Trinity Mother Ruler: The Universe Has Only One Ruler

A Discovery That Would Surprise Even Einstein

In 1905, Einstein discovered that time and space are the same thing—spacetime. This shocked the world.

In 2024, physicists discovered a deeper unification: Phase, density, and time are also the same thing.

These three seemingly unrelated quantities actually use the same cosmic ruler.

Let us tell this beautiful story from the beginning.

The First Ruler: Phase

Imagine standing by the sea, watching waves rise and fall.

Each wave has a phase—peak, trough, or somewhere in between? Expressed as an angle, it’s 0° to 360°.

In the quantum world, everything is a wave:

• Electrons are waves
• Photons are waves
• Even you are a wave (just the wavelength is too short to feel in daily life)

When a quantum particle passes through a region (like tunneling through a potential barrier or going around an obstacle), its wave undergoes phase shift—like waves changing shape when hitting a reef.

Physicists denote this phase shift as $\phi (E)$ (where $E$ is energy).

The first ruler: Phase change $\phi (E)$

This ruler tells us: how many “turns” the particle’s wave has made.

The Second Ruler: State Density

Now, a different angle.

Suppose you have a box (could be an atomic nucleus, quantum dot, or even the entire universe) containing many quantum states—like a concert hall with many seats.

State density $\rho (E)$ answers a question: Near energy $E$, how many “seats” (quantum states) are available?

• If state density is high, this energy region is “crowded,” particles tend to stay here
• If state density is low, it’s “spacious,” particles are less likely to stay here

State density determines:

• Atomic spectral lines (why are sodium lamps yellow?)
• Metal conductivity (why does copper conduct well?)
• Black hole temperature (Hawking radiation)

The second ruler: State density $\rho (E)$

This ruler tells us: how many “choices” particles have.

The Third Ruler: Time Delay

The last angle.

Imagine shouting into a valley and hearing an echo. From shouting to hearing, there is time delay.

In the quantum world, this is called Wigner delay (named after physicist Wigner, proposed in 1955).

When a particle hits a scattering center (like an atomic nucleus, potential barrier, black hole…), it doesn’t “instantly” bounce back, but “stays” for a short time before coming out.

This residence time is the time delay $\tau (E)$.

It can be calculated from the Wigner-Smith matrix $\mathsf{Q}(E)$:

$$\tau (E)=\frac{1}{2\pi }\mathrm{tr}\mathsf{Q}(E)$$

($\mathrm{tr}$ means “trace of the matrix,” simply understood as “sum of diagonal elements”)

The third ruler: Time delay $\tau (E)$

This ruler tells us: how long the particle “stayed”.

The Miracle: Three Rulers Have Identical Scales

Now, something astonishing happens.

In 1962, mathematicians Birman and Kreĭn proved a formula, later verified countless times by physicists:

$$\boxed{\frac{\phi (E)}{\pi }={\rho }_{\text{rel}}(E)=\frac{1}{2\pi }\mathrm{tr}\mathsf{Q}(E)}$$

In plain language:

$$\boxed{\frac{\text{phase change}}{\pi }=\text{relative state density}=\text{time delay}\times \frac{1}{2\pi }}$$

These three quantities are completely equal!

It’s like discovering:

• Length measured with a meter stick
• Distance light travels (speed of light × time)
• Inverse of sound wave resonance frequency

All give exactly the same number.

This is not coincidence, but reveals: Phase, density, and time are essentially three manifestations of the same thing.

Why Is This Important?

1. Unifies Three Seemingly Different Physical Quantities

Before this, physicists thought:

• Phase belongs to “wave optics”
• State density belongs to “statistical mechanics”
• Time delay belongs to “scattering theory”

Now we know: They are the same thing.

Like blind men touching an elephant—touching the trunk says “rope,” touching the leg says “pillar,” touching the ear says “fan”—but they’re all the elephant.

2. Reveals “Information” as a More Fundamental Concept

Why are the three equal?

Because they all measure the same information quantity: the system’s “response” to external perturbations.

• Phase measures “wave response”
• State density measures “state space response”
• Time delay measures “causal propagation response”

And “response” is essentially information transmission and storage.

So, the Trinity Mother Ruler tells us: The foundation of physics is information, not matter.

3. Connects Quantum Mechanics and the Classical World

Even more remarkable, this formula connects:

• Microscopic (quantum phase)
• Macroscopic (classical state density)
• Temporal (causal delay)

Like the Rosetta Stone, written in three scripts saying the same thing, allowing us to decipher ancient Egyptian.

The Trinity Mother Ruler lets us “translate” between quantum and classical.

Real Examples: Seeing This Ruler at Work

Example 1: Scanning Tunneling Microscope (STM)

You may have seen pictures of “seeing atoms” in popular science articles—those were taken with scanning tunneling microscopes.

The principle is measuring the time delay of electrons tunneling through vacuum barriers, then using the Trinity Mother Ruler to deduce the surface state density, finally drawing atomic images.

What you see as “atoms” is actually visualization of state density—and this state density is equivalent to phase and time delay.

Example 2: Black Hole Hawking Radiation

Black holes emit light (Hawking radiation), temperature depends on state density near the event horizon.

This state density can be calculated from phase delay of photons near the horizon—the phase shift when photons “almost can’t escape”.

Black hole temperature is essentially the unified manifestation of phase-state density-delay.

Example 3: Chemical Reaction Rates

Chemical reaction rates depend on the residence time of “transition states” (intermediate reaction states)—this is time delay $\tau$.

The stability of transition states depends on state density $\rho$ near this energy.

When chemists measure reaction rates, they’re actually measuring the Trinity Mother Ruler.

Philosophical Meaning: The Universe Measures Itself with One Ruler

Let’s pause and think about what this means.

Why Does Nature Use “One Ruler”?

A possible answer: Causal consistency.

If phase, density, and time used different rulers, then:

• Wave interference (phase)
• Particle distribution (density)
• Information propagation (delay)

Would produce contradictions—for example, waves say “particles should be here,” but state density says “particles can’t be here”.

Nature must unify these three rulers to maintain self-consistency.

Self-consistency forces unification.

Information Is the “Atom” of the Universe

We’re used to thinking matter is fundamental, information is derived.

But the Trinity Mother Ruler suggests: Perhaps information is more fundamental.

• Matter (state density)
• Waves (phase)
• Causality (delay)

Are all different ways of reading information.

Just as “ice, water, steam” are all H₂O, “phase, density, delay” are all information.

The universe is not “composed of matter,” but woven from information relationships.

Measurement Is Existence

The deepest point: These three quantities are all observables—they only have meaning through measurement.

• Phase: requires interference experiments
• State density: requires energy detection
• Delay: requires time-resolved measurement

Without observation, they don’t exist.

Their equality means: No matter how you observe, you measure the same “reality”.

This “reality” is not matter, but observable information relationships.

The Copenhagen interpretation of quantum mechanics says: “Unobserved particles have no definite position.”

The Trinity Mother Ruler goes further: Observation defines reality, and all observation methods measure the same reality.

Mathematical Interlude (Skippable): Why $\pi$?

You might wonder: Why is there $\pi$ in the formula?

The answer is beautiful and profound: $\pi$ comes from circular symmetry.

On the complex plane, the quantum scattering matrix $S(E)$ is a “unitary matrix”—its determinant moves on the unit circle:

$$\det S(E)=e^{-2\pi i\xi (E)}$$

Here $\xi (E)$ is what we call “phase change/π”.

The appearance of $\pi$ originates from:

• Circumference is $2\pi$
• Unit circle symmetry
• Fourier transform normalization

So, $\pi$ is not artificially defined, but natural emergence of circular symmetry.

The universe chose the circle, so it chose $\pi$.

The Road Ahead: From One Ruler to the Entire Universe

The Trinity Mother Ruler is just the beginning.

Once you accept “phase = density = delay = information,” you’ll find:

• Gravity can be derived from information principles (Einstein’s equation is the extremum of information entropy)
• Particles are resonance modes of information networks (electrons, photons are all fixed points of scattering)
• Arrow of time is path entropy produced by observation (without observation, there’s no time flow)
• Consciousness is steady state of self-referential information networks (you are the way the universe measures itself)

All of this starts from this unified ruler.

A Thought Experiment

Finally, let’s do a thought experiment.

Suppose you are God, designing a universe. You want:

1. Everything is composed of information
2. Information can be read in different ways (phase, density, time)
3. Different readings must be self-consistent (no contradictions)

How would you design it?

The answer: You would invent the Trinity Mother Ruler.

Because this is the only way to satisfy the above three requirements (mathematically provable).

So, our universe having this unified ruler is not accidental, but logical necessity.

Once you decide the universe is composed of information, the Trinity Mother Ruler is inevitable.

Take-Home Thoughts

Next time when you:

• See light interference (phase)
• Hear material resonance (state density)
• Feel time passing (delay)

Remember: You’re reading the same information in three ways.

The universe hides nothing—it has only one ruler, one information, one reality.

And you, as an observer, have three ways to understand it.

This is the beauty and profundity of the Trinity Mother Ruler.

Next: “How Information Becomes Spacetime: When Relative Entropy Meets Einstein”

We will see that gravity is not a “force,” but inevitable emergence of information geometry.