Probability as a Conservation Law in the Theory of Entropicity (ToE)

One of the most striking conceptual departures introduced by the Theory of Entropicity (Toe) is the re‑interpretation of probability itself. In classical physics and in the Kolmogorov framework, probability is defined axiomatically: the sum of all mutually exclusive outcomes must equal unity. This rule has no dynamical origin; it is not derived from physical principles, nor does it arise from the geometry of the underlying state space. It is simply imposed.

In contrast, ToE does not assume probability conservation. It derives it.

The starting point is the structural decomposition of the total Hilbert space into two orthogonal sectors,

$${\mathcal{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e,$$

where ${\mathcal{H}}_o$ represents the coherent observer sector and ${\mathcal{H}}_e$ represents the entropic sector. Under the combined unitary–entropic evolution generated by

$$U_{\mathrm{T}\mathrm{o}\mathrm{E}}(t)=e^{-iHt}{e}^{-Ct},$$

the total state decomposes as

$$\mathrm{\Psi }(t)={\psi }_o(t)+{\psi }_e(t),$$

with the orthogonality condition ${\psi }_o(t)\perp {\psi }_e(t)$. Norm conservation of the total state,

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=1,$$

implies the additive relation

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2\mathrm{.}$$

Defining the sectoral probabilities as

$$P_o(t):=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2,P_e(t):=\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2,$$

one obtains the entropic probability‑conservation law,

$$P_o(t)+P_e(t)=1.$$

Although this expression resembles the classical normalization rule, its meaning is fundamentally different. Classical probability partitions outcomes; ToE partitions reality. The quantities $P_o(t)$ and $P_e(t)$ are not probabilities of events but probabilities associated with two dynamically coupled, orthogonal sectors of the universe. The entropic operator $e^{-Ct}$ transfers amplitude from the observer sector into the entropic sector, generating irreversibility, decoherence, and the arrow of time. Yet the total probability is conserved across the full ToE evolution.

Thus, ToE elevates probability from an epistemic bookkeeping rule to an ontological conservation law. Probability becomes a structural invariant of the universe’s Hilbert‑space geometry and its entropic dynamics. This shift—from axiom to conservation principle—marks one of the most conceptually significant contributions of the Theory of Entropicity (ToE).

Scholium

The ToE probability law is about sectoral decomposition, not human observation

The law:

$$P_o(t)+P_e(t)=1$$

is not about what a person sees. It is about how the universe partitions amplitude between:

• the coherent (observer) sector

• the entropic sector

This is why ToE emphasises:

“${\psi }_o(t)\perp {\psi }_e(t)$” “${\mathcal{H}}_{\text{tot}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e$” “$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2$”

This is geometry, not psychology.

⭐ 1. ToE does not say “measurement requires a human observer”

ToE explicitly rejects the Copenhagen idea that:

• the Moon exists only when observed

• measurement requires consciousness

• physical reality depends on a human presence

ToE says the opposite:

The Moon is there whether or not anyone looks at it. Measurement is an entropic process, not a psychological one.

So far, no contradiction.

⭐ 2. ToE does say measurement is “observer‑dependent” — but in a technical sense

When ToE uses the term observer, it does not mean a person.

It means:

the coherent sector of the Hilbert space

$${\mathcal{H}}_o$$

This sector is defined by:

• coherence

• information accessibility

• low entropy

• ability to support classical records

This is a physical structure, not a mind.

Thus, “observer‑dependent” in ToE means:

dependent on which sector of the Hilbert space is coherent enough to register information.

Not dependent on a human.

⭐ 3. The entropic sector is the complement

$${\mathcal{H}}_e$$

This is the sector where:

• entropy increases

• coherence is lost

• information becomes inaccessible

• microscopic details dissipate

This is what ToE calls the entropic sector.

⭐ 4. The probability law is about sectoral decomposition, not human observation

The law:

$$P_o(t)+P_e(t)=1$$

is not about what a person sees. It is about how the universe partitions amplitude between:

• the coherent (observer) sector

• the entropic sector

This is why the uploaded page emphasises:

“${\psi }_o(t)\perp {\psi }_e(t)$” “${\mathcal{H}}_{\text{tot}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e$” “$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2$”

This is geometry, not psychology.

⭐ 5. So why does ToE say measurement is “observer‑dependent”?

Because measurement = projection onto the coherent sector:

$$\mathrm{\Psi }(t)\mapsto {\psi }_o(t)$$

This projection depends on:

• which degrees of freedom are coherent

• which are decohered

• which are accessible

• which are entropically suppressed

This is observer‑dependent in the same sense that:

• simultaneity is observer‑dependent in relativity

• electric vs magnetic field components are observer‑dependent

• kinetic vs potential energy is observer‑dependent

These are frame‑dependent, not human‑dependent.

⭐ 6. ToE’s position is therefore perfectly consistent

✔ The Moon exists without a human observer

Because the entropic threshold determines measurement, not consciousness.

✔ Measurement is observer‑dependent

Because the coherent sector depends on the physical structure of the system, not on a person.

✔ Probability is conserved across sectors

$$P_o(t)+P_e(t)=1$$

✔ The partition between sectors is relative

But the total is invariant.

No contradiction — just a subtle but powerful distinction.

⭐ The simplest way to understand ToE

ToE does not make reality depend on humans. It makes measurement depend on the structure of the Hilbert space.

The Moon is there whether or not anyone looks. But the sector in which its information resides determines what can be measured.

There is no contradiction once we distinguish:

(A) Information that becomes measurable to the classical sector

vs.

(B) Information that becomes inaccessible to the coherent quantum sector

These are two different flows happening at the "same time".

⭐ 1. What ToE says about classical measurement

When a system crosses the entropic threshold, ToE says:

• entropy increases

• coherence is lost

• microscopic quantum details dissipate

• information becomes measurable (i.e., classical records form)

This is the quantum → classical transition.

It is the moment when:

• decoherence has occurred

• the observer sector ${\mathcal{H}}_o$ receives a stable record

• the system becomes classical enough to be “measured”

This is the birth of classical information.

So far, everything is consistent.

⭐ 2. What ToE says about the entropic sector

The entropic sector ${\mathcal{H}}_e$ is defined as:

• high‑entropy

• decohered

• information‑inaccessible

• dynamically irreversible

This is where:

• microscopic details are lost

• coherence is destroyed

• amplitude flows under $e^{-Ct}$

• information becomes inaccessible to the coherent sector

This is the quantum → entropic transition.

It is the moment when:

• the system’s fine‑grained quantum information is no longer retrievable

• the observer sector cannot access the microstructure

• the entropic sector absorbs the lost coherence

This is the loss of quantum information.

⭐ 3. These two processes happen simultaneously

This is the key insight.

When a system decoheres:

✔ The observer sector gains classical information

(“information becomes measurable”)

✔ The entropic sector gains lost quantum information

(“information becomes inaccessible”)

These are not contradictory. They are two sides of the same entropic flow.

⭐ 4. Why the two descriptions look opposite

Because they refer to different kinds of information:

Classical information

• macroscopic

• coarse‑grained

• stable

• measurable

• accessible to ${\mathcal{H}}_o$

Quantum micro‑information

• fine‑grained

• phase‑sensitive

• coherence‑dependent

• lost to decoherence

• inaccessible to ${\mathcal{H}}_o$

• absorbed by ${\mathcal{H}}_e$

So:

• classical information increases

• quantum information becomes inaccessible

This is exactly what ToE predicts.

⭐ 5. The conservation law ties it all together

$$P_o(t)+P_e(t)=1$$

This means:

• the observer sector gains classical probability

• the entropic sector gains lost quantum probability

• the total is conserved

Thus:

• measurement is the transfer of coherence into entropy

• classical information emerges

• quantum information disappears into ${\mathcal{H}}_e$

No contradiction — just two complementary flows.

⭐ 6. The simplest way ToE declares it

ToE says measurement makes classical information accessible, but quantum information inaccessible.

Both statements are true. They refer to different layers of the same entropic process.

Reference

1. J. O. Obidi, “On the Discovery of New Laws of Conservation and Uncertainty, Probability and CPT-Theorem Symmetry-Breaking in the Standard Model of Particle Physics: More Revolutionary Insights from the Theory of Entropicity (ToE),” Cambridge University (CoE), 2025. https://doi.org/10.33774/coe-2025-n4n45

2. https://theoryofentropicity.blogspot.com/2026/04/entropic-probability-conservation-and.html

3. https://theoryofentropicity.blogspot.com/2026/04/probability-as-conservation-law-in.html

4. https://theoryofentropicity.blogspot.com/2026/04/scholium-sectoral-probability.html