E8 Spin8 Triality Alt9 Certificate
Author: Bernhard Mueller
First-party HTML and PDF publication page for E8 Spin8 Triality Alt9 Certificate in the OPH paper stack.
Section jump
Paper release: r1519
Released: July 8, 2026
Role in the OPH Stack
OPH uses finite observer-visible data, records, repair maps, quotient invariants, and public receipts as primitive evidence carriers. The present certificate belongs to that finite exact discipline on the exceptional-symmetry side: \[ \mathrm{Spin}(8)\ \text{triality} + E_8\ \text{lattice preservation} \quad\Rightarrow\quad \text{finite exceptional representation certificate}. \] It supports the \(E_8\)-type representation-closure lane used by compact-gauge and heterotic-local bookkeeping. It does not replace the recovered-core theorem stack, the MAR selection of the Standard Model quotient, the critical-edge CFT gate, or any physical carrier evidence rule.
Certificate Statement
Theorem 1 (Finite \(E_8/\mathrm{Spin}(8)\) triality certificate). The certificate target is an exact finite matrix construction with the following data.
An \(A_8\) root subsystem inside the \(E_8\) root lattice. Its Weyl group is \(W(A_8)\cong\mathrm{Sym}(9)\), and the even subgroup is \(\mathrm{Alt}(9)\).
The permutation involution \((12)(34)\in\mathrm{Alt}(9)\) lifts to \(\mathrm{Spin}(8)\) with square \(-1\). Hence the preimage of \(\mathrm{Alt}(9)\) is the nonsplit Schur double cover \(2\!\cdot\!\mathrm{Alt}(9)\), not \(2\times\mathrm{Alt}(9)\).
Under the positive half-spin representation \(\Delta^+\), the lifted group preserves an even unimodular determinant-one lattice, hence an \(E_8\) lattice. The spin-side copy therefore lands in \(\mathrm{Aut}(E_8)=W(E_8)\).
The vector and positive-half-spin mod-2 orbit fingerprints on \(E_8/2E_8\setminus\{0\}\) are different: \[ \mathrm{Alt}(9)_{\mathrm{vec}}:\{9,36,84,126\}, \qquad (2\!\cdot\!\mathrm{Alt}(9))_{\Delta^+}:\{120,135\}. \] Thus the two copies are not conjugate in \(O_8^+(2)\).
The outer triality automorphism of \(\mathrm{Spin}(8)\) permutes the vector and two half-spin eight-dimensional representations and identifies these otherwise nonconjugate presentations.
Remark 1 (Notation). Here \(A_8\) denotes the root subsystem and \(\mathrm{Alt}(9)\) denotes the alternating group. The shorthand \(A_9\) is avoided for the group because \(A_9\) also denotes a root system.
Remark 2 (Claim boundary). This is a finite algebraic certificate. It supports exceptional representation-closure and triality bookkeeping. It does not prove OPH, derive the Standard Model quotient, prove physical \(E_8\) realization, close the heterotic edge CFT, or count as a hardware receipt.
Remark 3 (Public receipt gate). For public verifier status,
the repository bundle must include the Sage source, exact matrix data,
lattice bases, mod-2 orbit computation, stdout or machine-readable check
receipts, and stable hashes under code/e8_triality/. Until
that bundle is populated, this note records the certificate statement
and its OPH claim placement rather than a standalone public reproduction
bundle.
Remaining Extension
This certificate concerns the nonsplit \(2\!\cdot\!\mathrm{Alt}(9)\) subgroup and its triality-fused vector and positive-half-spin presentations. The full Griess–Lam \(2.\mathrm{Sym}(9)\) double-cover construction requires adjoining odd Clifford lifts. Those odd lifts exchange the two half-spin modules and carry the associated \(\sqrt 2\)-normalization. That is a separate certificate gate.