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: Rₛolve (f) = (gₘin (Galf) + 1) /2 For generic degree‑7 polynomial: Galf = S6gₘin = 6→ Rₛolve = 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 M35bHeunHyperradical: 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.