M35 The ISHE Hyperradical and the Fano-Galois Correspondence: Explicit Solutions to Degree-Seven Polynomials via the Inter-Shelf Half-Étage at Rank R = 7/2

M35 constructs the ISHE hyperradical H7, an operational solving function for generic degree‑7 polynomials, completing the rank‑3.5 (ISHE) layer of algebraic solvability. Core ideaClassical algebra:degree ≤ 4 → radicalsdegree ≥ 5 → no general radical solution Operational programme:degree 5 → AGM hyperradical (R = 3/2) → Hermite 1858 → (M29a proposal)degree 6 → Heun hyperradical (R = 5/2) → M35bdegree 7 → ISHE hyperradical (R = 7/2) → M35, here M35 provides:H7 = new solving operation at R = 3.5 Main theorem (clean form)Using the Galois–rank relation:R_solve(f) = (g_min(Galf) + 1)/2 For generic degree‑7 polynomial:Galf = S6g_min = 6→ R_solve = 3.5 (ISHE) Result:Degree‑7 equations are solvable by ISHE hyperradical H7. Structure of H7The construction uses three aligned structures:(1) ISHE geometryR = 3.5 half‑rankmixed logarithmHyperzeug bridge → same layer as:Yang–Mills / ISHE / mass-gap geometry (2) Genus‑3 structureISHE has Picard–Fuchs equations of genus 3 → matches:degree‑7 solvability complexity (3) Fano-plane symmetryPSL(2,7) ≅ GL(3, F2) → acts as:7-fold symmetry of ISHE structure= Galois seam symmetry for degree 7 Mechanism (compressed)The solution chain:General degree‑7 polynomial→ Bring–Harriot reduction→ requires iterative solving inside ISHE layer→ solved via H7 So:H7 plays the role of "7th root",but at operational rank 3.5. Hyperradical ladder (important)From M35b_Heun_Hyperradical:H5 (AGM) → degree 5H6 (Heun) → degree 6H7 (ISHE) → degree 7 Bootstrap:H6 uses H5H7 uses H6 → recursive hyperradical hierarchy Conceptual interpretationM35 changes the classical statement: "No radical solution exists" into: "No solution exists at classical ranks, but solution exists at higher or fractional operational rank." So:unsolvability = insufficient rank Position in the programmeM29 → Galois–rank correspondenceM30 → geometry of ranksM34 → ISHE physics layerM35 → algebra at ISHE: solving equations So M35 is:arithmetic realization of ISHE Status discipline (important)Galois–rank relation → structural (from M29a)Genus matching → structuralH7 construction → explicit (main claim of M35)Bring reduction use → classical method So the strongest claim:H7 is constructed and solves Bring–Harriot form within the operational framework.