PFUSRC‑168: \sqrt{7} and the Welding Point — The Numerical Boundary of the Projection‑Layer Observational Grid, the Absence Signature of the Fourth Dimension, and the 55‑Coverage Positive Proof of the Bicone

“7 cannot be hidden in a cube” is a classic geometric non‑constructibility phenomenon. This paper fully localizes this phenomenon within the PFUSRC framework, delivering both a negative proof against the cube and a positive proof for the bicone. For the negative proof: the cube is not an ontologically necessary construction. Within PFUSRC, the true ontology is the continuous topological unfolding rhythm of flow variables. The bicone serves as its direct extrinsic form on the projection layer, whereas the cube is merely an orthogonal simplified grid constructed by observers on projection cross‑sections. By Lagrange’s four‑square theorem, 7 is the smallest positive integer that requires four squares for its representation, while the three‑dimensional projection‑layer observational grid only has three orthogonal directions and can accommodate at most three squared terms. This paper defines the welding point as the first positive integer in an n-layer projection observational grid that demands the (n+1)‑th square. The welding‑point sequence is 2, 3, 7, 15,\dots; for the 3‑D grid the welding point is 7. Grid failure at this welding point reveals the non‑ontological nature of the cubic orthogonal grid. For the positive proof via the 55‑coverage test: prior PFUSRC‑002 establishes 55 as the unique closed total of three‑dimensional projected all‑root numbers (19+17+19=55). An ontologically faithful projection structure must host \sqrt{1} through \sqrt{55}. The cube fails starting at \sqrt{7}, only covering \sqrt{1}–\sqrt{6}. In contrast, the bicone defines its numerical spectrum through angles and curvature rather than integer square‑sum decomposition, showing no exclusion across \sqrt{1} to \sqrt{55}. Therefore the bicone is the unique projection structure capable of supporting the complete 55‑anchor numerical spectrum. Using Legendre’s three‑square theorem, the full set of fourth‑dimensional absence signatures within 1–55 is extracted for multi‑number‑theoretic consistency checks. These phenomena belong to Class‑A “ontological‑touch” effects under the PFUSRC‑166B binary classification and admit topological localization verification. Number‑theoretic results are secondary numerical fingerprints produced when topological nodes project onto the natural‑number axis; ontological topological logic takes priority. Notably, 7 functions both as the 3‑D grid welding point and an ontological anchor prime under PFUSRC‑2026, enabling cross‑document mutual validation.