Algebraic Resolution of Divergent Topologies via Multilayered Discrete Fields and Contextual Zeros

Abstract & Research Overview A central axiom of classical continuous real analysis is the strict structural equivalence of null states, which reduces additive inverses to a featureless, absolute void. While functionally sufficient for linear arithmetic, this assumption fails catastrophically in reciprocal transformations. Evaluating the quotient function $f(x)=F/x$ as $x \to 0$ generates undefined topological singularities conventionally abstracted as absolute infinity ($\infty$). This paper formalizes the Multilayered Discrete Field ($\mathbb{A}$), predicated on the Unitary Symmetry Series. It rigorously reconstructs continuous real numbers as exact anchor points for discrete, base-dependent multiplicative layers governed by rational reciprocal operators, resolving long-standing topological paradoxes. Key Theoretical Frameworks & Mathematical Proofs The Axiom of Contextual Zero ($0_B$): Proves that mathematical null states are topologically distinct ($1-1 \neq 2-2$). Null states inherently retain the localized rational boundary constraints of their generating base rather than collapsing into an absolute void. The Theorem of Positional Averaging: Redefines classical singularities ($F/0$) by evaluating systemic tension across discrete boundaries. It derives deterministic, finite Signed Algebraic Pressures ($\mathcal{P}_s$) for null intersections (utilizing a $1/4$ geometric factor) and manifest scalar flows (utilizing a $1/2$ factor). Universal Normalization & Recursive Sub-Layering: Introduces formal mechanisms to resolve asymmetric topologies ($\alpha \cdot \beta \neq 1$) and degenerate states ($\beta = 1$) dynamically, without inducing topological collapse. Inter-Systemic Translation Bridge: Establishes a formal mathematical method to prevent dimensional contamination when transferring scalar magnitudes across isolated discrete universes with varying systemic densities ($\beta_{sys}$). Conclusion: Resolving Absolute Infinity By executing a rigorous mathematical translation bridge, this paper demonstrates that classical continuous calculus ($\mathbb{R}$) is strictly a degraded informational subset of the Multilayered Discrete Field. Absolute infinity is mathematically exposed not as an intrinsic algebraic singularity, but as an artifact of homomorphic information loss and irreversible metadata erasure resulting from the zero-limit. Official Documentation & Research Framework For extensive documentation, foundational research papers on the Unitary Symmetry Series, and institutional updates regarding this mathematical framework, please visit the official project domain at anadihilo.org. Keywords: Unitary Symmetry Series, Multilayered Discrete Field, Contextual Zero, Positional Averaging, Singularity Resolution, Asymmetric Normalization, Information Erasure, Inter-Systemic Translation, Topological Geometry.