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 and what is selected.
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.
Second, the first-principles status of $\varepsilon$ is therefore exactly the first-principles status of the gate. The note characterises the obstruction to deriving the gate as an arithmetic orthogonality: the selection of the $\sqrt{5}$-locus carrying the $3{:}2$ ratio requires the nontrivial element of $\mathrm{Gal}(\mathbb{Q}(\sqrt{5})/\mathbb{Q})$, the action $\sqrt{5} \mapsto -\sqrt{5}$. The only field automorphism supplied by the foundations is the Born–Infeld parity $c \mapsto q - c$ — the restriction of complex conjugation $\zeta_q \mapsto \zeta_q^{-1}$ — which fixes $\sqrt{5}$.
In the compositum $K_q = \mathbb{Q}(\zeta_q, \sqrt{5})$ the two actions lie in distinct direct factors of the Galois group, hence are arithmetically orthogonal. Parity non-injectivity is not a partial version of the required action — it is orthogonal to it.
Core contributions
- Conditional ADE split (Theorem). If the gate selects the central-lift class over the $\sqrt{5}$-locus (the parity class over the $A_5$ five-cycle 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. The selection of the $\sqrt{5}$-locus that carries the $3{:}2$ ratio requires the nontrivial element of $\mathrm{Gal}(\mathbb{Q}(\sqrt{5})/\mathbb{Q})$. The available foundations supply only the Born–Infeld parity, the restriction of complex conjugation, which fixes $\sqrt{5}$ pointwise.
- Central-lift tie-break and negative theorem (unconditional). In the compositum $K_q = \mathbb{Q}(\zeta_q, \sqrt{5})$ the two actions lie in distinct direct factors of the Galois group — they are arithmetically orthogonal. For the present corpus construction the no-go is unconditional: by Proposition (spin-stratum type-rigidity), the operations available at the spin stratum — inner conjugations, $-I$, $R_b = W(-I)$, $J_\Pi$, tensor functors of $\mathrm{Rep}(2I)$, Schur projection, A4 locking — each fix $\mathbb{Q}(\sqrt{5})$ pointwise. Their generated group lies in the $\sqrt{5}$-fixing subgroup of $\mathrm{Gal}(K_q/\mathbb{Q})$, and by closure cannot manufacture the outer automorphism that realises $\sqrt{5} \mapsto -\sqrt{5}$.
- Spin-stratum type-rigidity (Proposition, proved). Established by a bounded, reproducible character-table audit of $2I \cong \mathrm{SL}(2,5)$: the conjugacy-class structure is reconstructed from the group, the character table recovered by simultaneous diagonalisation of the class-sum matrices, validated against orthogonality, and the action of each spin-stratum operation on $\mathrm{Irr}(2I)$ is checked explicitly. The Galois action $\sqrt{5} \mapsto -\sqrt{5}$ induces the permutation $(2\,2')(3\,3')$; none of the available operations realises it.
- Full-tower type-rigidity (Lemma, open). Whether every conceivable recursive stratum above the base inherits the parity type, rather than introducing an action in the $\sqrt{5}$ factor, is open. A proof direction via entropy-monotonicity is indicated. This abstract generalisation is broader than the ADE gate requires, and Theorem (negative) does not depend on it.
- Chiral-lift corollary (Corollaries, proved). The same spin-stratum type-rigidity cuts the opposite way for the Lorentzian chiral lift of the ordered cascade step. The boost is Galois-internal (it extends the spin-stratum action only by the central character and the analytic complexification, both fixing $\sqrt{5}$), and the chiral weight $\sigma_L$ classes the Weyl branches $S_L$ vs. $S_R$ by complex-conjugation type ($\mathbf{2}$ vs. $\bar{\mathbf{2}}$), orthogonal to the outer automorphism separating $2$ from $2'$. Hence the lift induces no action on $\sqrt{5}$: it discharges the residual obstruction of the chiral orientation note and closes $[\mathrm{H\text{-}orient}]$ with $N_A = \theta_{\max} \neq 0$ unconditionally within the present spin stratum. This closes the frontier amplitude datum $N_A \neq 0$, not the spectral split $u \neq 0$, whose promotion remains the Schur transport.
- 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. For the present corpus construction, the gate is a no-go: by closure of the $\sqrt{5}$-fixing subgroup of the Galois group, no iteration, central lift, Schur projection, or A4 locking built from the available spin-stratum operations can manufacture the outer automorphism that the $\sqrt{5}$-locus requires. The obstruction is a closure argument, not an unresolved computation.
Settled (this version). The same closure argument, applied to the Lorentzian chiral lift, discharges the residual obstruction of the chiral orientation note: the lift induces no action on $\sqrt{5}$, so $[\mathrm{H\text{-}orient}]$ is closed and $N_A \neq 0$ unconditionally within the present spin stratum. The gate no-go and the chiral closure are opposite cuts of one spin-stratum type-rigidity.
Not settled. The abstract full-tower type-rigidity: whether some other, richer construction could supply at a recursive stratum a non-injectivity acting on the $\sqrt{5}$ factor. ENI's recursive non-injectivity constrains existence, not arithmetic type; a type-inheritance argument is indicated but not concluded. This is broader than the ADE gate requires, and the negative theorem does not depend on it.
Position in the programme
This note belongs to the fermionic matter sub-programme (Presentation Note 6). It complements the A4-Note (which fixes the amplitude mechanism as Born–Infeld saturation on the corpus-derived real cascade) 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 chiral-lift corollary it also closes the residual obstruction of the chiral orientation note, the gate no-go and the chiral closure being opposite cuts of a single spin-stratum type-rigidity.
References
Jérôme Beau. Spin-Stratum Type-Rigidity: Orthogonality of the ADE Gate and Closure of the Chiral Lift. Working paper, 2026. 10.5281/zenodo.20693084