(p, q)-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 M 4 with its usual null cones, we consider a background M 4×7 = M 4 × G2 and attach to each spacetime point x an internal G2-source fiber. Local field configurations are encoded by propositions in a fuzzy Heyting-Kripke semantics, and a localized action S(x) defines normalized truth-values σ(x, φ) ∈ [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 G2-source. On this background we introduce interstices = G2 ∩M 4 at (p, q) string junctions, regarded as higher-dimensional analogues of Chan-Paton defects. Each interstice supports a pair of Sdual field-families (FFs) (Φ, Ψ) with a coprime arithmetic grading that organizes primary, descendant, and auxiliary modes, and we show how a G2-inspired flux superpotential can be reinterpreted as a "modal" functional controlling the exponents in σ(x, φ) = 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 σ(x, φ), 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.