What question did this study set out to answer?

The goal is to establish that π is an emergent property linked to vacuum thermodynamics and information processing, rather than a fundamental constant.

February 21, 2026Open Access

The Emergence of Geometry: π as the Imaginary Phase of Modular Information

Key Points

The goal is to establish that π is an emergent property linked to vacuum thermodynamics and information processing, rather than a fundamental constant.
Derived the identity π = -i[ln ζ(0) + ln 2] using theoretical frameworks.
Validated the identity with arbitrary-precision arithmetic achieving 150-digit precision.
Utilized Python's mpmath library for numerical computations.
Confirmed the exact identity of π with zero absolute error within computational precision.
Reinterpreted Euler's identity to show a connection between arithmetic stability of the vacuum and geometric emergence.
Provided a prediction for Hubble's constant, suggesting it as a rate of phase creation.

Abstract

This work presents the final chapter of the Modular Substrate Theory (MST), demonstrating that the geometric constant π is not a fundamental axiom but an emergent property arising from the interaction between vacuum thermodynamics and binary information processing. Core Result The paper derives and validates the exact identity: π = -iln ζ (0) + ln 2 where: ζ (0) = -1/2 represents the ground state of the Riemann zeta function (the vacuum's thermodynamic potential) ln 2 represents the Shannon information unit (the binary bit) Validation The identity has been verified with 150-digit precision using arbitrary-precision arithmetic (mpmath in Python). The absolute error is zero within the computational precision, confirming that this is an exact structural relationship, not a numerical approximation. Theoretical Framework This result completes the three-level hierarchy of the Modular Substrate Theory: Level Domain Foundation Level 1 Arithmetic ℤ/6ℤ modular ring (hardware) Level 2 Thermodynamic Constants e, α, Rfund (software) Level 3 Geometric π as emergent phase (interface) Key Implications Euler's Identity Reinterpreted: The MST framework shows that e^ (iπ) = 2ζ (0). Therefore, Euler's formula e^ (iπ) + 1 = 0 is equivalent to the ground state condition 2ζ (0) + 1 = 0, demonstrating that the "beauty" of Euler conceals the arithmetic stability of the vacuum. Geometric Emergence: Space-time continuity is an artifact of phase accumulation; geometry is the "imaginary residue" of discrete information processing. Cosmological Resolution: The Hubble tension is explained as the rate of phase creation, predicting H₀ = 73. 45 km/s/Mpc for the local universe. Reproducibility All results are fully reproducible: Source code: Python scripts using mpmath for 150-digit validation Notebooks: Interactive Google Colab notebooks for immediate verification Data: Complete numerical outputs and validation logs Repository: https: //github. com/NachoPeinador/The-Emergence-of-Geometry Related Publications This work is part of a series developing the Modular Substrate Theory: Peinador Sala, J. I. (2026). The Genesis of e and the Unification of Fundamental Constants. DOI: 10. 5281/zenodo. 18673474 Peinador Sala, J. I. (2026). The Fine Structure of the Arithmetic Vacuum. DOI: 10. 5281/zenodo. 18611630 Peinador Sala, J. I. (2026). Modular Substrate Theory: Geometric Unification. DOI: 10. 5281/zenodo. 18609093 Peinador Sala, J. I. (2025). The Modular Spectrum of π: Algorithmic Hybridization in Z/6Z. DOI: 10. 5281/zenodo. 18485154 Peinador Sala, J. I. (2025). The Modular Spectrum of π: From Prime Channel Structure to Elliptic Supercongruences. DOI: 10. 5281/zenodo. 18417862 Version v1. 0. 0 | February 2026 License MIT License (software) | CC BY 4. 0 (documentation) Keywords modular substrate theory, pi, riemann zeta function, shannon entropy, emergence of geometry, fundamental constants, high precision validation, mpmath, Z/6Z, hubble tension, euler identity

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