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 $\epsilon=\tfrac{1}{10}$ is often presented as a derived prediction of the spectral stratigraphy of the corpus. This note isolates its exact logical status. First, $\epsilon=\tfrac{1}{10}$ 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 $\epsilon=\tfrac{1}{10}$ and $\kappa=\tfrac{5}{12}$ algebraically, with no further input. Second, the first-principles status of $\epsilon$ 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 $K_q=\mathbb{Q}(\zeta_q,\sqrt{5})$ the Born–Infeld parity $c\mapsto q-c$, the restriction of complex conjugation $\zeta_q\mapsto\zeta_q^{-1}$ (the candidate fibre involution of the foundations, taken here as a premise), fixes $\sqrt{5}$, whereas the Galois action $\sigma:\sqrt{5}\mapsto-\sqrt{5}$, which exchanges the two two-dimensional irreducible characters of the binary icosahedral group $2I\cong\mathrm{SL}(2,5)$, 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 $J_\Pi$ induce the identity permutation of $\mathrm{Irr}(2I)$, whereas $\sigma$ induces the permutation $(2\,2')(3\,3')$; under [H-WS] of the central reflection $R_b$ 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], $J_\Pi$-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 $\sigma$. This is a statement about $\sigma$ 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 $\sigma$ 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 $\sigma$. The Lorentzian chiral lift ($S_L\oplus S_R$, $\mathfrak{mp}(2,\mathbb{R})_{\mathbb{C}}\simeq\mathfrak{sl}_2(\mathbb{C})$) 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 $\mathrm{i}$ acting by complex conjugation; this is asserted, not derived, and the Galois group of $K_q(\mathrm{i})$ does contain elements moving $\sqrt{5}$), the lift induces no action on $\mathbb{Q}(\sqrt{5})$, so that the residual spin-Galois caveat of that note does not arise within the present spin stratum; the overlap $\rho_\chi=1$ holds there by definition of $\sigma_L$, and the conclusion $N_A\neq0$ is conditional on all of these, in particular on [B].