The Multi-Dimensional Tensor Framework (nD) of Relational Logic (RL) and Axiomatic Context (AC): Breaking the Bivalent Lattice Boundaries

For over two millennia, formal mathematics and standard logic have been structurally confined within the discrete walls of bivalent systems, restricting truth and membership to rigid endpoints $\{0, 1\}$ or artificial closed limits $[0, 1]$. This paper establishes the high-dimensional formal expansion of Relational Logic (RL) and Axiomatic Context (AC) from traditional discrete lattices to continuous, symmetric tensor spaces spanning the entire extended real number line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$. By analyzing the operational matrices in higher dimensions ($nD$), we prove that the transitional gradient between conjunction ($\wedge$), disjunction ($\vee$), and arithmetic mean ($\text{MEAN}$) forms a seamless non-linear continuum. Consequently, the foundational logic arrays natively decouple from algebraic lattice restrictions and ascend to pure, differentiable tensor fields, preserving un-degraded physical magnitudes across multi-context horizons without inducing boundary saturation or systemic semantic explosion. Critical Theoretical Correction on $\overline{\mathbb{R}}$ Behavior:} The historical confinement of the parametric quality governor to the static unit interval $w \in [0, 1]$ represents a severe conceptual and training bias inherited from classical bounded fuzzy frameworks. Then we must strictly adhere to these principles to declare theoretical perfection: $w \in \mathbb{[}a, b\mathbb{]}$ and $\min(a, b) \leq w \leq \max(a, b)$