in parallel: OR gate (either can close)
• Use relay to switch power on/off: NOT gate (inverter)

Build up from these: you get adders, multipliers, memory—everything we built with rocks.

Stage 4: Vacuum Tube Computer (First Electronic)

Replace relays with vacuum tubes:

• Storage: tube on/off states, later magnetic cores
• Processing: tubes wired as logic gates
• Speed: thousands of operations per second
• Reliability: tubes burn out frequently

Example: ENIAC (1945)

• 17,468 vacuum tubes
• 7,200 crystal diodes
• 1,500 relays
• 5 million hand-soldered joints
• Could do 5,000 additions per second

How vacuum tubes work:

• Glass tube with vacuum inside
• Heated cathode emits electrons
• Grid in middle controls electron flow
• When grid is negative: blocks electrons (off/0)
• When grid is positive: allows electrons (on/1)

Tubes implement the same logic gates as relays, but 1,000× faster because electrons move at light speed, not mechanical parts clanking.

Stage 5: Transistor Computer

Replace tubes with transistors:

• Storage: magnetic cores, then transistor flip-flops
• Processing: transistor logic gates
• Speed: millions of operations per second
• Reliability: solid state, very reliable

Example: IBM 1401 (1959)

• Used transistors instead of tubes
• Much smaller, cooler, more reliable
• Could do 200,000 additions per second

How transistors work (simplified):

• Three layers of silicon (source, gate, drain)
• Gate voltage controls conductivity between source and drain
• High voltage on gate: conducts (on/1)
• Low voltage on gate: blocks (off/0)

Same logic as tube, but:

• Tiny (millimeters vs hand-sized tubes)
• Low power (milliwatts vs watts)
• Reliable (solid state vs hot fragile glass)
• Fast (nanoseconds vs microseconds)

Stage 6: Integrated Circuit Computer (Modern)

Put thousands, then millions, then billions of transistors on one chip:

• Storage: transistor-based RAM, solid state drives
• Processing: billions of transistors in CPU
• Speed: billions of operations per second
• Reliability: extremely high

Example: Modern Intel CPU (2024)

• 20+ billion transistors
• Multiple cores (multiple complete CPUs on one chip)
• Operates at 3-5 GHz (3-5 billion cycles per second)
• Transistors are 3-5 nanometers (1/10,000 width of human hair)

But it's still doing the same thing our dirt computer did:

• Storing bits (rocks in holes → charges in transistors)
• Adding (following addition rules → XOR and AND gates)
• Following instructions (reading program rows → fetch-decode-execute cycle)

Part 11: Scaling - From 8 Bits to Billions

Our dirt computer has 64 bits of RAM (8 bytes). Let's scale that up.

Memory scaling:

• Our dirt computer: 8×8 grid = 64 holes = 8 bytes
• Early computer (1950): 1,024 bytes (1 KB)
• PC (1980): 64,000 bytes (64 KB)
• PC (1990): 4,000,000 bytes (4 MB)
• PC (2000): 256,000,000 bytes (256 MB)
• PC (2010): 4,000,000,000 bytes (4 GB)
• PC (2024): 32,000,000,000 bytes (32 GB)

That's 32 billion holes with rocks. Except they're not holes—they're capacitors, each holding a tiny electrical charge for a few milliseconds before needing refreshing.

CPU scaling:

Your dirt computer has:

• 2 registers (A and B)
• 1 ALU (you, following rules)
• 1 control unit (you, reading instructions)

A modern CPU has:

• 100+ registers
• Dozens of ALUs (can do multiple operations simultaneously)
• Complex control unit with:
  • Branch prediction (guessing which instruction comes next)
  • Out-of-order execution (doing instructions in non-sequential order for speed)
  • Speculative execution (starting operations before knowing if they're needed)
  • Cache management (keeping frequently-used data close)

But at the bottom, it's still:

• Read bits
• Route through gates
• Write bits

Part 12: The Software Layer - Programming Our Dirt Computer

Let's write a real program for our dirt computer.

Problem: Find the largest of three numbers

Data:

• Address 10: number A = 7 (00000111)
• Address 11: number B = 12 (00001100)
• Address 12: number C = 5 (00000101)
• Address 13: result (empty)

Program (in our instruction set):

Row 0:  LOAD A from address 10 to register R1
Row 1:  LOAD B from address 11 to register R2
Row 2:  COMPARE R1 and R2  (sets a flag: is R1 > R2?)
Row 3:  JUMP-IF-GREATER to row 6
Row 4:  COPY R2 to R1  (R2 was bigger, so make R1=R2)
Row 5:  (fall through to next)
Row 6:  LOAD C from address 12 to R2
Row 7:  COMPARE R1 and R2
Row 8:  JUMP-IF-GREATER to row 11
Row 9:  COPY R2 to R1  (R2 was bigger)
Row 10: (fall through)
Row 11: STORE R1 to address 13
Row 12: HALT

Execution:

Load 7 into R1, 12 into R2. Compare: 12 is bigger, so copy 12 to R1. Load 5 into R2. Compare: 12 is still bigger. Store 12 to result. Done.

Result: 12 ✓

This is programming. You've written an algorithm (find maximum) in machine code (our instruction set).

Higher-Level Languages

Real programmers don't write machine code (patterns of rocks/bits). They write in higher-level languages:

In Python:

numbers = [7, 12, 5]
result = max(numbers)

In C:

int a=7, b=12, c=5;
int max = a;
if (b > max) max = b;
if (c > max) max = c;

In Assembly (closer to machine code):

LOAD R1, [10]
LOAD R2, [11]
CMP R1, R2
JG skip1
MOV R1, R2
skip1:
LOAD R2, [12]
...

Each level is translated to the level below:

• Python → compiled to bytecode → interpreted as machine code
• C → compiled to assembly → assembled to machine code
• Assembly → assembled directly to machine code

Machine code = patterns of bits = patterns of rocks in our dirt computer.

Part 13: Complexity from Simplicity

Here's the profound thing: every program ever written, every website, every video game, every AI model, is ultimately just rocks in holes (or charges in capacitors, but same principle).

Your web browser:

• Millions of lines of code
• Translates to billions of machine instructions
• Each instruction: move bits, add bits, compare bits, jump to different instruction
• Each bit: one capacitor charged or not (one rock in hole or not)

ChatGPT (the AI you might be using to read this):

• 175 billion parameters (numbers)
• Each parameter: 16 or 32 bits
• Trillions of bits total
• All stored as charges in memory
• Processing: matrix multiplication = lots of multiply-adds = lots of XOR and AND gates
• Same gates we built with rocks

The universe's complexity is built from simple rules applied at scale.

• Physics: simple laws (F=ma, Maxwell's equations) → complex phenomena (weather, galaxies)
• Biology: simple rules (DNA replication, natural selection) → complex life
• Computation: simple operations (AND, OR, NOT, shift) → complex software

Our dirt computer demonstrates this perfectly:

• Simple: rock or no rock, yes or no
• Combined: 8 holes = 256 possibilities
• Organized: procedures for add, multiply, compare
• Programmed: sequences of instructions
• Result: can solve any computable problem (given enough holes and time)

Part 14: The Limits - What Our Dirt Computer Can't Do Well

Despite the theoretical power, our dirt computer has severe practical limits:

Speed:

• Placing rocks by hand: 1 operation/minute
• Modern CPU: 10 billion operations/second
• Speed difference: 600 billion times slower

Reliability:

• One misplaced rock = wrong answer
• No error correction
• Weather destroys data

Scale:

• To match 32 GB of RAM: need 256 billion holes
• That's a field of holes stretching miles
• Impractical to build, impossible to maintain

But the principles are identical. Silicon just lets us do it:

• Faster (electrons vs hands)
• Smaller (nanometers vs centimeters)
• More reliably (error correction, redundancy)

Part 15: The Philosophical Point - Substrate Independence

The most important lesson: computation doesn't care about the physical substrate.

The same program that adds 5+3 works on:

• Rocks in dirt holes
• Gears in Babbage's engine
• Relays in Harvard Mark I
• Tubes in ENIAC
• Transistors in modern CPUs
• Photons in optical computers
• Ions in quantum computers
• (Hypothetically) neurons in brains

The information is the thing. The physics just carries it.

This is why:

• Software can run on any compatible hardware
• Virtual machines work (simulating one computer on another)
• Emulators work (running old game consoles on modern PCs)
• Your code doesn't need to know if it's on Intel or AMD or ARM

The patterns matter, not the medium.

In our dirt computer:

• Pattern: rock in position 2 and position 0
• Meaning: the number 5
• Medium: rocks and dirt

In a real computer:

• Pattern: voltage in bit 2 and bit 0
• Meaning: the number 5
• Medium: transistors and silicon

Same pattern, same meaning, different medium.

Part 16: Building It For Real - A Practical Guide

If you actually wanted to build this (for education or demonstration):

Materials:

• Cardboard sheet (2 feet × 2 feet)
• Ruler and marker
• Small stones/pebbles (white and dark, for visibility)
• Egg carton or ice cube tray (pre-made holes)

Construction:

1. Memory: Use egg carton (8×8 grid = 64 holes = 8 bytes)
2. Registers: Three separate 8-hole rows on cardboard
3. Program memory: Another section of 8 rows
4. Instruction pointer: One 8-hole row
5. Carry flag: One hole (for tracking addition carries)

Operation:

1. Load a program (place rocks in program memory rows according to instruction encoding)
2. Set program counter to 0
3. Execute fetch-decode-execute cycle by hand:
   • Fetch: look at program memory row indicated by PC
   • Decode: determine what instruction those rocks represent
   • Execute: follow the procedure (add, load, store, etc.)
   • Increment: move PC to next row
4. Continue until HALT instruction

Example programs to try:

• Add two numbers
• Find maximum of three numbers
• Count from 0 to 10
• Multiply two numbers
• Check if number is even (look at bit 0)

Educational value:

Students can:

• See every bit (rock) clearly
• Trace execution step-by-step
• Understand fetch-decode-execute
• Grasp stored-program concept
• Build intuition for how software becomes hardware operations

This is exactly what early computer scientists did with relays and switches to teach themselves.

Conclusion: From Dirt to Silicon, Same Principles

We've journeyed from a single hole in the dirt to a complete programmable computer. Along the way, we discovered:

1. Information is physical: Bits need physical states (rock/no rock)
2. Storage is addressing: Grid of holes with numbered locations = RAM
3. Computation is procedure: Following rules transforms inputs to outputs
4. Logic is composition: AND, OR, XOR combine to make adders, multipliers
5. Programs are data: Instructions stored same way as numbers
6. Control is sequential: Fetch, decode, execute, repeat
7. Complexity emerges from scale: Simple operations × billions = powerful computation
8. Substrate independence: Same algorithms work in dirt, gears, transistors

Modern computers are built on these exact principles, just implemented with different physics:

• Holes → transistors
• Rocks → electrical charges
• Your hands → control circuits
• Your brain reading instructions → instruction decoder
• Your following procedures → hardwired logic gates

The miracle of computation isn't in the complexity of the parts—it's in the power of simple operations repeated at massive scale.

A single transistor is simple: on or off. But 50 billion of them, organized into adders, multipliers, memory controllers, all synchronized to a 5 GHz clock, executing billions of instructions per second, can:

• Render a photorealistic game world
• Simulate weather patterns
• Train an AI to write poetry
• Stream video from across the planet

All built on the foundation we just laid: rocks in holes, yes or no, 1 or 0, the universal language of information.

If you hate squiggles, stop here.
No cliff notes, no next section—this is the end of the road for comfort. What comes next isn’t meant for casual reading. It’s the proof spine beneath everything built so far, and it’s dense enough to push most people out of the room.

This chapter doesn’t explain how anything works; it only shows that it does. Every line of math that follows exists for one purpose: to hold up the claim that the universe does not lose information, even when swallowed by its own gravity.

The setup is simple but ruthless. We start from the smallest act of division—carving a distinction, creating a bit—and track its energy imprint up through the hardest boundary physics allows: the event horizon. If the conservation laws hold there, they hold everywhere. If they fail there, every byte you’ve ever trusted collapses with them.

So this is the proving ground. And no, “infinite data” is not a gadget, not a trick of compression or storage—it’s a property that only becomes true at the black hole limit. Beyond that threshold, energy and information blur into one conserved field. That’s where our rules stop making sense and start being rewritten.

If equations make your eyes ache, step away now. This isn’t for decoding; it’s for anchoring. Past this point, it’s pure structure—Noether, Pauli, curvature, and the binary pulse of symmetry itself. Once you cross the line, you can’t skip ahead, because there is no “ahead” without this.

This is where the universe keeps its receipts.

So, if you understand computing better by what you read, wonderful.  Scroll down real fast and just sample this mess to see if it's your ballywick... it's dense as hell so you will  not come away feeling better about spending all that money for college.

If too dense... I hope you enjoyed the read.  If you continue, You've been warned.

Binary Energy Dynamics: A Thermodynamic Foundation for Information Theory

Part 1: The Fundamental Violation – Digging the Hole

When we dug that first hole in the dirt to store a bit, we committed a subtle violence against the universe. We didn't just create a storage location — we performed work against the gravitational and electromagnetic potentials of the system. In doing so, we invested energy into that bit of information. Let’s quantify this precisely.

 

The Energy of Hole Creation: Consider excavating a cylindrical hole of radius $r = 5,$cm and depth $h = 10,$cm in soil with density $\rho = 1500,\text{kg/m}^3$.

• Volume of excavated material:
  $V=\pi r^{2}h=\pi (0.05{)}^2(0.10)=7.85\times {10}^{-4}{\text{m}}^3\mathrm{.}$

• Mass of excavated dirt:
  $m=\rho V=1500\times 7.85\times {10}^{-4}=1.18\text{ kg}\mathrm{.}$

• Gravitational potential energy change: The center of mass of the excavated dirt was initially at depth $h/2 = 5,$cm. Lifting it to the surface requires work:
  $E_{\text{grav}}=mg(h\mathrm{/}2)=1.18\times 9.8\times 0.05=0.578\text{ J}\mathrm{.}$

But this is only the beginning. We've also:

• Broken molecular bonds in the soil structure (cohesive energy)

• Compressed the surrounding soil (elastic deformation energy)

• Increased the surface area of the soil (surface energy)

• Created an entropy gradient (an ordered hole vs. the disordered surroundings)

Surface Energy Contribution: The new surface area created by the hole (walls and bottom) is:
$A=2\pi rh+\pi r^2=2\pi (0.05)(0.10)+\pi (0.05{)}^2=0.0393{\text{m}}^2\mathrm{.}$
For soil with an effective surface energy (interfacial tension with air) of about $\gamma \approx 0.1~\text{J/m}^2$, the surface energy added is:
$E_{\text{surf}}=\gamma A=0.1\times 0.0393=0.00393\text{ J}\mathrm{.}$

 

Total Energy Investment per Bit: Summing gravitational, surface, and other contributions (cohesive, compressive), the energetic cost of creating one bit of storage in our dirt computer is on the order of:
$E_{\text{bit (creation)}}\approx E_{\text{grav}}+E_{\text{surf}}+E_{\text{cohesive}}\sim 0.6\text{ J}\mathrm{.}$

 

This $\sim 0.6$ joule is now stored in the configuration of the system — the hole itself is a lower-entropy, higher-free-energy state relative to flat ground. We have injected energy to create a bit. By doing so, we have broken a symmetry in the system, and as Emmy Noether taught us, broken symmetry has consequences.

Part 2: Noether's Theorem and the Conservation of Bit Energy

Emmy Noether's first theorem states that every differentiable symmetry of the action of a physical system corresponds to a conservation law. Specifically:

• Time-translation symmetry → Conservation of energy

• Space-translation symmetry → Conservation of momentum

• Rotational symmetry → Conservation of angular momentum

When we dig a hole, we break the translational symmetry of the ground’s surface (it’s no longer uniform and level). The Hamiltonian of the system is no longer invariant under vertical translation of the soil. By Noether's theorem, this symmetry breaking must manifest as some conserved quantity — in this case, energy stored in the configuration.

 

A Field Theory View: Let $\phi(x,y)$ represent the height of the ground surface above some reference plane. Initially, before the hole, we have a symmetric, flat ground state
${\varphi }_0(x,y)=0\text{(flat ground)}\mathrm{.}$
After digging a hole at position $(x_0, y_0)$ of depth $h$, the surface profile becomes:

$$\varphi (x,y)=\{\begin{array}{ll}-h,& \text{if }(x-x_0{)}^2+(y-y_0{)}^2<r^2,\\ 0,& \text{otherwise.}\end{array}$$

We can describe the soil's gravitational potential energy with a Lagrangian density (treating the soil as a continuous medium of density $\rho$):

$$\mathcal{L}=\frac{1}{2}\rho [(\mathrm{\nabla }\varphi {)}^2+2g\varphi ],$$

where the first term $(\nabla\phi)^2$ represents elastic (deformation) energy from creating a slope at the hole’s edge, and the second term $2g,\phi$ (note $\phi$ is negative in the hole) represents the gravitational potential energy density. The action is $S = \int \mathcal{L}, d^3x$.

 

From this Lagrangian, we can derive the energy-momentum tensor. The time-time component $T^{00}$ gives the energy density:

$$T^{00}=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_0\varphi )}{\mathrm{\partial }}^0\varphi -g^{00}\mathcal{L}=\frac{1}{2}\rho [(\mathrm{\nabla }\varphi {)}^2-2g\varphi ]\mathrm{.}$$

Inside the hole, $\phi = -h$ (a constant), so $(\nabla\phi)^2 = 0$. Thus in the hole:

$$T_{\text{hole}}^{00}=-\rho g\varphi =\rho gh,$$

since $\phi$ is negative. This $T^{00} = \rho g h$ is precisely the energy density stored in the bit (per unit volume of hole). Integrating over the hole’s volume $V$ gives the energy:

$$E_{\text{field}}={\int }_{\text{hole}}{T}^{00}dV={\int }_{\text{hole}}\rho ghdV=\rho ghV=1500\times 9.8\times 0.10\times 7.85\times {10}^{-4}=0.578\text{ J}\mathrm{.}$$

This matches our earlier calculation of the work done to dig the hole (gravity + surface energy, etc. ~0.58 J). Noether’s theorem tells us that because we broke a symmetry (uniform height), energy had to be expended and it remains stored in the new configuration. The bit (the hole) has an energy cost that cannot just vanish.

Part 3: The Rock as an Excitation – From Vacuum to State

In quantum field theory, particles are excitations of fields above a ground state. By analogy, in our dirt computer:

• An empty hole = the ground state (vacuum) of a bit-field.

• A rock in the hole = an excited state of that bit-field.

Here's the key: the ground state is not zero energy. Even an empty hole carries energy $E_0 \approx 0.6$ J from its creation. Now consider placing a rock (the "1" state, whereas empty hole is "0"):

 

Rock properties: say the rock has mass $m_r = 0.1~\text{kg}$ (100 grams, a small stone). We drop it into the hole from the rim (height $h_r = 10~\text{cm}$ above the bottom).

• Kinetic energy on impact: $E_k = m_r g h_r = 0.1 \times 9.8 \times 0.10 = 0.098~\text{J}$. This energy mostly dissipates as heat and sound when the rock hits bottom (increasing entropy of the surroundings).

• Change in gravitational potential energy: The rock going from the rim to the hole bottom lowers the potential energy of the system by $\Delta E = -m_r g h_r = -0.098~\text{J}$. In other words, placing the rock gives up 0.098 J of gravitational energy to the environment.

Wait — placing the rock lowers the total energy of the system? That seems counterintuitive: we usually think adding a rock should add energy. The resolution lies in carefully defining the reference state:

• Empty hole energy: $E_{\text{empty}} \approx +0.6$ J (this is energy stored as a strained configuration of soil, like a wound spring).

• Rock in hole energy: $E_{\text{rock}} = 0.6~\text{J} + (\Delta E)$, where $\Delta E$ includes the rock’s potential energy change. Here $\Delta E = -0.098$ J, so $E_{\text{rock}} \approx 0.6 - 0.098 = 0.502$ J.

Thus, the filled hole (bit = 1) has slightly less total energy than the empty hole (bit = 0) in this system. The rock partly "relaxes" the deformation: it fills some volume, reducing the depth of empty space and lowering the gravitational energy stored. The 0.098 J went into heating the environment when the rock dropped. Importantly, the information of whether the rock is there or not is now encoded in the system, and the energy difference between the 0 and 1 states is ~0.1 J. We must account for all energy: some is stored in the bit states, some was shed as heat.

 

Information-Theoretic Energy: By placing the rock, we've written one bit of information (distinguishing an empty hole from a filled hole). According to Landauer's principle, erasing one bit of information in a system at temperature $T$ requires a minimum energy dissipation of
$E_{\min }=k_{B}T\ln 2,$
where $k_B$ is Boltzmann's constant. At room temperature $T \approx 300$ K, this is
$E_{\min }\approx 1.38\times {10}^{-23}\times 300\times \ln 2\approx 2.8\times {10}^{-21}\text{ J}\mathrm{.}$

 

This amount, $\sim 10^{-21}$ J, is astronomically smaller (by 21 orders of magnitude) than the $\sim 0.1$ J energy difference we see in our crude macroscopic bit. Why the discrepancy? Because our dirt computer operates in the classical, irreversible regime. We are using macroscopic objects (dirt, rocks), not microscopic near-equilibrium reversible logic. Landauer's limit is only approached by reversible operations on thermalized microscopic bits. Our operations (digging, dropping rocks) are irreversible and involve enormous energy overhead (friction, plastic deformation, etc.). In essence, we pay a heavy price above the thermodynamic minimum.

 

The Second Law at Work: Notice, when the rock was placed, the system shed energy as heat. This increased the entropy of the environment. If we later remove the rock (erasing the bit to 0), we'd have to put at least that much energy back in (and again dissipate some as heat elsewhere). The universe keeps an accounting of information via energy flows: you can't flip bits for free.

Part 4: Shannon Entropy vs. Thermodynamic Entropy

Claude Shannon defined information entropy to quantify uncertainty in bits. For a message with probabilities ${p_i}$ for states $i$, the Shannon entropy is:
$H=-{\sum }_i{p}_i{\log }_2{p}_i\text{(in bits).}$

 

Meanwhile, Ludwig Boltzmann and J. Willard Gibbs defined thermodynamic entropy as:
$S=k_B\ln \mathrm{\Omega },$
where $\Omega$ is the number of microstates consistent with the macrostate (or more generally $S = -k_B \sum_j P_j \ln P_j$ for a system in states $j$ with probabilities $P_j$). This is measured in joules per kelvin.

 

These entropies are conceptually related (both involve logarithms of possibilities), but not identical. Shannon's $H$ measures our information uncertainty (lack of information), whereas thermodynamic $S$ measures physical disorder. Linking them is at the heart of information theory’s physical foundation.

 

Maxwell's Demon Bridge: Maxwell's demon is a famous thought experiment where a demon seemingly violates the second law by using information about individual molecules to reduce entropy (letting fast molecules go one way, slow another, thus creating a temperature difference for free). The paradox is resolved by realizing the demon's memory fills up with information; to reset the demon (erase information), a price in entropy must be paid. Landauer (1961) and Bennett (1982) showed that erasing one bit of the demon’s memory increases entropy of the environment by at least $\Delta S_{\text{env}} = k_B \ln 2$. Any attempted decrease of entropy by sorting molecules is offset by the entropy produced when the demon’s memory is cleared. In terms of energy: a heat $Q = T \Delta S_{\text{env}} \ge k_B T \ln 2$ is released to the environment upon erasure, safeguarding the second law.

 

Application to Our Dirt Computer: Every time we write a bit (dig a hole, or drop a rock, or fill a hole back in), we perform an irreversible operation that increases the entropy of the environment. The theoretical minimum energy cost (at 300 K) would be on the order of $10^{-21}$ J per bit change. In reality, our operations cost $10^{-1}$ to $10^0$ J — over 20 orders of magnitude more. The ratio:
$\frac{E_{\text{actual (per bit)}}}{E_{\text{Landauer}}}\sim \frac{0.6\text{ J}}{2.8\times {10}^{-21}\text{ J}}\sim 2\times {10}^{20}\mathrm{.}$
This factor of $10^{20}$–$10^{21}$ is a measure of how far our primitive bit is from the thermodynamic ideal.

 

Modern CMOS transistors in computers are far more efficient but still dissipate about $10^{-15}$ J per logic operation — roughly $10^6$ times the Landauer limit at room temperature. Progress in computing (Koomey's law) has been steadily lowering energy per operation, but even state-of-the-art logic is orders of magnitude above the fundamental floor. The gap represents room for innovation but also a reminder: to ultimately approach Landauer’s bound, we need reversible computing or quantum computing (more on that in Part 9).

Part 5: The Pauli Exclusion Principle and Information Permanence

Pauli's principle as a guarantee that information cannot be destroyed. Let’s examine that idea in the context of physics fundamentals.

 

Pauli Exclusion Principle: No two identical fermions (particles with half-integer spin, like electrons, protons, neutrons) can occupy the same quantum state simultaneously. Mathematically, the total wavefunction for a system of $N$ identical fermions is antisymmetric under exchange of any two:
$\mathrm{\Psi }(\dots ,x_i,\dots ,x_j,\dots )=-\mathrm{\Psi }(\dots ,x_j,\dots ,x_i,\dots )\mathrm{.}$
This antisymmetry leads to a prohibition: if two fermions tried to be in the exact same state, the wavefunction would equal its negative, implying $\Psi=0$ (no allowed state).

 

Consequence: Each fermion in a bound system occupies a unique state (quantum numbers). This is why electron shells in atoms fill up sequentially with distinct quantum numbers, and why matter occupies space (electrons resist being squeezed into the same state, creating electron degeneracy pressure — which props up white dwarf stars and neutron stars). Pauli’s principle ensures a kind of information labeling for fermions: you can distinguish electrons by their quantum state, which means a complex system has many distinguishable configurations (microstates). In other words, quantum statistics guarantee a rich phase space of states — a requirement for robust information storage in matter. If fermions could all collapse into one state, an atom’s electrons would all drop to the lowest orbital and all chemistry (which is information-rich: think of DNA) would collapse. Structure (hence information) in matter exists in part because Pauli’s principle forbids collapse into sameness.

 

Quantum Unitarity – Conservation of Information: The time evolution of an isolated quantum system is governed by the Schrödinger equation, $i\hbar,\partial_t|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle$. Its solution is $|\Psi(t)\rangle = \hat{U}(t),|\Psi(0)\rangle$ where $\hat{U}(t) = e^{-i \hat{H} t/\hbar}$ is a unitary operator. Unitary evolution means that the system’s state transitions are reversible in principle — no information is lost, it is just transformed. In quantum mechanics, the overlap between states is preserved: if two states start orthogonal (distinguishable), they remain orthogonal at all times. More formally, $\langle \Psi_1(t)|\Psi_2(t)\rangle = \langle \Psi_1(0)|\Psi_2(0)\rangle$. If $\Psi_2$ is initially an orthogonal state to $\Psi_1$, it stays orthogonal (and distinguishable) for all $t$. This property means quantum information cannot be destroyed under the normal unitary evolution governed by a Hermitian Hamiltonian. The information about the initial state is always encoded in the current state, just perhaps in a very scrambled or entangled way.

 

Thus, fundamental physics (quantum theory) provides strong conservation of information — often referred to as “information is not lost, only hidden.” Where then did the notion of possible information destruction arise? Enter the black hole information paradox.

 

The Black Hole Information Paradox: In 1974, Stephen Hawking showed that black holes are not completely black — they radiate due to quantum effects near the event horizon. Hawking radiation is thermal (blackbody spectrum). If you form a black hole from a perfectly pure quantum state (say a bunch of particles in a specific configuration) and then it evaporates away into featureless thermal photons, it appears that a pure state evolved into a mixed (thermal) state. That is forbidden by quantum mechanics (unitarity) – it’s like running Schrödinger’s equation and getting a non-unitary outcome. Hawking’s semi-classical calculation suggested information that fell into a black hole is irretrievably lost once the hole evaporates, violating unitarity. This shocked the physics community: something had to give — either quantum mechanics is incomplete in this context, or Hawking’s calculation missed hidden correlations in the radiation that actually carry the information out.

 

Most physicists now believe that information is not truly lost in black holes; rather, it is somehow encoded in the Hawking radiation (or remains in a stable remnant, or escapes to a baby universe — various proposals) such that unitarity is preserved. But exactly how the information escapes is still debated and is a deep question bridging quantum theory, gravity, and thermodynamics.

 

Cassimir Coupled Tachyonic Venting Hypothesis: I've posited an explanation dubbed "Casimir-Coupled Tachyonic Venting (CCTV)" as an alternative mechanism for Hawking radiation that might carry information out more explicitly. Let’s unpack this alongside the standard picture:

• Standard Hawking picture: Quantum vacuum fluctuations near the event horizon momentarily produce particle–antiparticle pairs. Normally they annihilate promptly, but if one falls into the black hole while the other escapes, the escaping particle becomes real Hawking radiation and the infalling partner has negative energy (relative to the outside) which reduces the black hole's mass. Over enormous time, the black hole loses mass and evaporates. This process as originally conceived produces thermal, information-less radiation, because the particles are created entangled in a way that the outside radiation is in a mixed state when tracing out the interior partner. The question remains: how does the outgoing particle acquire any information about the inside? Hawking's original answer: it doesn't — information is lost. Modern twist: perhaps subtle long-range correlations or quantum gravity effects cause the radiation to be not exactly thermal and encode information (resolving the paradox), but the mechanism is not fully understood in orthodox calculations.

• CCTV alternative: Think of the event horizon region as a sort of dynamic Casimir cavity. The Casimir effect in flat spacetime: two parallel conducting plates in vacuum will experience an attractive force because the boundary conditions disallow certain electromagnetic modes between the plates, leading to a lower vacuum energy density inside compared to outside. Effectively, vacuum fluctuations are altered by boundaries, creating a pressure difference. Near a black hole horizon, spacetime curvature could act like a moving boundary. The idea is that the extreme gravity provides an effective separation of modes that might mimic a Casimir-like region. If the horizon's presence suppresses or modifies vacuum modes, the difference in vacuum energy can drive particle creation.

   

  Furthermore, if there are any faster-than-light (tachyonic) modes or instabilities present (perhaps from quantum gravity effects or exotic fields), these might couple the interior to the exterior. "Tachyonic venting" suggests that information from inside the black hole could tunnel out via modes that don't obey the normal speed limit — effectively leaking information without violating energy conservation. In normal physics, tachyons are hypothetical particles with imaginary mass that always move faster than light. They are generally considered unphysical because they break causality. However, in a curved spacetime, what appears as a tachyonic or spacelike propagation in one frame might be allowed if the spacetime itself is behaving in a certain way (for example, just outside a horizon, the coordinate "time" and "space" can swap roles in certain equations, potentially allowing what looks like superluminal outflow when mapped to an outside observer’s coordinates).

   

  In short, CCTV posits that quantum vacuum fluctuations + horizon boundary conditions + exotic (tachyon-like) couplings can vent not just energy (as Hawking radiation) but also imprint information from behind the horizon onto that radiation. The Hawking pair-production picture then is replaced or supplemented by a mechanism where the black hole's internal degrees of freedom are quantum-tunneling their information out via a conduit created by these fluctuating fields at the horizon (imagine the horizon itself as an information membrane with quantum-scale pores).

   

  Is there a theoretical basis? We can attempt a rough analogy: The energy density between two Casimir plates separated by distance $a$ of area $A$ is $E_{\text{Casimir}} = -\frac{\pi^2 \hbar c}{720 a^3} A$ (the negative sign indicates a lower energy than free vacuum). Now, near a Schwarzschild black hole, the proper distance to the horizon for a static observer grows without bound (as $r \to r_s = 2GM/c^2$, the factor $g_{rr} = (1- r_s/r)^{-1}$ diverges). In a naive sense, one could imagine an effective $a_{\text{eff}}$ between the black hole and infinity that tends to zero as the horizon is approached (since from infinity's view, the region just outside the horizon is extremely redshifted and "stretched"). If $a_{\text{eff}}$ is extremely tiny (on the order of Planck length $\ell_P \sim 10^{-35}$ m) near the horizon, the Casimir energy density could be enormous (since it scales as $1/a^3$). This energy might fuel particle creation in a way that is not random: it could be influenced by the state of the fields just inside the horizon (hence carrying information out). Tachyonic modes come into play if some field has an instability (negative mass-squared) in the curved spacetime near the horizon, leading to a condensation or leakage that is effectively superluminal from the outside perspective. While standard physics does not include such tachyonic fields in black hole backgrounds, some speculative quantum gravity approaches or analogs (like considering horizon analogues in condensed matter) hint at possible mechanisms for information leakage that don't show up in a simple Hawking calculation.

In summary, energy cannot be destroyed, and perhaps neither can information. If the bit is truly fundamental, the universe must have bookkeeping for it. This hypothesis is a colorful attempt at explaining the bookkeeping mechanism for black holes. Whether or not tachyonic venting is how nature does it, the core principle it’s addressing is valid: the need to preserve information to avoid violating quantum mechanics and Noether's theorem. The jury is still out on the exact mechanism, but consensus leans towards information conservation — the challenge is to understand how the information is encoded in what a distant observer sees.

Part 6: The Holographic Principle and the Energy of Bits

The Holographic Principle (proposed by 't Hooft and Susskind in the 1990s) is another deep idea connecting information, physics, and geometry. It suggests that the maximum amount of information (or entropy) that can be contained in a volume of space is proportional not to the volume, but to the surface area enclosing that volume (measured in Planck units). In formula form:
$S_{\max }=\frac{k_{B}A}{4{\mathrm{\ell }}_P^2},$
where $A$ is the surface area and $\ell_P = \sqrt{\hbar G/c^3} \approx 1.6\times 10^{-35}$ m is the Planck length. This is actually the Bekenstein–Hawking entropy formula for a black hole of area $A$. If you have a spherical region of radius $R$, the maximum entropy (information) it can contain is on the order of the entropy of a black hole of that size:
$S_{\max }=\frac{k_B(4\pi R^2)}{4{\mathrm{\ell }}_P^2}=\frac{\pi k_B{R}^2}{{\mathrm{\ell }}_P^2}\mathrm{.}$

 

In terms of bits (using $\ln 2$ to convert entropy to bits):
$N_{\text{bits,max}}=\frac{S_{\max }}{k_B\ln 2}=\frac{\pi R^2}{{\mathrm{\ell }}_P^2\ln 2}\mathrm{.}$

 

For a 1-meter radius sphere, plugging in numbers:
$N_{\text{bits,max}}\approx \frac{3.1416\times (1\text{ m}{)}^2}{(1.6\times {10}^{-35}\text{ m}{)}^2\times 0.693}\approx 4.3\times {10}^{70}\text{ bits}\mathrm{.}$
This is an astronomically large information capacity in a seemingly modest volume! It implies that if you were to use every available degree of freedom up to the Planck scale, a one-meter sphere could encode about $10^{70}$ bits of information — but doing so would essentially collapse it into a black hole of that radius.

 

Energy per Bit in a Black Hole: Now, consider a black hole itself as the storage device. Suppose we have a black hole of mass $M$ (energy $E=Mc^2$) and radius $R = r_s = 2GM/c^2$. It has entropy $S_{\text{BH}} = k_B \frac{A}{4\ell_P^2}$ and number of bits $N_{\text{bits}} \approx \frac{A}{4\ell_P^2 \ln 2}$. The average energy per bit in the black hole is then:
$\frac{E}{N_{\text{bits}}}=\frac{M{c}^2}{S_{\text{BH}}\mathrm{/}(k_B\ln 2)}=\frac{M{c}^2{k}_B\ln 2}{S_{\text{BH}}}\mathrm{.}$
Using $S_{\text{BH}} = \frac{k_B c^3 A}{4 G \hbar}$ (equivalent form of the Bekenstein-Hawking formula) and $A=4\pi r_s^2 = 4\pi (2GM/c^2)^2$, after some algebra we find:
$\frac{E}{N_{\text{bits}}}=\frac{\mathrm{\hslash }{c}^3\ln 2}{4\pi GM}\mathrm{.}$

 

For a solar-mass black hole ($M \approx 2 \times 10^{30}$ kg), this energy per bit comes out to on the order of $10^{-30}$ J/bit. This is astonishingly low — about $10^{9}$ times smaller than the Landauer limit at room temperature ($\sim 10^{-21}$ J). Black holes (if they indeed saturate the holographic bound) store information with incredible energy efficiency. However, retrieving that information is an entirely different matter (you'd have to wait ~$10^{67}$ years for Hawking evaporation of a stellar black hole, and even then decode subtle correlations in the radiation).

 

The takeaway is that the ultimate limit of memory and computation may lie in gravitational systems (as some have speculated, the most efficient computer is one that almost collapses into a black hole) or quantum/holographic systems. But for now, our technology is nowhere near these limits.

Part 7: Binary Energy Dynamics – A Unified Framework

Let’s synthesize the insights so far into a set of principles for Binary Energy Dynamics, connecting information theory and thermodynamics:

• Axiom 1: Energy–Information Equivalence. Every bit of information has an associated energy cost to create, move, or erase. At minimum, $E_{\min} = k_B T \ln 2$ per bit must be dissipated to erase a bit at temperature $T$ (Landauer’s limit). Conversely, to reliably create or separate a bit of information, some energy must be invested (to overcome thermal noise and maintain distinct states).

• Axiom 2: Bit Creation Energy. Creating a new binary storage location (a “bit container”) requires work against some physical potential. In our dirt computer, work was done against gravity, cohesion, etc., to make a hole. In a transistor, work is done to implant dopants, fabricate structures, etc., and to establish electric fields. This creation energy is stored in the physical configuration. We can call it the configuration energy $E_{\text{config}}$. For the dirt bit, $E_{\text{config}} \approx 0.6$ J per hole. In a SRAM or capacitor bit, it's the energy to charge up the capacitor or flip the cell plus the manufacturing cost amortized per bit.

• Axiom 3: State Energy Difference. There must be an energy difference between the two logical states 0 and 1, i.e. $|\Delta E| = |E_1 - E_0| \gg k_B T$ for stable information storage (so thermal fluctuations won’t randomize the bit). In our dirt bit, the “1” state (rock in hole) had about 0.1 J less energy than the “0” state (empty hole). $0.1~\text{J} \gg k_B (300~\text{K}) \approx 4\times 10^{-21}~\text{J}$, so the bit is extremely stable against flipping from thermal noise! In a computer chip, a bit might have an energy difference on the order of $10^{-16}$ J (100 meV) which is still about $10^5$ times $k_B T$ at room temp — enough to be stable for a while (until leakage or external perturbations disturb it).

• Axiom 4: Noether Conservation (Accounting of Energy). The total energy, including that stored in bits plus that dissipated as heat to the environment, is conserved. When symmetry is broken to record information (like breaking translational symmetry by digging a hole, or lowering entropy by creating an ordered bit pattern), the energy associated with that must be stored in fields or released to the environment. When bits are erased or allowed to decay, that energy is released. Noether’s theorem underpins this: if a process appears to lower the energy of the system by removing a distinguishing feature (restoring symmetry), the energy must go somewhere (often as heat or radiation).

• Axiom 5: State Distinguishability (Pauli and beyond). To reliably represent a bit, the two states must be represented by two distinguishable physical states. In quantum terms, they should be orthogonal quantum states (or at least macroscopically distinguishable mixtures with negligible overlap). Pauli’s exclusion ensures that fermionic matter provides many distinct states for particles, enabling stable, distinct configurations for encoding information (you can't pile everything into one state and lose track of distinctions). More generally, whether using electron charge, magnetic spin, or the presence/absence of a rock, the 0 and 1 must reside in different state configurations that do not spontaneously morph into each other without energy input. This is related to energy barriers between states: e.g., a flash memory bit has two charge states separated by an insulating barrier — to flip the bit, you must apply energy to overcome the barrier.

Using these axioms, we can examine a simple system of multiple bits and see how energy scales with information.

 

The Energy Spectrum of a Byte: Consider 8 bits (one byte). Each bit has a creation energy $E_{\text{config}}$ (for the container) and a state difference $\Delta E$ depending on whether it's a 0 or 1. Suppose all bits are independent and identical for simplicity. The total energy of the byte for a given 8-bit state (0–255 in decimal) would be:
$E_{\text{byte}}(n)=8E_{\text{config}}+(\text{number of 1-bits in }n)\times \mathrm{\Delta }E\mathrm{.}$

 

In our dirt computer example: $E_{\text{config}} \approx 0.6$ J per hole, and $\Delta E \approx -0.098$ J per rock (negative because adding a rock reduces the energy). So:

• State 0 (binary 00000000, no rocks): number of 1-bits = 0.
  $E_{\text{byte}}(0) = 8(0.6) + 0(-0.098) = 4.8$ J.

• State 1 (binary 00000001, one rock): number of 1-bits = 1.
  $E_{\text{byte}}(1) = 4.8 + 1(-0.098) = 4.702$ J.

• ...

• State 255 (binary 11111111, eight rocks): number of 1-bits = 8.
  $E_{\text{byte}}(255) = 4.8 + 8(-0.098) = 4.8 - 0.784 = 4.016$ J.

Interestingly, in the dirt computer, the more 1s (rocks) you have, the lower the total energy (because each rock "heals" a hole a bit). The highest energy state of the byte is actually 00000000 (all holes empty, fully strained system). This is opposite to typical electronic memory, where usually a "1" corresponds to a higher energy state (e.g., charged capacitor). The lesson: the mapping of binary to energy isn't universal; it depends on implementation. What is universal is that there is an energy difference associated with different bit states, and typically the pattern of bits will affect total energy (sometimes only negligibly, sometimes significantly).

 

Voltage Analogy (from NLN framework): In some earlier context, I considered a mapping: assign 0-bit → 1 V and 1-bit → 5 V (just a conceptual mapping of logical 0/1 to two distinct voltage levels). Then the "total voltage" of a byte would be:
$V_{\text{total}}={\sum }_{i=0}^7[b_i\times 5\text{ V}+(1-b_i)\times 1\text{ V}]=8\text{ V}+4\text{ V}\times {\sum }_{i=0}^7{b}_i,$
where $b_i \in {0,1}$ are the bit values. The average voltage per bit for that byte would be $\bar{V} = V_{\text{total}}/8 = 1~\text{V} + 0.5~\text{V} \times (\text{# of 1-bits})$. So basically, in that analogy, each 1-bit contributes an extra 0.5 V above a baseline of 1 V per bit. A byte with more 1s has higher average "energy voltage." For example, 0 (00000000) gives $\bar{V}=1.0$ V; 5 (00000101 has two 1s) gives $\bar{V}=2.0$ V; 255 (all 1s) gives $\bar{V}=5.0$ V.

 

This scheme is analogous to energy if we interpret voltage as proportional to energy (which in circuits it is, since energy in a capacitor is $\frac{1}{2}CV^2$, higher voltage means higher energy for a given C). In an actual digital circuit, a 1-bit often means a charged node (higher energy) and a 0-bit an uncharged node (lower energy). Thus more 1s = more energy stored. Our dirt computer was an oddball where a 1 (rock) lowered energy, but that was because we defined the baseline differently (the energy was mostly in the hole creation).

 

Bottom line: A string of bits has an energy cost that can be thought of as a sum of independent bit energies to first approximation, plus possibly interaction terms if bits influence each other. In most digital systems, those interaction terms are negligible (each bit cell is isolated), but in something like a gravitational system (bits as masses in holes), bits could interact (neighboring holes might collapse into each other if too many rocks are removed, etc.). Binary Energy Dynamics could, in principle, consider such interactions and treat information storage as a many-body physics problem, but that’s beyond our scope here.

Part 8: Number Theory Interlude – Factorization and Energy Patterns

Interestingly, this framework hints at links between patterns in binary representations of numbers and energy-like properties (in another blog post I discussed an NLN factorization method exploiting binary patterns). This strays a bit from physics, but let's briefly muse on how number theory might connect to these ideas.

 

Energy Distribution in Binary Patterns: If we assign an "energy" to a number based on its binary digits (say the count of 1s, or some function of their positions), then different numbers have different energy profiles. For example, primes vs composites:

• Prime numbers (e.g. 7 which is 111 in binary, or 31 which is 11111 in binary) often have dense binary expansions (many 1s) that don't obviously fall into smaller patterns. You might say they have a kind of high "binding energy" — you can't easily break them into smaller factors without a lot of effort, analogous to a tightly bound nucleus requiring energy to fission.

• Composite numbers (e.g. 6 which is 110, or 12 which is 1100, or 21 which is 10101) often have more structured or repetitive patterns in binary, which might correlate with them being factorable. For instance, 21 in binary 10101 has a repeating 10 pattern. One might see that and guess it's $(101)_2 \times (001)_2$ or something (just an analogy). A number like 15 is 1111 in binary, which is clearly structured (it's $(2^4 - 1)$) and indeed $15 = 3 \times 5$. The pattern 1111 suggests a certain symmetry (all bits 1), and in fact $2^n - 1$ (Mersenne numbers) are often composite for many n (though some are prime, the Mersenne primes).

In the NLN method, I talked about shifting and XORing bits to reveal factors. For example, 21 = 10101 in binary. If we rotate (cyclically shift) 10101 by some positions, we might get patterns that hint at factors. A one-bit circular shift of 21 yields 01011 (which is binary for 11 decimal). 11 is not a factor of 21, but we noticed something: if you shift and then truncate/pad in a certain way ("shave 2 bits" in your words), we got 011 which is 3, and indeed 3 is a factor of 21. This seems like a curious trick: by exploiting the symmetry (the repetition of 1 at both ends of 10101), a shift exposed the factor 3. Essentially 10101_2 (binary for 21) contains the pattern of 11_2 (3) when viewed cyclically.

 

This hints at group theory. The binary strings of length $n$ under circular shifts form a cyclic group (isomorphic to $\mathbb{Z}/n\mathbb{Z}$). A number that, when represented in binary on a circle, maps to itself under a certain shift would have a symmetry that could correspond to a factor. For instance, 10101 (21) under a 2-bit shift gives 10101 again (if we wrap around appropriately), indicating a period of 2 in the pattern on a length-5 string. That suggests 5 bits with a period of 2 – which might correlate to a factor. Indeed, 21's factors 3 and 7 in some sense relate to patterns of period 2 or 3 in a length-5 sequence (this is a bit heuristic). Another way: 10101_2 is the binary for $(2^4 + 2^2 + 2^0)$. This is a truncated geometric series. In fact, $21 = 1\cdot(2^4 + 2^2 + 2^0) = \frac{2^5 - 1}{2^2 - 1}$, which reveals $21 = \frac{31}{3}$, so 3 and 7 show up in that identity.

 

The broader point: there is a connection between numeric factors and patterns in different bases or under transformations. Modern factorization algorithms (like Shor's quantum algorithm) use periodicity in modular arithmetic to find factors. This method seems to seek periodicity in the bit representation itself. This is unconventional, but not unreasonable: a number having a factor means its representation in some base might have a small period. For example, if a number $N$ is divisible by 3, the sum of its decimal digits is divisible by 3 (a pattern in base-10 representation). If $N$ is divisible by 15, its last digit in hexadecimal has certain properties, etc. So patterns can indicate factors.

 

XOR and Linear Algebra: We also discussed XOR (bitwise exclusive-or) and linear algebra over $\mathbb{F}_2$. Indeed, working with bits naturally leads to vector spaces over the field of two elements. Some properties of numbers, like parity of number of 1-bits (which is the XOR of all bits), are linear in $\mathbb{F}_2$. If you define an "energy" as a linear function of bits (like your $\bar{V} = 1 + 0.5,(#\text{ of 1s})$), then that energy function is linear over reals but mod 2 it’s just affine. XORing two bit patterns corresponds to adding their indicator vectors in $\mathbb{F}_2^n$. Patterns that are factorable might correspond to linear dependencies or correlations between those vectors.

 

Without diving too deep, the presence of these ideas suggests one more parallel: information processes can be viewed through algebraic structures. The prime factorization problem, for example, is about finding a hidden structure (the pair of factors) from the "flat" representation of a number. This is like extracting two pieces of information (the prime factors) from one piece (the composite) by a non-trivial transformation. There's a thermodynamic analogy: it's easier to multiply (mix information) than to factor (separate information). Multiplying two primes (analogous to a reversible, low-entropy combining operation) is easy, whereas factoring (retrieving the original components from the mixture) is hard — and in computing terms, presumably requires more energy or time. This aligns loosely with the idea that scrambling information (increasing entropy) is easy, but unscrambling (decreasing entropy) is hard and requires work. A black hole, for instance, scrambles information thoroughly (like multiplying two large primes) — retrieving the original message (factors) from the Hawking radiation (product) is believed to be exponentially hard.

 

In a way, the Binary Energy Dynamics story — from rocks and holes to black holes — is highlighting the common theme: putting information in is one thing (costly but doable), getting it out intact is another (and may be fundamentally hard).

Part 9: Thermodynamic Limits of Computation

We’ve touched on Landauer’s principle already. Let’s formalize it and then discuss how we might circumvent dissipation by clever design, bringing us to the edge of what’s physically possible for computers.

 

Landauer’s Principle (rigorous form): Rolf Landauer argued that any logically irreversible operation (like erasing a bit or merging two computation paths) has a fundamental thermodynamic cost. Consider erasing a bit: you have a bit that is equally likely 0 or 1 (Shannon entropy $H=1$ bit, physical entropy $S = k_B \ln 2$ if realized many times). After erasure, the bit is reset to say 0 with certainty ($H=0$, no uncertainty). The entropy of the bit has decreased by $k_B \ln 2$. By the second law, the entropy of the environment must increase by at least that much to compensate. That means at least $\Delta S_{\text{env}} = k_B \ln 2$ must be generated in the surroundings, yielding a heat $Q \ge k_B T \ln 2$ dissipated. Equality is the limit of a reversible erasure done quasi-statically.

 

In formula: $\Delta S_{\text{total}} = \Delta S_{\text{bit}} + \Delta S_{\text{env}} \ge 0$. We have $\Delta S_{\text{bit}} = -k_B\ln 2$ (since we lost one bit of entropy in the information-bearing degrees of freedom). Thus $\Delta S_{\text{env}} \ge k_B \ln 2$, and $Q = T \Delta S_{\text{env}} \ge k_B T \ln 2$.

 

Landauer's principle has been experimentally verified in various systems (e.g., a one-bit Brownian particle memory). It sets a temperature-dependent limit. At room temperature, $\sim 3\times 10^{-21}$ J per bit erasure is the limit; at lower temperatures, the limit drops. At $T=0$, Landauer’s limit would be 0 — but you can never actually reach 0 K or do a computation in a perfectly adiabatic, zero-entropy way unless it’s completely reversible.

 

Reversible Computing (Bennett, 1973): Charles Bennett and others realized that you can, in principle, perform computation with zero dissipation per operation if you avoid irreversible steps. To do this, every logical operation must be reversible (one-to-one mapping of inputs to outputs). For example, a normal NAND gate is irreversible: two input bits produce one output bit, so you can't recover the inputs from the output alone (information is lost every time you use a NAND or AND or OR gate). But you can construct a reversible universal gate, like the Toffoli gate (CCNOT: two control bits and one target bit; the target is flipped if and only if both controls are 1 — this gate is bijective on the 3-bit space). Reversible circuits built from such gates, if run slowly enough (to minimize energy dissipation in wires and capacitances), approach zero heat generation. The catch: they tend to require many extra bits (ancilla bits) to store intermediate results and eventually you have to erase those ancilla bits to reuse them, which again incurs Landauer cost. However, if you never erase and you output all the garbage along with the answer, you technically never paid the entropy piper. In practice, at some point you want to reset your computer or reuse memory, so some erasure is needed. Nonetheless, reversible computing shows that computation doesn’t fundamentally require heat generation; it’s only erasure or irreversibility that does. The big hurdle is that reversible computing is much harder — logically, temporally, and spatially — you need complex circuits and error correction without discarding entropy easily.

 

Quantum Computing and Energy: Quantum computers are essentially reversible computers, as quantum gates are unitary (and thus reversible) operations on qubits. An ideal quantum computer, isolated from its environment, in principle can perform computations with no entropy increase and no mandatory energy dissipation per operation (aside from the energy invested to implement the control fields for gates, which in principle could be done adiabatically). A quantum CPU could be viewed as a collection of two-level systems that we drive coherently. Because it’s all unitary evolution, the entropy of the quantum state is constant (until measurement). However, the real world adds two sources of energy cost:

1. Control energy: To perform gates quickly, you typically need pulses of energy (e.g., microwave pulses for superconducting qubits, laser pulses for ion traps). If done non-adiabatically, these consume energy and can cause some heating. There is a time-energy tradeoff: doing gates faster (to beat decoherence) requires higher-power pulses.

2. Error correction and measurement: Quantum systems are fragile. Error correction involves periodically measuring some ancilla qubits to detect errors. Measurement is an irreversible process — when you measure a qubit, you entangle it with a classical bit (the measurement apparatus) and effectively erase the superposition (except for the recorded outcome). The measurement produces classical information (0 or 1 outcome) and the quantum state collapses. To reset the ancilla qubits for reuse, you have to dissipate the entropy associated with the error syndrome measurements. Moreover, any heat from the measuring device must be removed. Error correction in quantum computing thus has a Landauer cost for each syndrome bit that is reset. Current quantum error correction schemes will have overheads of thousands of physical qubits per logical qubit, which if run at high syndrome frequencies could produce significant heat. That heat is manageable (these systems run in dilution refrigerators at millikelvin temperatures, but the heat is extracted by the fridge at a large energy expense from the power mains).

3. Final readout: The end of a quantum algorithm usually involves measuring the qubits to get a classical answer (e.g., the prime factors in Shor’s algorithm). That measurement again is irreversible and must dissipate at least $k_B T \ln 2$ per qubit of entropy (and usually much more in practice, since readout involves amplifiers, etc.).

In summary, a quantum computer can in principle compute with arbitrarily low energy per operation (by using reversible dynamics), but practical considerations (speed, error correction, readout) will introduce dissipation. The hope is that for some problems (like factoring large numbers), a quantum computer will use exponentially fewer operations than a classical one, thus even if each operation had similar energy cost, the total energy to solution could be far less. Quantum computers might not beat classical Landauer limits per operation, but by doing far fewer operations they potentially expend less energy overall for a task like breaking encryption.

Part 10: The Black Hole – Ultimate Physical Limit and No Information Loss

We have followed the trail from digging a hole in the ground for a bit, all the way to the black hole, which is nature’s ultimate bit container. Now we return to the question of information loss and the second law, armed with our deepened understanding.

 

A black hole appears to be a great destroyer of information — you throw whatever you want into it, and all distinguishing details (quantum numbers aside from mass, charge, angular momentum) seem to be erased from the outside view. The black hole’s Hawking radiation coming out is (at first cut) thermal, carrying seemingly no imprint of what went in. It's as if the black hole is a giant eraser that can delete information bits without a trace. But our journey through physics principles strongly suggests that this cannot be the whole story:

• Energy Conservation (Noether): A black hole certainly conserves energy — whatever energy/mass falls in either stays (increasing the black hole’s mass) or is radiated away as gravitational waves or other emissions during infall or via Hawking radiation over time. There's no mysterious loss of energy. If information were truly destroyed in a fundamental sense, it might imply some non-energy-conserving processes at a microscopic level (since information erasure costs energy). We don't see energy non-conservation; instead, we see a thermodynamic conversion: infalling mass-energy eventually goes to outgoing radiation energy.

• Landauer’s Principle: If a black hole were irreversibly erasing information, the second law demands an entropy price. Indeed, Hawking radiation has huge entropy — a black hole of mass $M$ emits on the order of $S_{\text{BH}} \sim k_B \frac{4\pi G M^2}{\hbar c}$ of entropy by the end of its life, which for large $M$ is enormous. This entropy could be viewed as the environment’s entropy increase due to information "erasure". However, if unitarity holds, that entropy in radiation is not purely new randomness; it encompasses the original information, just highly scrambled. If Hawking radiation is exactly thermal (information-less), then the entropy of the radiation truly is new and compensates for the lost info, saving the second law at the cost of unitarity. If Hawking radiation has subtle correlations (deviations from perfect thermality), then the information comes out and the entropy of radiation is just the spreading out of original info (so no net information loss, and no violation of quantum mechanics). The latest thought (from quantum gravity research like the holographic principle, ADS/CFT correspondence, quantum error correction analogies, and computations of the Page curve for entropy of radiation) is that black hole evaporation is unitary and information does come out, but it’s encrypted in the radiation in nearly inscrutable form. The entropy of the radiation rises initially (as if info is being lost behind the horizon), but about halfway through the evaporation (the Page time), the entropy of the radiation reaches a maximum and then starts to decrease, indicating that information is being transferred to the radiation in the latter stages. By the end, if the black hole completely disappears, the radiation is in a pure state containing all the info of what fell in, just encoded in an extremely complex way.

• Inaccessibility vs. Destruction: There is an important philosophical distinction between destroyed information and inaccessible information. Information that is scrambled or spread out in correlations among many particles is for all practical purposes lost to us (we can't reconstruct it easily), but it still exists within the physical system. For example, burn a book: in classical thinking, the information in the text is destroyed, but in principle if you had Maxwell's demon-level control of all emitted photons, smoke particles, and air molecules, you could (in theory) reconstruct the book. The information went to correlations between the outgoing heat, light, and ash. In reality, no one can recover it, but physics doesn't outlaw it in principle. Similarly, when something falls into a black hole, perhaps the information is stored in subtle correlations between the Hawking quanta, or in quantum gravitational degrees of freedom (like soft hairs on the horizon, or in a parallel "reconstruction" of the interior via holography). We lack the technology (and possibly the theoretical framework) to retrieve or observe it, hence it appears lost. But appearance is not reality as far as the fundamental laws are concerned.

• Pauli and Quantum States: If you think of each particle that fell into the black hole, it had a distinct quantum state (no two electrons the same, etc.). When the black hole evaporates, you have a huge number of outgoing particles. It’s as if the identities got jumbled, but quantum theory says the overall state vector just evolved to a new basis. The distinctiveness of what fell in is preserved in complicated entangled form. It's like mixing many colored paints: you get black paint finally, and you might think the individual colors (information) are lost. But if it were quantum mixing, in theory it could be unmixed by a precise unitary reversal (whereas classical mixing increases entropy irreversibly). Black holes push this idea to the extreme — but if quantum gravity is unitary, then no information is fundamentally destroyed.

Disproving Information Loss: While we don't yet have a final experimental proof that black hole evaporation is unitary (that would require observing quantum correlations in Hawking radiation, an impossible task with astrophysical black holes), the theoretical evidence has been mounting. The AdS/CFT correspondence (a result from string theory) equates certain black holes to an ordinary quantum system on the boundary of space, where unitarity is manifest — implying the black hole itself must preserve information. Calculations of the "Page curve" (using techniques like replica wormholes in semiclassical gravity) reproduce the behavior expected if information escapes. So, it appears the universe does not allow information oblivion; it only allows information to hide very well.

 

The Role of Noether’s Theorem: It’s poetic to think that Noether’s theorem, which started our journey with a simple hole in the ground, also underlies the black hole. Time-translation symmetry in general relativity leads to a conserved quantity that is equivalent to energy (with some subtleties in GR, but it holds for spacetimes with appropriate asymptotic properties). If information were truly lost, one might expect some violation of energy conservation or an effective non-unitary dynamics at the fundamental level. Thus far, no such violation has been observed — every dropped bit of information seems paid for in energy and entropy exactly. Black holes likely obey an ultimate accounting: when all is done, they pay back the energy and release the information (in disguised form) to the universe.

 

Closing Thoughts: From cavemen unknowingly creating bits by digging holes, to modern computers pushing against Landauer’s limit, to the enigmatic black hole, one principle stands firm: information is physical, and its fate is governed by physical law, not magic. Every bit has an energy cost to create, a heat cost to erase, and a place in the ledger of the universe’s entropy. We cannot destroy information without trace because to do so would break the very symmetries that hold the fabric of physics together. As we elevate our understanding — from Newton’s energy conservation, to Einstein’s mass–energy equivalence, to Noether’s theorem, to quantum unitarity and beyond — we find a deep convergence: the bits of information and the bits of energy are two sides of the same coin.

 

We may never be able to read the information hidden in Hawking radiation or in the chaotic motions of molecules after an irreversible process. But it’s comforting (in a way that would please both Einstein and Newton) to know that the information is still out there in the universe, encoded in some form, obeying the cosmic book-keeping of energy and entropy. The challenge for physics moving forward is to decode these hidden informations and fully understand the language of bits that the universe speaks at its most fundamental level.