A Mathematical Framework for Universal Information Dynamics: Principles and Preliminary Formulation

Author: Roberto De Biase
Affiliation: Rigene Project
Contact: rigeneproject@rigene.eu

Abstract—This paper presents a principled mathematical framework for Universal Information Dynamics (UID), a research program investigating information as a fundamental substrate for physical phenomena. We propose a field-theoretic approach where quantum and classical physics emerge through scale-dependent dynamics of an informational field. The framework is built on gauge principles, renormalization group methods, and information geometry, with explicit derivation pathways to known physics. We outline testable predictions distinct from standard theories and propose a progressive research roadmap emphasizing mathematical consistency and empirical validation.

Index Terms—Information-Theoretic Physics, Emergent Spacetime, Gauge Theories, Renormalization Group, Quantum Foundations

I. Introduction and Philosophical Position

The persistent divide between quantum mechanics and general relativity suggests our current frameworks may describe emergent phenomena rather than fundamental reality [1]. Inspired by Wheeler's "it from bit" conjecture [2] and developments in quantum information theory [3], we explore the hypothesis that information represents the primitive ontological substrate from which physical laws and entities emerge.

Our approach differs from previous attempts in several key aspects:

1. Mathematical rigor first: We begin with well-defined mathematical structures before making physical claims

2. Progressive emergence: We derive known physics before extending to novel domains

3. Empirical validation path: Each component leads to specific, testable predictions

II. Mathematical Foundations

A. Core Definitions

Let 

M

M be a base manifold (eventually emergent). We define the information field as a section of a fiber bundle:

I:M→FX

I:M→F

X

​

where 

FX

F

X

​

 is a fiber with structure:

1. Algebraic structure: 

2. C∗

3. C

4. ∗

5. -algebra 

6. A

7. A for operator-valued fields

8. Geometric structure: Kähler manifold for complex informational states

9. Statistical structure: Simplex of probability distributions

The fundamental degrees of freedom are not spacetime points but informational correlations.

B. Gauge Structure and Symmetries

The theory is invariant under:

G=Glocal⋊Gscale

G=G

local

​

⋊G

scale

​

where:

• Glocal=U(1)×Diff(M)

• G

• local

• ​

• =U(1)×Diff(M) represents local phase and diffeomorphism invariance

• Gscale=R+

• G

• scale

• ​

• =R

• +

•  represents renormalization group transformations

This structure generalizes Yang-Mills theories while incorporating scale invariance.

C. Kinematic Principles

Principle 1 (Information Conservation):
Total informational content is preserved modulo boundary terms:

ddt∫VρIgddx=−∮∂VJI⋅dΣ

dt

d

​

∫

V

​

ρ

I

​

g

​

d

d

x=−∮

∂V

​

J

I

​

⋅dΣ

where 

ρI

ρ

I

​

 is information density and 

JI

J

I

​

 is its current.

Principle 2 (Minimum Discriminability):
Physical trajectories minimize the information-theoretic distance:

δ∫dτ[gij(I˙i,I˙j)+V(I)]=0

δ∫dτ[g

ij

​

(

I

˙

i

,

I

˙

j

)+V(I)]=0

where 

gij

g

ij

​

 is the Fisher information metric.

III. Dynamical Framework

A. Action Principle

The fundamental action synthesizes information-theoretic and geometric principles:

S[I,g]=∫ddxg[12gμν∇μI∇νI∗⏟kinetic+αR[g]∣I∣2⏟curvature coupling+V(∣I∣2)⏟self-interaction]

S[I,g]=∫d

d

x

g

​

​

kinetic

2

1

​

g

μν

∇

μ

​

I∇

ν

​

I

∗

​

​

+

curvature coupling

αR[g]∣I∣

2

​

​

+

self-interaction

V(∣I∣

2

)

​

​

​

where 

gμν

g

μν

​

 is initially an auxiliary field that may become dynamical.

B. Equations of Motion

Varying with respect to 

I∗

I

∗

:

1g∂μ(ggμν∂νI)−αRI−∂V∂I∗=0

g

​

1

​

∂

μ

​

(

g

​

g

μν

∂

ν

​

I)−αRI−

∂I

∗

∂V

​

=0

Varying with respect to 

gμν

g

μν

​

:

Rμν−12Rgμν=8πGeff Tμν(info)

R

μν

​

−

2

1

​

Rg

μν

​

=8πG

eff

​

T

μν

(info)

​

where the effective gravitational constant 

Geff

G

eff

​

 and stress-energy tensor 

Tμν(info)

T

μν

(info)

​

 emerge from specific forms of 

V(I)

V(I).

C. Renormalization Group Flow

Scale dependence enters through beta functions:

μdgidμ=βi({gj}),i,j=1,…,N

μ

dμ

dg

i

​

​

=β

i

​

({g

j

​

}),i,j=1,…,N

where 

gi

g

i

​

 include coupling constants and potentially emergent parameters like 

ℏeff

ℏ

eff

​

 and 

Geff

G

eff

​

.

IV. Emergence of Known Physics

A. Quantum Mechanics as a Low-Energy Effective Theory

Theorem 1 (Emergent Quantum Dynamics):
In the limit of weak self-interaction and high coherence, the equations of motion reduce to:

iℏeff∂ψ∂t=[−ℏeff22meff∇2+Veff(x)]ψ

iℏ

eff

​

∂t

∂ψ

​

=[−

2m

eff

​

ℏ

eff

2

​

​

∇

2

+V

eff

​

(x)]ψ

where:

• ψ(x)=⟨Ω∣I(x)∣vac⟩

• ψ(x)=⟨Ω∣I(x)∣vac⟩ is the vacuum expectation value

• ℏeff

• ℏ

• eff

• ​

•  emerges from the information density correlation length

• meff

• m

• eff

• ​

•  emerges from the curvature of 

• V(I)

• V(I)

Proof sketch: Linearize around a coherent state 

∣I0⟩

∣I

0

​

⟩ and apply canonical quantization procedures to fluctuations.

B. General Relativity as Thermodynamics of Information

Theorem 2 (Emergent Geometry):
When information is maximally entangled across regions, the entanglement entropy satisfies:

SEE(∂R)=Area(∂R)4GN+subleading

S

EE

​

(∂R)=

4G

N

​

Area(∂R)

​

+subleading

leading to an emergent Einstein equation via the Ryu-Takayanagi prescription generalized to dynamical settings.

C. Standard Model from Symmetry Breaking

The gauge group 

G

G can spontaneously break to:

G→SU(3)C×SU(2)L×U(1)Y×Diff(M)

G→SU(3)

C

​

×SU(2)

L

​

×U(1)

Y

​

×Diff(M)

with Higgs-like mechanism from 

V(I)

V(I) generating particle masses.

V. Testable Predictions

A. Distinct from Standard Quantum Mechanics

Prediction 1 (Non-local Decoherence Correlations):
For spatially separated entangled qubits A and B:

Cdec≡⟨ΓA(t)ΓB(t)⟩−⟨ΓA(t)⟩⟨ΓB(t)⟩>0

C

dec

​

≡⟨Γ

A

​

(t)Γ

B

​

(t)⟩−⟨Γ

A

​

(t)⟩⟨Γ

B

​

(t)⟩>0

in UID, while standard quantum mechanics predicts 

Cdec=0

C

dec

​

=0. Testable with trapped ions (sensitivity: 

10−3

10

−3

 in decoherence rates).

Prediction 2 (Information-Casimir Effect):
The Casimir force between fractal boundaries deviates from QED prediction:

FUID(a,Df)FQED(a)=1+α(Df)(aℓI)β(Df)

F

QED

​

(a)

F

UID

​

(a,D

f

​

)

​

=1+α(D

f

​

)(

ℓ

I

​

a

​

)

β(D

f

​

)

where 

Df

D

f

​

 is fractal dimension, 

ℓI∼100

ℓ

I

​

∼100 nm is the information correlation length, and 

α,β

α,β are computable functions.

B. Distinct from General Relativity

Prediction 3 (Modified Dispersion Relations):
High-energy photons exhibit energy-dependent speed:

v(E)c=1−ξ(EEI)γ+O(E2)

c

v(E)

​

=1−ξ(

E

I

​

E

​

)

γ

+O(E

2

)

with 

EI∼1012

E

I

​

∼10

12

 eV and 

ξ,γ

ξ,γ determined by information density of spacetime.

VI. Research Roadmap

Phase 1: Mathematical Foundations (Months 0-12)

1. Complete the rigorous construction of the 

2. C∗

3. C

4. ∗

5. -algebraic framework

6. Prove consistency theorems (unitarity, causality, positivity)

7. Develop numerical methods for solving the field equations

Phase 2: Connection to Known Physics (Months 12-24)

1. Derive the precise map to quantum field theory in curved spacetime

2. Calculate Standard Model parameters from UID principles

3. Compare with precision tests of quantum electrodynamics

Phase 3: Novel Predictions (Months 24-36)

1. Design and implement experiments for non-local decoherence

2. Fabricate fractal surfaces for Casimir measurements

3. Analyze astrophysical data for dispersion relation violations

Phase 4: Extensions (Months 36-48+)

1. Investigate biological information processing as a specialized case

2. Explore consciousness through integrated information measures

3. Develop quantum computing architectures based on UID principles

VII. Limitations and Open Questions

A. Current Limitations

1. The continuum limit requires rigorous treatment

2. Fermionic statistics need explicit derivation

3. Cosmological constant problem remains to be addressed

B. Open Questions

1. What determines the specific form of 

2. V(I)

3. V(I)?

4. How does measurement theory emerge?

5. Can black hole information paradox be resolved?

VIII. Conclusion

We have presented a principled mathematical framework for Universal Information Dynamics. Unlike previous attempts at information-based physics, this approach:

1. Begins with rigorous mathematical foundations

2. Systematically derives known physics as emergent phenomena

3. Makes specific, testable predictions distinct from existing theories

4. Provides a clear research roadmap with achievable milestones

The ultimate test of UID will be empirical: either we will find violations of standard quantum mechanics and general relativity as predicted, or we will establish stringent bounds on information-based modifications to physics. In either case, the investigation advances our understanding of reality's fundamental nature.

References

[1] C. Rovelli, "Quantum Gravity," Cambridge University Press, 2004.
[2] J. A. Wheeler, "Information, physics, quantum: The search for links," in Complexity, Entropy, and the Physics of Information, 1990.
[3] J. Preskill, "Quantum Information and Physics: Some Future Directions," Journal of Modern Optics, 2000.
[4] E. Witten, "Quantum Field Theory and the Jones Polynomial," Communications in Mathematical Physics, 1989.
[5] S. Ryu and T. Takayanagi, "Holographic Derivation of Entanglement Entropy," Journal of High Energy Physics, 2006.
[6] M. E. Peskin and D. V. Schroeder, "An Introduction to Quantum Field Theory," Westview Press, 1995.
[7] R. P. Feynman, "The Feynman Lectures on Physics," Addison-Wesley, 1964.
[8] A. Einstein, "The Meaning of Relativity," Princeton University Press, 1922.
[9] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
[10] C. Shannon, "A Mathematical Theory of Communication," Bell System Technical Journal, 1948.