M35b The Heun Hyperradical: Degree-Six Polynomials, Grand Unification, and Lambert's Orbital Equation

M35b constructs the Heun hyperradical H6, a solving operation for generic degree‑6 polynomials, located at rank R = 5/2 (Genus‑2 / Heun layer). 1. Position in the hyperradical ladderM35b is the middle element of the ladder: 5 3/2 AGM H5 (Hermite 1858, M29a - general insight) 6 5/2 Heun H6 (M35b) 7 7/2 ISHE H7 (M35) Key statement: Solvability is not lost after degree 4. It shifts to higher operational ranks. 2. Core mathematical claimGeneric degree‑6 polynomials are solvable at rank R = 5/2 via the Heun hyperradical H6. Mechanism: Galois complexity → genus 5 → intermediate structure → R = 2. 5 3. Construction mechanismThe solving chain is: General sextic→ Bring–Harriot reduction→ intermediate equations of degree ≤ 5→ solved using AGM hyperradical H5→ lifted via Heun structure to full solution So structurally: H6 = H5 + Heun layerImportant property: H6 is not independent: it bootstraps on H5. 4. Nature of the Heun layerThe Heun level (R = 5/2) is characterised by: - genus‑2 / genus‑5 (depending on perspective) - Heun differential equations (generalization of hypergeometric) - multi‑branch monodromy structureInterpretation: AGM → single elliptic curveHeun → interacting elliptic structures / higher genusThus: Heun = next complexity level of algebraic solvability after elliptic (AGM). 5. Bootstrap structure (central insight) M35b establishes: H5 → solves degree 5H6 → requires H5H7 → requires H6So: Hyperradicals form a recursive hierarchy. This is the first place in the corpus where: solvers are not independent, but form a nested computational chain. 6. Conceptual shiftClassical statement: Degree ≥ 5 → unsolvable by radicalsOperational reinterpretation: Each degree requires a fractional and higher rank. Specifically: degree 5 → first non-radical → first hyperradical (AGM) degree 6 → next obstruction → HeunSo: unsolvability = crossing a rank boundary 7. Structural role in the programmeM25 → builds analytic structure (genus / Picard–Fuchs) M29 → maps Galois group → rankM35b → constructs the solver at that rankM35 → completes next level (ISHE) Thus: M35b = arithmetic realization of the Heun layer. 8. Orbital mechanics, GUT and predictions The same object appears in two apparently distant settings: GUT: degree-6 symmetry-breaking equation→ solved by H6 Orbital mechanics: degree-6 critical-orbit equation→ solved by H6 The common reason is the rank: both problems cross the same Heun seam. They are not identical physical systems, but they occupy the same operational layer. In the GUT reading, the CatHeun Hermit unittauCH = exp (W₀ (i*pi) ) is the operational marker of the Heun layer. The document identifies this unit with a GUT scaleEGUT ≈ 1. 86 × 10¹6 GeV and records the associated proton lifetime estimatetauₚ ≈ 5. 4 × 10³6 yr.