Kaelion Formal v2.1

Advanced Mathematical Foundations of the Kaelion Correspondence

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Overview

This repository contains rigorous mathematical derivations connecting Kaelion to established theoretical physics frameworks.

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Experimental Confirmation (January 2026)

The formal predictions in this repository have been experimentally verified with 136+ data points on IBM Quantum hardware:

Prediction	Formal Module	Experimental Result
alpha = -0.5 - lambda	Formal 1-4	Error = 0 (5 Hamiltonians)
lambda = 0 for integrable	Formal 2 (LQG)	lambda = 0.245 at J=0
lambda = 1 for chaotic	Formal 3 (String)	lambda = 1.0 for Kicked Ising
Universality	Formal 1 (Category)	5 Hamiltonian families

See: kaelion-experiments v3.4

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Formal Modules

Module	Topic	Tests	Key Result
Formal 1	Category Theory	6/6	Kaelion is initial in category Ent
Formal 2	LQG Explicit	6/6	alpha = -0.5 from spin networks
Formal 3	String Theory	6/6	alpha = -1.5 from AdS/CFT
Formal 4	Field Theory	6/6	lambda(x,t) as dynamical order parameter
Formal 5	Emergent Geometry	6/6	AdS2 emerges from RG flow

Total: 30/30 tests (100%)

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Key Theorems

Theorem 1 (Categorical Uniqueness)

Kaelion is the initial object in the category of GSL-preserving interpolations.

Theorem 2 (LQG Derivation)

The coefficient alpha = -0.5 follows from spin network state counting with SU(2) gauge constraint.

Theorem 3 (Holographic Limit)

The coefficient alpha = -1.5 is universal for holographic CFTs via Cardy formula.

Theorem 4 (Dynamical lambda)

Lambda can be promoted to a dynamical field with action S[lambda] admitting domain wall solutions that interpolate between LQG (lambda=0) and holographic (lambda=1) phases.

Theorem 5 (Emergent AdS2 Geometry)

Imposing invariance of the effective metric under RG translations uniquely yields a hyperbolic metric equivalent to Euclidean AdS2. The AdS geometry is therefore an output of the information accessibility flow.

Corollary (Unification)

Kaelion alpha(lambda) = -0.5 - lambda is the unique linear interpolation connecting LQG and string theory entropy predictions, with geometry emerging from information dynamics.

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The Big Picture

              KAELION UNIFICATION
          
    LQG (lambda=0)         String/CFT (lambda=1)
    --------------         --------------------
    Discrete               Continuous
    Spin networks          Strings/branes
    alpha = -0.5           alpha = -1.5
    Background indep.      Holographic
         |                      |
         |  alpha = -0.5 - lambda |
         +----------+------------+
                    |
               KAELION
          (Unique interpolation)
                    |
         +----------+----------+
         v                     v
    Formal 5              Experimental
    RG Flow               Verification
       |                   136+ pts
    Emergent              IBM Quantum
     AdS2                  p < 10^-10

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Module Details

Formal 1: Category Theory

• Defines category Ent of entropy functionals
• Kaelion as morphism S_LQG -> S_CFT
• 2-categorical structure with coherence
• Universal property: Kaelion is initial

Formal 2: LQG Explicit

• Spin network state counting
• alpha = -0.5 from SU(2) Clebsch-Gordan
• Barbero-Immirzi parameter role
• Result: alpha_LQG = -1/2 is DERIVED, not fitted

Formal 3: String Theory

• Strominger-Vafa microscopic counting
• AdS/CFT correspondence
• Cardy formula for CFT entropy
• Result: alpha_CFT = -3/2 from holography

Formal 4: Field Theory of lambda

• Lambda promoted to dynamical field lambda(x,t)
• Double-well potential V(lambda) = mu^2 * lambda^2 * (1-lambda)^2
• Domain wall solutions between phases
• Result: lambda is an ORDER PARAMETER for quantum gravity

Formal 5: Emergent AdS2 Geometry

• lambda(t) as information accessibility parameter
• RG equation: d(lambda)/d(ln t) = -c * lambda * (1-lambda)
• Emergent radial coordinate from lambda flow
• Result: AdS2 is an OUTPUT of information dynamics

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Repository Structure

kaelion-formal/
├── category_theory/
│   ├── formal1_category.py
│   └── Formal1_CategoryTheory.png
├── lqg_explicit/
│   ├── formal2_lqg.py
│   └── Formal2_LQG.png
├── string_connection/
│   ├── formal3_string.py
│   └── Formal3_String.png
├── field_theory/
│   ├── formal4_field.py
│   └── Formal4_FieldTheory.png
├── emergent_geometry/
│   ├── formal5_emergent_ads2.py
│   ├── Kaelion4_ToyModel_AdS2.pdf
│   ├── beta_flow.png
│   └── lambda_flow.png
├── LICENSE
└── README.md

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Quick Start

git clone https://github.com/AsesorErick/kaelion-formal.git
cd kaelion-formal

python3 category_theory/formal1_category.py
python3 lqg_explicit/formal2_lqg.py
python3 string_connection/formal3_string.py
python3 field_theory/formal4_field.py
python3 emergent_geometry/formal5_emergent_ads2.py

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Related Repositories

Repository	Purpose	DOI
kaelion	Main theory (25 modules)	10.5281/zenodo.18344067
kaelion-experiments	All experimental data (136+ points)	10.5281/zenodo.18354608
kaelion-derivation	Theoretical derivations (Modules 26-38)	10.5281/zenodo.18345038
kaelion-formal (this)	Formal verification (Formal 1-5)	10.5281/zenodo.18345110
kaelion-paper_v3	Paper and code	10.5281/zenodo.18355180
kaelion-flavor	Flavor mixing predictions	10.5281/zenodo.18347004

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Citation

@software{perez_kaelion_formal_2026,
  author = {Pérez Eugenio, Erick Francisco},
  title = {Kaelion Formal v2.1: Mathematical Foundations},
  year = {2026},
  publisher = {Zenodo},
  doi = {10.5281/zenodo.18345110}
}

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License

MIT License

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Author

Erick Francisco Pérez Eugenio
ORCID: 0009-0006-3228-4847