M35c The Solvability Ladder, Rebuilt - Refutation of the Galois-Rank Seam and of the Hyperradicals, and What Rank 5/2 Can Actually Solve

Refutation of the Galois-Rank Seam and of the Hyperradicals, and What Rank 5/2 Can Actually Solve This monograph supersedes M35, M35b, and the Galois-Rank seam section of M29a. The septic and sextic hyperradicals proposed in M35 and M35b are refuted and must not be used as solving functions. M35c is the authoritative replacement. It is also a companion to F02, whose rank-genus dictionary and corrected period conjecture remain explicitly conjectural. The earlier solvability programme assigned a transcendental solving rank to a polynomial from the minimal genus on which its Galois group acts. It then proposed explicit “hyperradicals” at ranks 5/2 and 7/2 for solving the general sextic and septic. M35c performs a complete audit of that construction and finds that the underlying Galois-Rank seam fails in all three forms printed across the corpus. The generic groups were shifted by one degree, the tabulated genera did not match the stated invariant, and the invariant fails to distinguish solvable from nonsolvable equations in either direction. No rescaling of the seam formula repairs this problem. The monograph also refutes the proposed hyperradicals themselves. The septic construction of M35 begins from an incorrectly attributed normal form, assigns the generic septic the wrong group, invokes a cyclic order-six action unavailable in PSL(2,7), and reports numerical roots that do not satisfy the stated equations. Its ISHE iteration has no claimed interior fixed point. The sextic construction of M35b uses an alternating iteration that diverges, conflates several distinct units, assigns the generic sextic the wrong Galois group, and incorrectly identifies generic genus-two periods with solutions of a Heun equation. The associated SU(5), Lambert-orbit, mass-gap, and bootstrap applications are consequently withdrawn. The negative verdict is therefore definitive: the three printed Galois-Rank seam formulas are refuted; the septic and sextic hyperradicals are refuted; the bootstrap chain between successive polynomial degrees is refuted; the proposed rank-5/2 “triple coincidence” is refuted; M35 and M35b are obsolete as sources of polynomial-solving mechanisms. This refutation does not eliminate the original ladder’s rank sequence. The assignments quintic -> rank 3/2sextic -> rank 5/2septic -> rank 7/2 reappear after the mechanism is replaced. They no longer arise from the minimal genus of a surface on which the Galois group acts. Instead, M35c reconstructs the ladder through level structures on abelian varieties, following the classical route from Hermite and Klein to contemporary work on modular functions and resolvent problems. For a hyperelliptic curve of genus g, the permutation of its 2g+2 Weierstrass points acts on the two-torsion of its Jacobian: S_(2g+2) -> Sp(2g,F_2). This action is injective for g >= 2, and at genus two gives the exceptional isomorphism: S_6 = Sp(4,F_2). M35c uses this structure to prove a radical-free solvability ladder. For a generic squarefree polynomial of degree at most 2g+2, the roots are rational functions of the coefficients and of level-two Siegel modular values. The roots are recovered from Rosenhain invariants, expressed through even theta constants by Thomae’s formula. No radical is required once the relevant period matrix and level structure are available. This gives the corrected genus ladder: Genus one: the cubic; special quintics after adjoining accessory irrationalities, through A_5 = PSL(2,5) and Klein’s icosahedral route. Genus two: the generic quartic, quintic, and sextic; A_6 equations; the 27 lines on a cubic surface; and the setting of Hilbert’s sextic conjecture. Genus three: generic equations of degrees seven and eight, together with the classical geometry of the 28 bitangents. Reading genus g as operational rank g+1/2 recovers the earlier numerical rungs. The genus and modular-cover columns are classical, while their interpretation as operational ranks remains the corpus’s rank-genus conjecture. The radical-free reconstruction is proved in M35c. Thus the old mechanism is killed, but the ladder survives as a successful reformulation with corrected groups and a new geometric foundation. The paper then asks what rank 5/2 can actually do. At genus two, the level-two Siegel cover is versal for S_6. M35c verifies Thomae’s formula to twelve digits, gives Rosenhain formulas in an explicit basis, and numerically reconstructs three S_6 sextics and two S_5 quintics from their coefficients alone to approximately 10^-15. These computations demonstrate the classical coefficient-to-period-to-theta-to-root pipeline without invoking the discarded hyperradical. The situation for the septic is sharply separated. M35c embeds A_7 explicitly into PSp(4,7), but proves that the resulting level-seven genus-two cover is not versal. The natural attempt to descend the general septic to genus two, and therefore to rank 5/2 under the rank-genus dictionary, fails. Whether an appropriate cover becomes E-versal after accessory irrationalities remains open. The paper therefore does not claim that rank 5/2 solves the general septic. The corpus’s rank-5/2 commutative object is renamed from the Symmetric Heun Operation to the Symmetric Hyperelliptic Operation, retaining the acronym SHO; its mean is correspondingly the Symmetric Hyperelliptic Mean, or SHM. The rename is substantive. Generic genus-two periods are not naturally governed by a second-order Heun equation with four singular points. Calling the object “Heun” incorrectly built an elliptic Picard-Fuchs interpretation into its name. “Hyperelliptic” states the intended geometric programme without pretending that the required identification has already been proved. M35c derives the SHM’s first-order departure from the AGM as an exact orbit series. It does not prove that the SHM solves sextics. A branch-point period machine cannot in general be supplied with the required pairing of six roots by radicals, because selecting one of the fifteen pairings carries a nonsolvable S_6 action. The paper therefore introduces the Inverter Conjecture: a rank-5/2 construction must compute the genus-two period matrix directly from symmetric coefficient data, such as the Igusa-Clebsch invariants. Only such an invariant-to-period inverter would make “rank 5/2 solves the sextic” a theorem internal to the Theory of Operations. Finally, the paper places Lambert W correctly within the polynomial hierarchy. For the root near one of n x^(n+1) - n x^n = z, the scaled displacement n(x-1) tends to W(z) as n tends to infinity. Lambert W is therefore an infinite-degree limit of a trinomial family whose degree-five member is related to the Bring-Jerrard quintic. It is a limit object, not a universal finite-degree radical. M35c replaces an unsupported formula-and-hyperradical theory with a narrower but rigorous programme: polynomial solvability is organised through modular level structures, period matrices, theta constants, and versal covers. The rank labels remain conjectural; the genus geometry and the radical-free recovery mechanism do not. Supersession notice M35 is obsolete: its septic hyperradical, normal form, Galois-group assignment, iteration, numerical roots, and physical applications are refuted. M35b is obsolete: its sextic hyperradical, alternating iteration, unit identifications, Heun bridge, applications, and bootstrap chain are refuted. The Galois-Rank seam section of M29a is obsolete: the formula fails and its supporting group and genus claims are corrected. M35c is the replacement source for the solvability ladder, the status of rank 5/2, and the corrected relationship among polynomial degree, genus, modular level structure, and operational rank. F02 remains a companion source: its rank-genus dictionary and corrected SHM period interpretation are not promoted beyond their stated conjectural status.