(<em>p</em>, <em>q</em>)-String Junctions as Interstitial Fields on a Modal Lightcone

We develop a framework in which lightcones, $(p,q)$ strings, and M-theory compactifications are equipped with an explicit layer of modal structure. Starting from Minkowski spacetime $\mathbb{M}^4$ with its usual null cones, we consider a background $\mathbb{M}^{4\times 7} = \mathbb{M}^4 \times G_2$ and attach to each spacetime point $\mathfrak{x}$ an internal $G_2$-source fiber. Local field configurations are encoded by propositions in a fuzzy Heyting--Kripke semantics, and a localized action $S(\mathfrak{x})$ defines normalized truth-values $\sigma(\mathfrak{x},\varphi) \in [0,1]$ which we interpret as a ``modal radius'' on a refined lightcone. Nilpotent extensions of the truth-value space capture unphysical ``eternal photon'' modes as purely internal, non-observable directions in the $G_2$-source. On this background we introduce \emph{interstices} $\varcurlywedge = G_2 \cap \mathbb{M}^4$ at $(p,q)$ string junctions, regarded as higher-dimensional analogues of Chan--Paton defects. Each interstice supports a pair of S-dual field-families (FFs) $(\Phi,\Psi)$ with a coprime arithmetic grading that organizes primary, descendant, and auxiliary modes, and we show how a $G_2$-inspired flux superpotential can be reinterpreted as a ``modal'' functional controlling the exponents in $\sigma(\mathfrak{x},\varphi) = n^{-z_i}$. We work out explicit examples: a three-leg $(p,q)$ junction and a minimal two-junction web with an internal $(1,1)$ segment, including concrete numerical assignments for $\sigma(\mathfrak{x},\varphi)$, an S-duality reshuffling of the FFs, and a simple propagation rule for modal data along the internal edge. The resulting picture can be phrased in terms of a small Kripke frame whose accessibility relation is determined by the (p,q) web geometry. Taken together, these constructions suggest that junctions, defects, and compactification data in string/M-theory admit a natural description in terms of modal structure on lightcones, with logical semantics providing a controlled way to track which field configurations are realized, suppressed, or confined to nilpotent internal sectors.