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.
A status sharpening makes the gate concrete as a generator-set fork inside the binary icosahedral group $2I$. 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 therefore the selection of the order-five locus over the neutral order-four one — not a harmless convention change — and the tempting fermionic shortcut does not supply it, the spinorial grading read by the chiral lift being common to both two-dimensional irreducibles and inducing no action on $\mathbb{Q}(\sqrt{5})$.
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 split parameter $u$ and the angular datum $\varepsilon$ are therefore one functional of a single ratio, and 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 carrier-preserving recursion cannot supply the order-five value. Same functional, distinct carriers, value-level equality only through the ADE dictionary.
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, Proposition, Corollary; established for admissible fibre operations). Whether every conceivable recursive stratum above the base inherits the parity type is settled in the corpus notion of an admissible fibre operation. The correct invariant is the Galois-module type of the fibre identification, not its entropy (which records only the multiplicity of identifications). A realisation audit shows the horizontal Galois exchange $\sqrt{5} \mapsto -\sqrt{5}$ does not preserve the binary-icosahedral carrier — it maps $2I$ onto its opposite-chirality copy, exchanging $2 \leftrightarrow 2'$ — whereas the Born–Infeld parity (complex conjugation) preserves the carrier and fixes $\sqrt{5}$. Hence every carrier-preserving admissible fibre identification is vertical, and a finite recursive tower preserves the binary-icosahedral arithmetic type. This is a scope hypothesis on what counts as an admissible operation, not an absolute impossibility of any external equivalence of character tables; the negative theorem 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.
Settled (this version), in a precise sense. The abstract full-tower type-rigidity is established for admissible fibre operations. ENI's recursive non-injectivity constrains existence, not arithmetic type; the correct invariant is the Galois-module type of the fibre identification. A realisation audit shows the horizontal exchange $\sqrt{5} \mapsto -\sqrt{5}$ does not preserve the binary-icosahedral carrier, whereas every admissible (carrier-preserving) identification does and fixes $\sqrt{5}$, so a finite recursive tower preserves the arithmetic type. The former open dependency lock is thus replaced by a scope hypothesis — carrier-preserving admissible fibre operation — not a closure by fiat: the result does not assert that no external equivalence of abstract character tables exists, only that the horizontal Galois exchange is not realised by an admissible operation preserving the binary-icosahedral carrier. 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