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Modules

Rowan Brad Quni edited this page May 24, 2026 · 1 revision

Mathematical Modules (M1-M12)

QWAV's technical framework is organized into 12 mathematical modules. Each module addresses a specific aspect of ultrametric quantum physics.

M1-M4: Mathematical Foundations

M1: Ratio-Based Valuation Theory

Establishes the mathematical framework for ultrametric valuations. Defines ratio-based encoding where physical qubits carry valuation ratios that encode their position in the hierarchical tree structure.

M2: Bruhat-Tits Tree Construction

Constructs the Bruhat-Tits tree — the central geometric object of QWAV. This infinite regular tree encodes the structure of p-adic numbers and provides the geometric substrate for ultrametric quantum computation.

M3: Vladimirov Operator

Defines the Vladimirov operator — the p-adic analog of the Laplace operator. This operator governs dynamics on ultrametric spaces, analogous to how the Laplacian governs dynamics in Euclidean quantum mechanics.

M4: Adelic Theory for General Number Fields

Extends the framework to adelic spaces, unifying behavior across all primes simultaneously. Provides a complete description of quantum states on ultrametric spaces using adelic methods from number theory.

M5-M8: Quantum Error Correction & Computation

M5: Ratio-Based QEC

Implements quantum error correction using the ratio-based encoding from M1. Errors are confined to individual branches of the Bruhat-Tits tree, providing intrinsic fault tolerance without active syndrome measurement.

M6: Quantum Gate Theory

Defines quantum gates that operate on ultrametric-encoded qubits. Gates respect the tree's hierarchical structure, ensuring that operations on one branch do not disturb others.

M7: Thermodynamic Limits with Scaling Ratios

Analyzes the thermodynamic limits of ultrametric quantum computation. Derives scaling laws for energy, entropy, and computational capacity as functions of tree depth and branching factor.

M8: Wheeler-DeWitt on Ratio-Based Trees

Applies the Wheeler-DeWitt equation to ultrametric quantum geometries. Investigates quantum gravitational aspects of tree-based computation, connecting to quantum cosmology.

M9-M12: Physical Implications & Synthesis

M9: Emergent Lorentz Symmetry

Demonstrates that Lorentz symmetry emerges naturally from the ultrametric structure at appropriate scales. The continuous spacetime of special relativity arises as a limiting case of the underlying discrete tree geometry.

M10: Cosmological Dynamics

Explores cosmological implications of ultrametric quantum physics. Investigates how tree-based geometries might inform our understanding of early-universe physics and cosmic structure formation.

M11: Monna Map

Defines the Monna map — a mapping between p-adic and real numbers that connects ultrametric and Euclidean descriptions. Provides a bridge between tree-based and continuous mathematical frameworks.

M12: Synthesis

Integrates all modules into a unified framework. Demonstrates the consistency and completeness of the ultrametric quantum computing paradigm across all 11 preceding modules.

See Also

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