Scholium: Sectoral Probability, Measurement, and Dual Information Flow from the Law of Conservation of Probability in the Theory of Entropicity (ToE)

The probability law of the Theory of Entropicity (ToE) is frequently misunderstood when interpreted through the lens of classical or Copenhagen‑style measurement theory. In ToE, the relation

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

does not refer to what a human observer sees, nor does it presuppose the presence of a conscious agent. Instead, it expresses a sectoral decomposition of the total Hilbert space, reflecting how the universe partitions amplitude between two orthogonal components:

• the coherent (observer) sector ${\mathcal{H}}_o$, and

• the entropic sector ${\mathcal{H}}_e$.

This decomposition is encoded in the structural relations

$${\psi }_o(t)\perp {\psi }_e(t),{\mathcal{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e,\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2\mathrm{.}$$

These are statements of geometry, not psychology.

1. Measurement in ToE is not human‑dependent

ToE explicitly rejects the Copenhagen claim that physical reality depends on human observation. It does not require consciousness, perception, or an experimenter to bring phenomena into existence. The Moon exists whether or not anyone looks at it. Measurement, in ToE, is an entropic process, not a mental act.

Thus, ToE is fully consistent with an observer‑independent external world.

2. Measurement is “observer‑dependent” only in a technical, sectoral sense

When ToE refers to an “observer,” it does not mean a person. It means the coherent sector ${\mathcal{H}}_o$ of the Hilbert space: the subspace capable of supporting stable, classical records. This sector is defined by:

• coherence,

• information accessibility,

• low entropy, and

• the ability to retain classical information.

“Observer‑dependent” therefore means:

dependent on which degrees of freedom remain coherent enough to register information.

It does not mean dependent on a human presence.

3. The entropic sector is the complementary domain

The entropic sector ${\mathcal{H}}_e$ is characterized by:

• increasing entropy,

• loss of coherence,

• dynamical irreversibility, and

• inaccessibility of fine‑grained quantum information.

This is the sector into which microscopic details dissipate under the entropic evolution operator $e^{-Ct}$.

4. The probability law expresses sectoral conservation, not subjective observation

The relation

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

is a conservation law describing how amplitude flows between ${\mathcal{H}}_o$ and ${\mathcal{H}}_e$. It is not a statement about what a person sees. It is a structural identity arising from the orthogonal decomposition of the total state.

Thus, the ToE probability law is sectoral, not psychological.

5. Why ToE calls measurement “observer‑dependent”

Measurement in ToE is the projection of the total state onto the coherent sector:

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

This projection depends on:

• which degrees of freedom remain coherent,

• which have decohered,

• which are accessible to ${\mathcal{H}}_o$, and

• which have been entropically suppressed into ${\mathcal{H}}_e$.

This is analogous to:

• simultaneity in relativity,

• electric vs. magnetic field components,

• kinetic vs. potential energy.

All are frame‑dependent, not human‑dependent.

6. The consistency of ToE’s position

ToE therefore asserts:

• The Moon exists without a human observer. Measurement is determined by entropic thresholds, not consciousness.

• Measurement is observer‑dependent because the coherent sector is defined by the physical structure of the system.

• Probability is conserved across sectors

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

• The partition is relative, but the total is invariant.

There is no contradiction—only a precise distinction between physical sectors and human observers.

7. Dual information flow: classical accessibility vs quantum inaccessibility

The apparent tension between “information becomes measurable” and “information becomes inaccessible” dissolves once we distinguish two kinds of information:

Classical information (accessible to ${\mathcal{H}}_o$)

• macroscopic

• coarse‑grained

• stable

• measurable

Quantum micro‑information (lost to ${\mathcal{H}}_o$)

• fine‑grained

• phase‑sensitive

• coherence‑dependent

• absorbed by ${\mathcal{H}}_e$

Thus, when a system crosses the entropic threshold:

• classical information becomes accessible (birth of a classical record),

• quantum information becomes inaccessible (loss of coherence).

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

8. Conservation unifies the two flows

The conservation law

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

expresses that:

• the observer sector gains classical probability,

• the entropic sector gains lost quantum probability,

• the total remains conserved.

Measurement is therefore the transfer of coherence into entropy, producing classical information while dissipating quantum microstructure.

9. The ToE declaration

ToE states:

Measurement makes classical information accessible, while quantum information becomes inaccessible.

Both statements are true. They describe 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