A Galois Group of Order 5^649: The Fourth Graded Layer Is Longitude-Blind

Abstract

Let K = Q(ζ₅) and let G be the Galois group of the maximal pro-5 extension of K unramified outside the five places above 5 and 11. The preceding paper determined G/λ₄(G), of order 5¹³⁹, from four Koch relators and a fully named arithmetic Frobenius longitude. This note determines the next layer. The fourth graded piece gr₄(G) has dimension 510, the quotient G/λ₅(G) has order 5⁶⁴⁹, a 454-digit number, and the degree-four relator ideal is the span of 234 explicit columns of rank 218 inside the 728-dimensional layer gr₄(F) of the free group. The load-bearing result is a blindness theorem: because the 32 cubic generators of the ideal have vanishing syzygy module, every degree-four column is a function of the four quadratic relation rows alone. The Frobenius longitude, which at class three was the entire arithmetic content, does not enter; above the class-two centre the tower absorbs no new arithmetic at the graded level, and gr₄(G) is already determined by the class-two certificate. The action of Gal(K/Q) on gr₄(F) is semisimple with isotypic multiplicities (179, 186, 186, 177), splitting as (54, 55, 55, 54) on the ideal and (125, 131, 131, 123) on the quotient; none of the three modules is free. At the second full-basis prime q = 61 every one of these invariants is reproduced identically, while the two ideals are transverse to the maximal extent allowed: all 234 columns differ, and the intersection of the two 218-dimensional ideals — generically zero in dimension 728 — is exactly 16-dimensional, a free F₅[Gal(K/Q)]-module of rank four supported away from the triple-power block. The two towers first become distinguishable inside one shared graded module. Every constant is an exact machine-verified integer computation.