Search for a command to run...
The Problem: A central axiom of continuous real analysis is the strict structural equivalence of null states (e.g., a-a = b-b = 0), which reduces additive inverses to a featureless, absolute void. While sufficient for linear arithmetic, this assumption fails in reciprocal transformations, generating undefined topological singularities ($F/0 \rightarrow \infty$). Proposed Framework: This paper formalizes the Multilayered Discrete Field (A), predicated on the Unitary Symmetry Series (USS). In this field, continuous real numbers are reconstructed as anchor points for discrete, base-dependent multiplicative layers governed by rational reciprocal operators (5/4 and 4/5). Key Innovation: We introduce the axiom of the Contextual Zero (0_B), proving that mathematical null states are topologically distinct and retain the localized geometric boundaries of their generating base. Methodology: By applying the Theorem of Positional Averaging, we derive finite, base-dependent Signed Algebraic Pressures for both null intersections and manifest scalar flows. This allows classical singularities (like 5/0) to be resolved into exact, finite rational states based on the tension of the discrete boundaries. Expansion & Translation: The framework resolves asymmetric topologies ($\alpha \cdot \beta \neq 1$) via Universal Normalization and formalizes an Inter-Systemic Translation Bridge to prevent dimensional contamination across varying universal densities. Conclusion: The paper executes a rigorous 'One-Way Bridge' proof, demonstrating that classical absolute infinity is not an intrinsic property but an artifact of homomorphic information loss and irreversible metadata erasure. Keywords Unitary Symmetry Series, Multilayered Discrete Field, Contextual Zero, Positional Averaging, Singularity Resolution, Asymmetric Normalization, Information Erasure.