Spin-Stratum Type-Rigidity: An Obstruction to the Galois Action on the Binary Icosahedral Stratum, with Conditional Consequences for the Chiral Lift

The generation-split value ε=1/10 is often presented as a derived prediction of the spectral stratigraphy of the corpus. This note isolates its exact logical status. First, ε=1/10 is a theorem modulo the ADE case-selection gate: once a case is selected whose class-complete spectrum carries the external level ratio $3:2$, the generation-deficit normal form forces ε=1/10 and κ=5/12 algebraically, with no further input. Second, the first-principles status of ε is therefore exactly the first-principles status of the gate, which this note does not settle. We then examine a candidate arithmetic obstruction. In the compositum Kq=Q(ζq,√5) the Born–Infeld parity c↦ q-c, the restriction of complex conjugation ζq↦ζq⁻¹ (the candidate fibre involution of the foundations, taken here as a premise), fixes √5, whereas the Galois action σ:√5↦-√5, which exchanges the two two-dimensional irreducible characters of the binary icosahedral group 2I(2,5), lies in the other direct factor of the Galois group: the two are arithmetically orthogonal. A character-table audit of $2I$, computed from the group, proves that inner conjugations, the central element $-I$ and the projective involution J_Π induce the identity permutation of Irr(2I), whereas σ induces the permutation $(2\,2')(3\,3')$; under [H-WS] of the central reflection Rb acts as $-I$ does. The remaining generators of the spin stratum enter as stated clauses of different status: the Schur-projection and tensor-functor clauses are modelling assertions, and the A4-locking clause holds under the A4 hypotheses ([H-Gen], J_Π-covariance, a locked pair) and is also a modelling assertion. The completeness of this generator list is a modelling premise, not a proved fact. Modulo these clauses and that premise, no operation of the spin stratum realises σ. This is a statement about σ and not about the ADE gate: the outer automorphism fixes the order-four class and maps each of the order-five and order-ten loci to itself, so a link between the gate and σ is not established here, and no conclusion on the gate is drawn. The extension to a recursive tower of admissible fibre operations is conditional on the same modelling premise [M] (completeness of the list, which implies that every admissible operation is carrier-preserving) and, for the central reflection, on [H-WS]: recursive non-injectivity then multiplies the vertical parity type and cannot supply σ. The Lorentzian chiral lift (SL⊕ SR, mp(2,R)C₂(C)) is an input that this note does not construct; its status is that of the spin-solder hypothesis [H-Spin] of Q14, which no source supplies. Under [H-Spin], [H-WS], the modelling premise, the inherited central-phase convention of the chiral orientation note and a further explicit premise [B] (the lift supplies, beyond the spin-stratum operations, only the central character and the complexification unit i acting by complex conjugation; this is asserted, not derived, and the Galois group of Kq(i) does contain elements moving √5), the lift induces no action on Q(√5), so that the residual spin-Galois caveat of that note does not arise within the present spin stratum; the overlap ρ_χ=1 holds there by definition of σL, and the conclusion NA≠0 is conditional on all of these, in particular on [B].