On the Homology of Algebra, Number Theory and Information Theory under the Multi-Origin Curvature Framework

This paper adopts the geometric system of Multi-Origin Curvature (MOC) as a unified perspective, attempting to sort out the homologous underlying structures of algebra, number theory and information theory. Traditional theories are self-consistent and valid within their respective scopes of application. This paper does not negate existing systems, and merely offers a geometric interpretation compatible with classical theories.

I. Curvature Generalization of Algebra

Classical algebra is built upon a single zero element and global axioms. Its operations are governed by universal constraints such as commutativity and associativity. The structure is well-organized and widely applicable, yet relatively fixed in form.

From the MOC perspective, algebraic structures arise from the coupling of multiple origins. Each curvature origin may possess a local identity element and local operational reference, rather than relying on one unique global origin.

The symmetry of operations is no longer absolute. When the curvatures of origins are close, the system approximately recovers the rules of classical algebra. Where the gradient of curvature differs substantially, operations exhibit non-commutative and non-associative behaviour.

The classical equivalence of homomorphism and isomorphism may be replaced by curvature projection equivalence. Two algebraic structures are identical if their curvature configurations coincide within the projection space. Ideals and quotient structures can be understood as closed substructures formed by dimensional collapse of the curvature space.

Based on the above, rudimentary generalized curvature algebras can be constructed. A multi-origin group defines multiplication weighted by curvature gradients; a curvature ring distinguishes addition from same-layer superposition and multiplication from cross-layer coupling; a projection field may degenerate into a classical field under specific mappings.

When the curvature of all origins tends to uniformity, gradients vanish, and the projection becomes an identity map, MOC algebra fully reverts to classical groups, rings and fields.

In short: classical algebra is the flat special case of multi-origin curvature algebra.

II. Geometric Curvature Interpretation of Number Theory

Classical number theory is established on integer lattices, with divisibility, factorization and primality as its core foundations, forming a discrete arithmetic system.

The MOC framework embeds integer lattices into a high-dimensional curvature space. Numerical features are essentially projections of the arrangement of curvature origins.

A prime number may be reinterpreted as an irreducible origin within the curvature space. Its corresponding curvature configuration cannot be decomposed into the superimposed projections of several non-trivial curvature substructures, possessing the simplest, indivisible form. Composite numbers are composite lattice points formed by the coupling and superposition of multiple curvature origins.

Accordingly, the distribution of prime numbers corresponds to the projection density of irreducible points in high-dimensional space. The Riemann zeta function and L-functions can be regarded as frequency-domain statistical expansions of the curvature space. Problems of integer partition may be interpreted as the superimposed projection of connecting lines between distinct prime curvature origins.

When the space becomes flat, the lattice uniform, and the projection identity, the curvature-based definitions naturally reduce to conventional definitions of integers and primes.

Classical number theory is the degenerate result of discrete curvature geometry on flat lattices.

III. Curvature Uncertainty Representation of Information Theory

Classical information theory is modelled on probability spaces. It employs entropy, mutual information and channel capacity to describe uncertainty and transmission limits, boasting mature applications and concise computation.

Within the MOC framework, information can be fully interpreted geometrically.

Information entropy corresponds to the uncertain volume of states in a curvature system. A richer set of curvature configurations implies greater uncertainty. Mutual information represents the overlapping projected volume of two sets of curvature structures, measuring the correlation between systems. Channel capacity equals the maximum number of distinguishable independent projection regions in curvature space.

Coding compression is the controlled collapse of redundant curvature. Error-correcting codes rely on cross-checking of curvature across multiple origins to restore distorted structures.

In a flat, isotropic space subject to additive Gaussian noise, curvature measure is equivalent to probability measure. Geometric limits naturally reduce to Shannon entropy and the Shannon channel capacity formula.

Classical information theory is a special case of curvature geometric information theory under flat space and standard noise assumptions.

IV. Overview of Unification

Algebra describes local operational rules and coupling structures within curvature systems.

Number theory describes quantitative features and distribution patterns of discretely arranged curvature.

Information theory describes the uncertainty and distinguishability limits of curvature systems.

Though they appear in different forms, the three share a common origin:

Algebraic structures, number-theoretic units and information measures can all be attributed to local properties, discrete projections and global limits of the same multi-origin curvature space.

Remarks

This paper presents a theoretical discussion aiming to establish a cross-disciplinary homologous framework. It does not modify or replace classical theories. All new MOC constructions can revert to traditional mathematics and information theory under corresponding flat, uniform and identity-projection conditions, ensuring the compatibility and self-consistency of the system. Detailed formal construction and quantitative verification may be developed in follow-up research.