Overview
The generation-split value $\varepsilon = 1/10$ is often presented as a derived prediction of the spectral stratigraphy of the corpus. This note draws the line precisely between what is forced, what is proved and what is only conditional.
First, $\varepsilon = 1/10$ 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 $\varepsilon = 1/10$ and the common scale $\kappa = 5/12$ algebraically, with no further input. The first-principles status of $\varepsilon$ is therefore exactly that of the gate, which this note does not settle.
Second, the note examines a candidate arithmetic obstruction, and the result it proves is an obstruction concerning the Galois automorphism $\sigma : \sqrt{5} \mapsto -\sqrt{5}$, not a result on the gate. In the compositum $K_q = \mathbb{Q}(\zeta_q, \sqrt{5})$ the Born–Infeld parity $c \mapsto q - c$, the restriction of complex conjugation $\zeta_q \mapsto \zeta_q^{-1}$, fixes $\sqrt{5}$ (proved) and lies in a distinct direct factor of the Galois group from $\sigma$; its status as the base-fibre involution is that of a candidate hypothesis of the foundations, which the note takes as a premise. A character-table audit of the binary icosahedral group $2I \cong \mathrm{SL}(2,5)$, computed from the group, proves that inner conjugations, the central element $-I$ and the projective involution $J_\Pi$ act trivially on $\mathrm{Irr}(2I)$, whereas $\sigma$ induces the permutation $(2\,2')(3\,3')$.
The other generators of the spin stratum enter as clauses of different status. The central reflection $R_b$ acts as $-I$ under [H-WS] (the scalar $-1$ on the chiral carrier, supplied by no source); the A4 locking contributes only a parity-odd sign under the A4 hypotheses; the tensor-functor and Schur-projection clauses are modelling assertions that are not audited. The completeness of this generator list is itself a modelling premise [M], which implies that every admissible operation is carrier-preserving. Modulo these clauses and that premise, no listed operation of the spin stratum realises $\sigma$.
No link between the gate and $\sigma$ is established. The outer automorphism fixes the order-four class and maps each of the order-five and order-ten loci onto itself, so $\sigma$ would be needed to tell the two order-five classes apart, not to choose between the order-four and the order-five locus. The gate therefore stays open, and no obstruction to the gate and no no-go for the gate follows from this note.
A status sharpening makes the gate concrete as a generator-set fork inside $2I$. Once the order-five $\sqrt{5}$-locus is selected, the ratio $3{:}2$ is not a fit but a forced Cayley-spectral invariant ($R = 3/2$, outer deficits $\{1/6, 1/4\}$); the neutral order-four convention instead gives $R = 5/3$ and $\varepsilon = 1/8$, not $1/10$. The remaining lock is the selection of the order-five locus over the neutral order-four one, which is not a harmless convention change and which this note leaves open.
Downstream mass-split gate. The same datum governs the projected-Yukawa mass split. The order-five Cayley ratio $R_{\mathrm{ADE}} = 3/2$ gives $\varepsilon = (R-1)/(2(R+1)) = 1/10$, and the very same formula rewrites the free diagonal split parameter $u$ through $R_\Pi = (\tfrac12+u)/(\tfrac12-u)$, so that $\varepsilon(R_\Pi) = u$ identically. The equality $u = \varepsilon = 1/10$ is exactly the dictionary identification $R_\Pi = R_{\mathrm{ADE}}$: the representation $\mathrm{Sym}^2(\mathbb{C}^2)$ leaves $u$ free, and the recursion of admissible operations (carrier-preserving by [M], and under [H-WS] for $R_b$) does not supply $\sigma$, no link between $\sigma$ and the selection of the order-five value being established.
Core contributions
- Conditional ADE split (Theorem). If the gate selects the central-lift class over the $\sqrt{5}$-locus (whose class-complete Cayley spectrum has $|\lambda_3 - 1|/|\lambda_1 - 1| = 3/2$), then the generation-deficit normal form forces $\varepsilon = 1/10$ and $\kappa = 5/12$ algebraically. The lemma is general and case-independent: all of the first-principles content lives in the upstream selection of the case.
- Arithmetic organising lemma (proved). In the compositum $K_q = \mathbb{Q}(\zeta_q, \sqrt{5})$ the Born–Infeld parity (restriction of complex conjugation) fixes $\sqrt{5}$ (proved) and lies in a distinct direct factor of the Galois group from $\sigma$; that this parity is the base-fibre involution is a candidate hypothesis of the foundations, taken as a premise.
- Character-table audit (proved). Computed from the group, inner conjugations, $-I$ and $J_\Pi$ induce the identity permutation of $\mathrm{Irr}(2I)$, whereas $\sigma$ induces $(2\,2')(3\,3')$.
- Spin-stratum type-rigidity (Proposition; clauses of different status). (a) the audit above; (b) $R_b$ acts as $-I$ under [H-WS]; (c) the A4 locking contributes only a parity-odd sign under the A4 hypotheses ([H-Gen], $J_\Pi$-covariance, diagonal form, a locked pair); (d) tensor functors and the Schur projection fix $\mathbb{Q}(\sqrt{5})$ as modelling assertions, not audited. Relative to the modelling premise [M] that the generator list is complete (which implies carrier-preservation of every admissible operation), no listed operation realises $\sigma$.
- No link between the gate and $\sigma$ (not established). The outer automorphism fixes the order-four class and preserves each order-five and order-ten locus; the gate stays open.
- Full-tower statement (relative to [M] and [H-WS]). Under [M], every admissible fibre operation is carrier-preserving; since $R_b$ belongs to the list that [M] makes complete, the statement inherits [H-WS]. A finite recursive tower then multiplies the vertical parity type and cannot supply $\sigma$. The premise [M] is not proved, and the statement is not unconditional.
- Chiral-lift consequences (conditional). 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 the note does not construct: it is the spin-solder hypothesis [H-Spin] of Q14, supplied by no source. Under [H-Spin], [H-WS], [M], the boost premise [B] and the inherited central-phase convention of the chiral orientation note, the lift induces no action on $\mathbb{Q}(\sqrt{5})$, so the residual spin-Galois caveat of that note does not arise within the present spin stratum; $\rho_\chi = 1$ holds there by definition of $\sigma_L$, and $N_A \neq 0$ is conditional on all of these. This concerns the frontier amplitude datum $N_A$, not the spectral split $u \neq 0$.
- Downstream consequence. Observable masses route through the explicit operator $E_\Pi$ and the Yukawa sector regardless of how the gate is resolved.
What this note settles, and what it does not
Settled. The value $\varepsilon = 1/10$ is forced, by elementary algebra on the generation-deficit normal form, as soon as the case-selection gate produces the external $3{:}2$ ratio. The common scale $\kappa = 5/12$ follows on the same step.
Settled (audit). Inner conjugations, $-I$ and $J_\Pi$ act trivially on $\mathrm{Irr}(2I)$, while the Galois automorphism $\sigma$ induces $(2\,2')(3\,3')$.
Conditional. No listed spin-stratum operation realises $\sigma$, under [H-WS] for $R_b$, the A4 hypotheses for the locking, modelling assertions for the tensor functors and the Schur projection, the modelling premise [M] that the generator list is complete, and the candidate base-fibre Born–Infeld parity taken as a premise. The extension to a recursive tower is relative to [M] and, for $R_b$, to [H-WS].
Conditional (chiral lift). Taking the Lorentzian chiral lift as an input ([H-Spin] of Q14, supplied by no source and not constructed here), and under [H-WS], [M], the boost premise [B] and the inherited central-phase convention of the chiral orientation note, the lift induces no action on $\sqrt{5}$, so that the residual spin-Galois caveat of that note does not arise within the present spin stratum, with $\rho_\chi = 1$ by definition of $\sigma_L$.
Not established. That the ADE gate requires $\sigma$. No link between the gate and the Galois action is established, so the first-principles status of the gate, and hence of $\varepsilon = 1/10$, is unchanged: it remains open. The completeness premise [M], the boost premise [B] (the lift supplies, beyond the spin-stratum operations, only the central character and the complexification unit, acting by complex conjugation; it supplies no automorphism of $K_q(\mathrm{i})$ moving $\sqrt{5}$ although the Galois group contains such elements; asserted, not derived) and the hypotheses [H-WS] and [H-Spin] are not proved.
Position in the programme
This note belongs to the fermionic matter sub-programme (Presentation Note 6). It complements the A4-Note (which states the saturation margin of a supplied two-form and selects no amplitude mechanism) and the PRS reduction by locating where the residual quantitative content of the split value comes from, and why it is not yet a first-principles output. Through its conditional chiral-lift consequences, under [H-Spin] (as input), [H-WS], [M], the boost premise [B] and the inherited central-phase convention, it also bears on the residual caveat of the chiral orientation note.
References
Jérôme Beau. Spin-Stratum Type-Rigidity: An Obstruction to the Galois Action on the Binary Icosahedral Stratum, with Conditional Consequences for the Chiral Lift. Working paper, 2026. 10.5281/zenodo.20693084