Mass–Energy–Entropy–Information Proof

Mathematical

Foundations of Scale-Invariant Mass–Energy–Entropy–Information

Equivalence

Author: Rowan Brad Quni-Gudzinas

Affiliation: QNFO Contact:

rowan.quni@qnfo.org 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{kB c^3 A}{4G\hbar} = \frac{4\pi k_B G m^2}{\hbar

c}.

\] By Axiom 1, \(S{\text{BH}} = kB

\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: \[

TH = \frac{\hbar c^3}{8\pi G m kB}.

\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 kB TH}.

\tag{4}

\] Substitute (1) into (4): \[

m = \frac{\hbar c^3}{8\pi G kB TH} = \frac{\hbar c^3}{8\pi G kB TH}

\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): \[

TH = \frac{\hbar c^3}{8\pi kB m} \cdot \frac{4\pi m^2}{\hbar c \kappa}

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

\] Rearrange: \[

m = \frac{2 kB TH}{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 \(TU = \hbar a / (2\pi c kB)\). 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{kB T \kappa}{c^2} = \frac{2G kB

\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 \(V(\kappa) = \rho c^2 = kB T \kappa\),

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 = -kB \text{Tr}(\rho \log \rho) = kB

\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

Quantity |

Expression |

Scale Transformation |

Information |

\(\kappa = K(s)/K0 = S/kB\) |

\(\kappa \mapsto \kappa\) |

Mass |

\(m = \dfrac{k_B T}{c^2}

\kappa\) |

\(m \mapsto \lambda^{-1} m\) |

Energy |

\(E = \hbar \omega \kappa\) |

\(E \mapsto \lambda^{-1} E\) |

Entropy |

\(S = k_B \kappa\) |

\(S \mapsto S\) |

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**.