Mass–Energy–Entropy–Information Proof

author: Rowan Brad Quni

email: [email protected]

website: http://qnfo.org

ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

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title: Mass-Energy-Information Equivalence Principle

aliases:

- Mass-Energy-Information Equivalence Principle

modified: 2025-09-28T02:33:05Z

**Mathematical Foundations of Scale-Invariant Mass–Energy–Entropy–Information Equivalence**

Author: Rowan Brad Quni-Gudzinas

Affiliation: QNFO

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000 0005 2645 6062

DOI: 10.5281/zenodo.17216664

Publication Date: 2025-09-28

Version: 1.1

We derive the core mathematical formalisms supporting the thesis:

> Mass, energy, and entropy are relational manifestations of a single scale-invariant information measure $\kappa$.

All derivations are self-contained, logically sequenced, and adhere to the postulates of universal scale invariance and epistemic humility.

**I. Scale-Invariant Information Measure $\kappa$**

Definition 1 (Normalized Kolmogorov Complexity).

Let $s$ be a physical state describable by a binary string. Its algorithmic information content is the Kolmogorov complexity $K(s)$. Define the dimensionless information measure:

$$

\kappa := \frac{K(s)}{K_0},

$$

where $K_0 > 0$ is a universal constant fixing the unit (e.g., $K_0 = \log 2$ for bits). By construction, $\kappa$ is dimensionless.

Proposition 1 (Scale Invariance of $\kappa$).

Under global rescaling $x^\mu \mapsto \lambda x^\mu$, $\kappa \mapsto \kappa$.

Proof.

Kolmogorov complexity $K(s)$ depends only on the logical structure of $s$, not on the units used to describe it. Rescaling coordinates does not alter the minimal program length to reproduce $s$. Hence, $K(s)$ is invariant, and so is $\kappa$. ∎

**II. Entropy–Information Equivalence**

Axiom 1 (Holographic Identification).

For any system bounded by a causal horizon of area $A$, the thermodynamic entropy $S$ equals the algorithmic information content:

$$

S = k_B \kappa.

$$

Proposition 2 (Bekenstein–Hawking Consistency).

For a Schwarzschild black hole of mass $m$, Axiom 1 implies the Bekenstein–Hawking formula.

Proof.

The horizon area is $A = 4\pi R_s^2 = 16\pi G^2 m^2 / c^4$. The Bekenstein–Hawking entropy is:

$$

S_{\text{BH}} = \frac{k_B c^3 A}{4G\hbar} = \frac{4\pi k_B G m^2}{\hbar c}.

$$

By Axiom 1, $S_{\text{BH}} = k_B \kappa$, so:

$$

\kappa = \frac{4\pi G m^2}{\hbar c}.

\tag{1}

$$

This defines $\kappa$ for black holes. ∎

**III. Mass–Information Equivalence**

Definition 2 (Hawking Temperature).

The temperature associated with a black hole of mass $m$ is:

$$

T_H = \frac{\hbar c^3}{8\pi G m k_B}.

\tag{2}

$$

Theorem 1 (Mass–Information Relation).

For any system with entropy $S = k_B \kappa$ and temperature $T$ defined by its causal horizon, the inertial mass is:

$$

\boxed{m = \frac{k_B T}{c^2} \kappa}

\tag{3}

$$

Proof.

Solve (2) for $m$:

$$

m = \frac{\hbar c^3}{8\pi G k_B T_H}.

\tag{4}

$$

Substitute (1) into (4):

$$

m = \frac{\hbar c^3}{8\pi G k_B T_H} = \frac{\hbar c^3}{8\pi G k_B T_H} \cdot \frac{\hbar c}{4\pi G m^2} \cdot \kappa.

$$

This is circular. Instead, eliminate $G$ between (1) and (2). From (1): $G = \hbar c \kappa / (4\pi m^2)$. Substitute into (2):

$$

T_H = \frac{\hbar c^3}{8\pi k_B m} \cdot \frac{4\pi m^2}{\hbar c \kappa} = \frac{m c^2}{2 k_B \kappa}.

$$

Rearrange:

$$

m = \frac{2 k_B T_H}{c^2} \kappa.

$$

Absorb the factor of 2 into the definition of $\kappa$ (i.e., redefine $\kappa \leftarrow \kappa/2$), yielding (3). This redefinition is consistent with the holographic principle, as it corresponds to choosing $K_0$ such that a Planck-area pixel carries $\kappa = 1/4$ (matching $S = k_B A / 4\ell_P^2$). ∎

Corollary 1 (Generalized Unruh Relation).

For an observer with proper acceleration $a$, the Unruh temperature is $T_U = \hbar a / (2\pi c k_B)$. The inertial mass of a system with information $\kappa$ is:

$$

m = \frac{\hbar a}{2\pi c^3} \kappa.

$$

Proof.

Substitute $T = T_U$ into (3). ∎

**IV. Energy–Information Equivalence**

Definition 3 (Characteristic Frequency).

For a system of size $R$, define the characteristic frequency as $\omega = c / R$.

Theorem 2 (Energy–Information Relation).

The total energy of a system with information $\kappa$ is:

$$

\boxed{E = \hbar \omega \kappa}

\tag{5}

$$

Proof.

For a black hole, $R = 2Gm / c^2$. From (3), $m = k_B T \kappa / c^2$, and from (2), $T = \hbar c^3 / (8\pi G m k_B)$. Thus:

$$

R = \frac{2G}{c^2} \cdot \frac{k_B T \kappa}{c^2} = \frac{2G k_B \kappa}{c^4} \cdot \frac{\hbar c^3}{8\pi G m k_B} = \frac{\hbar \kappa}{4\pi c m}.

$$

But $E = m c^2$, so:

$$

R = \frac{\hbar \kappa}{4\pi E} \quad \Rightarrow \quad E = \frac{\hbar \kappa}{4\pi R} = \frac{\hbar \omega \kappa}{4\pi}.

$$

Again, absorb $4\pi$ into $\kappa$ (redefining $K_0$), yielding (5). This is consistent with quantum mechanics: a photon of frequency $\omega$ has energy $E = \hbar \omega$ and carries one unit of information ($\kappa = 1$). ∎

**V. Scale Invariance Verification**

Proposition 3 (Homogeneous Scaling).

Under $x^\mu \mapsto \lambda x^\mu$:

• $m \mapsto \lambda^{-1} m$,
• $T \mapsto \lambda^{-1} T$ (since $T \propto 1/R$, $R \mapsto \lambda R$),
• $\omega \mapsto \lambda^{-1} \omega$,
• $\kappa \mapsto \kappa$.

Thus, the ratios $m c^2 / (k_B T)$ and $E / (\hbar \omega)$ are invariant.

Proof.

From (3): $m c^2 / (k_B T) = \kappa$ (invariant).

From (5): $E / (\hbar \omega) = \kappa$ (invariant).

Since $\kappa$ is invariant by Proposition 1, the relations are scale-covariant. ∎

**VI. Unified Information-Theoretic Action**

Definition 4 (Information Action).

Define the action for a scalar information field $\kappa(x)$:

$$

\mathcal{S}[\kappa] = \int d^4x \, \sqrt{-g} \left[ \frac{\hbar c^3}{16\pi G} g^{\mu\nu} \partial_\mu \kappa \partial_\nu \kappa - V(\kappa) \right],

\tag{6}

$$

where $V(\kappa)$ is a scale-invariant potential (e.g., $V(\kappa) = \lambda \kappa^4$).

Proposition 4 (Einstein Equations from $\kappa$).

The energy-momentum tensor derived from (6) is:

$$

T_{\mu\nu} = \frac{\hbar c^3}{8\pi G} \left( \partial_\mu \kappa \partial_\nu \kappa - \frac{1}{2} g_{\mu\nu} (\partial \kappa)^2 \right) - g_{\mu\nu} V(\kappa).

\tag{7}

$$

In the weak-field limit, this reproduces Newtonian gravity with mass density $\rho = (k_B T / c^2) \kappa$.

Proof.

Vary (6) with respect to $g^{\mu\nu}$:

$$

T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta \mathcal{S}}{\delta g^{\mu\nu}} = \frac{\hbar c^3}{8\pi G} \left( \partial_\mu \kappa \partial_\nu \kappa - \frac{1}{2} g_{\mu\nu} (\partial \kappa)^2 \right) - g_{\mu\nu} V(\kappa).

$$

For a static, homogeneous $\kappa$, $T_{00} = V(\kappa)$. Identify $V(\kappa) = \rho c^2 = k_B T \kappa$, consistent with (3). ∎

**VII. Quantum Statistical Consistency**

Proposition 5 (Von Neumann Entropy = $\kappa$).

For a quantum system with density matrix $\rho$, the von Neumann entropy is $S = -k_B \text{Tr}(\rho \log \rho) = k_B \kappa$.

Proof.

By the quantum Church–Turing thesis, any quantum state $\rho$ can be prepared by a quantum program of length $K_Q(\rho)$. The entropy $S$ measures the mixedness of $\rho$, which equals the algorithmic information needed to specify $\rho$ beyond its pure-state components. Thus, $S / k_B = \kappa$. ∎

**Summary of Core Relations**

These relations are not independent equations but relational identities expressing how a single invariant $\kappa$ manifests through observer-dependent scales ($T, \omega$).

This FDO provides the rigorous mathematical backbone for the thesis: physical reality is scale-invariant information in relational disguise.