Published February 17, 2026 | Version v1
Preprint Open

The Genesis of e and the Unification of Fundamental Constants from the Z/6Z Modular Substrate

Authors/Creators

  • 1. Independent Researcher

Description

*"Nature is an arithmetic orchestration from the ℤ/6ℤ modular substrate."*

This repository provides the complete source code, 110-digit precision numerical audit, and full manuscript for the Modular Substrate Theory (MST) — a unified framework that derives the fundamental constants of physics from a single algebraic principle.

📄 Abstract

The Modular Substrate Theory proposes that the space-time continuum is an emergent property of a discrete informational processing layer. By reconciling ternary volume logic (Bulk) with binary surface encoding (Boundary), we derive the Fundamental Impedance of the Vacuum:

R_fund = 1/(6 log_2 3) = ln 2/(6 ln 3) ≈ 0.1051549589

This single constant — the thermodynamic cost of projecting ternary information onto binary degrees of freedom — generates all subsequent physical relationships.

🔷 The Master Identities

From R_fund emerge four master identities connecting physics and mathematics:

 
 
Identity Physical Significance Equation
Genesis of e Emergence of the continuum e^(6 R_fund ln 3) = 2
Fine Structure QED Coupling (α⁻¹) α⁻¹ = (4π³ + π² + π) - (R_fund³)/4 - (1 + 1/(4π)) R_fund⁵
Hubble Tension Cosmological expansion H_local = H_global · (1 - κ_info)^(-1/2) = 73.45 km/s/Mpc
Zeta-Riemann Arithmetic unitarity e^(iπ - ln 2) = ζ(0) = -1/2

where κ_info = (3/2) R_fund = ln 2/(4 ln 3) is the information-expansion coupling constant.

💎 The Five Conceptual Pearls

The theory distills into five core relationships that unify disparate domains:

 
 
# Pearl Formula
1 Fundamental Identity e^(6 R_fund ln 3) = 2
2 Origin of the 1/4 factor (Bekenstein-Hawking entropy) 1/4 = κ_info · (1/log_2 3) · (3/4) + Δ_quantum
3 Fine Structure α⁻¹ = (4π³ + π² + π) - (1/4)R³ - (1 + 1/(4π))R⁵
4 Connection with ζ(0) e^(iπ - ln 2) = ζ(0) = -1/2
5 SNR Saturation in Riemann SNR_sat = 2/κ_info ≈ 12.68

📊 Key Results & Validations

All predictions are validated against experimental data with high precision (up to 110 digits using mpmath):

 
 
Phenomenon Theoretical Value Experimental Reference Discrepancy
Identity of e 2.000... (100 digits) 2.0 (Exact) < 10⁻¹⁰⁰
Fine Structure 137.035999206... 137.035999206(11) (CODATA 2022) 0.00000011 ppb
Hubble H₀ 73.45 km/s/Mpc 73.04 ± 1.04 (SH0ES) < 0.5σ
Riemann SNR 12.68... 12.69 ± 0.01 < 0.1%
Ξcc⁺⁺ Mass 3619 MeV 3621 MeV (LHCb) < 0.06%
Prime Resonances f_n = n · R_fund Spectral analysis of 6×10⁶ primes > 99.5% significance

🔢 Number Theory Connections

The theory reveals deep links to the Riemann Hypothesis and prime number distribution:

  • Spectral resonances in prime gaps at frequencies f_n = n · R_fund with statistical significance > 99.5%

  • The Riemann Hypothesis as a unitarity condition: The stability of the quantum vacuum requires Re(ρ) = 1/2 for all non-trivial zeros of ζ(s)

  • SNR saturation in Riemann zeros at SNR_sat = 2/κ_info, directly linking number theory to cosmology

🌠 Cosmological Implications

The theory resolves major cosmological tensions without free parameters:

 
 
Tension MST Prediction Observation Status
Hubble (H₀) 73.45 km/s/Mpc 73.04 ± 1.04 (SH0ES) ✓ Resolved
S₈ (structure) 0.766 ± 0.014 0.76-0.79 (DES, eROSITA) ✓ Compatible
Local bubble (D_c) ≈ 70.2 Mpc CosmicFlows-4 kinematic limit ✓ Saturated

⚛️ Hadronic Physics

MST predicts flavor blindness — mass as a geometric property of the substrate rather than constituent composition. Airy scaling with factor β = 3/4 predicts the Ξcc⁺⁺ baryon mass:

M(d**) ≈ M(d*) × (z₂/z₁)^β ≈ 2380 × 1.520 ≈ 3619 MeV

This matches the LHCb observation of 3621 MeV (< 0.06% error), suggesting that mass is a geometric property of the substrate, not just of the constituents.

🌍 Philosophical Implications

MST suggests a worldview where:

  • The discrete is fundamental, the continuous emergent

  • Information is substantial: the thermodynamic cost of processing information (R_fund) is a geometric property of the vacuum

  • Mathematical constants (e, π, γ) are not axioms, but consequences of the underlying arithmetic structure

  • Physics and mathematics are one and the same: prime numbers are excitations of the vacuum, the Riemann Hypothesis is a condition of cosmic stability

🛠️ Reproducibility

All results are fully reproducible via Google Colab notebooks:

 
 
Domain Key Calculations Colab Link
Alpha & e 110-digit audit of e^(6 R_fund ln 3) = 2 and α⁻¹ https://colab.research.google.com/github/NachoPeinador/The-Genesis-of-e/blob/main/Notebooks/The-Genesis-of-e.ipynb
Cosmology Hubble tension resolution, local bubble calculation https://colab.research.google.com/github/NachoPeinador/Modular-Substrate-Theory/blob/main/Notebooks/MST_Cosmology.ipynb
Hadrons Hexaquark mass prediction, Airy scaling https://colab.research.google.com/github/NachoPeinador/Modular-Substrate-Theory/blob/main/Notebooks/MST_Hadronic.ipynb
Prime gaps Spectral resonance analysis https://colab.research.google.com/github/NachoPeinador/Modular-Substrate-Theory/blob/main/Notebooks/Harmonic_primes.ipynb
Riemann zeros SNR saturation, modular phase coherence https://colab.research.google.com/github/NachoPeinador/RIEMANN_Z6/blob/main/Notebooks/Spectral_Arithmetic_Duality.ipynb

 

📚 Related Publications

  • Peinador Sala, J. I. (2026). The Fine Structure of the Arithmetic Vacuum: Exact Derivation of α⁻¹ via Modular Renormalization. Zenodo. DOI: 10.5281/zenodo.18611630.

  • Peinador Sala, J. I. (2026). Modular Substrate Theory: Geometric Unification of Cosmology and Hadronic Spectroscopy. DOI: 10.5281/zenodo.18609093

  • Peinador Sala, J. I. (2025). Spectral-Arithmetic Duality: Modular Phase Coherence and the Riemann-GUE Ensemble. DOI: 10.5281/zenodo.18485154

 

🔑 Keywords

modular-substrate-theory, fine-structure-constant, hubble-tension, riemann-hypothesis, prime-numbers, theoretical-physics, cosmology, number-theory, mpmath, high-precision, fundamental-constants, alpha, euler-number, zenodo, open-science, reproducible-research

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Additional details

Additional titles

Alternative title
A Derivación from First Principles with Cosmological and Arithmetic Implications

Related works

Is described by
Preprint: 10.5281/zenodo.18485154 (DOI)
Is supplemented by
Preprint: 10.5281/zenodo.18611630 (DOI)
Preprint: 10.5281/zenodo.18609093 (DOI)

Software

Repository URL
https://github.com/NachoPeinador/The-Genesis-of-e/tree/main
Programming language
Python
Development Status
Active

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