E₈ from the extended Hamming code
Original ledger classification: Exact [E] code/lattice identities + [C] typed compiler ties (integer combinatorics; Python-only per GATE.WOLFRAM.02).
Recorded statement: The E8 code round (one module, 7 checks, ~2 s; discovery probe e8_code_probe.py 7/7, verdict E8-IS-A-CODE). Construction A on the self-dual extended Hamming code [8,4,4] yields the even unimodular rank-8 lattice = E8 (shells 240/2160 = the…
Statement, assumptions and sources
Paper context: input 5 tfpt 3 e8 audit bootstrap
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E8.CODE.01
Original ledger classification: Exact [E] code/lattice identities + [C] typed compiler ties (integer combinatorics; Python-only per GATE.WOLFRAM.02).
Original claim text
The E8 code round (one module, 7 checks, ~2 s; discovery probe e8_code_probe.py 7/7, verdict E8-IS-A-CODE). Construction A on the self-dual extended Hamming code [8,4,4] yields the even unimodular rank-8 lattice = E8 (shells 240/2160 = the v625 theta); every single-bit error is exhaustively correctable (16 x 8, unique nearest codewords); the compiler ties (8 = rank, 4 = d = |mu4|, 16 = half-spinor, 14 = 2 x 7 parabolic) are typed observations. [2026-08-02 XLI: the code reading sharpens to an exact dictionary -- v638: the naive v626 placement itself is DEAD for the family 3-cycle (only 2/30 placements carry both compiler symmetries), and the unique equivariant placement C* is Reed-Muller RM(1,3) on AG(3,2) with information bits one per mu4 pair (3 families + 1 anchor), a syndrome flag carrying the sigma 3-cycle, and decode = projection verbatim; the v626 theorems are placement-independent, the SEMANTICS is not; see E8.CODESEM.01.] [2026-08-02 XLV: v664 types 14 = A_4 = 2 x 7 parabolics and 16 = |C| as FORCED (mathematically determined by the unique self-dual [8,4,4] code -- no free coincidence mass; their value is the theorem, not the number); LEAN ROUND 3: the code layer is machine-certified (HammingCode.lean: self-duality, weight distribution {0:1, 4:14, 8:1}, coset-leader weight distribution {0:1, 1:8, 2:7} with covering radius 2, all by kernel decide on explicit witnesses); lake build green.]
Original classification detail
E8-IS-A-CODE: Construction A(Hamming [8,4,4]) = E8 exactly; single-bit errors correctable; the error-correction reading is a theorem; no physics claim, no marker moves.
Dependencies as recorded
PRIME.SHADOW.01; EXTREV.LATTICE.01; v1; v625
Open the original ledger source, row 641Location recorded in the ledger
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- E8 audit and bootstrap
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- Front matter / supportE8 audit and bootstrap
- Front matter / supportE8 audit and bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
Navigation through a shared verification script
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- Scope and outlookE8 and the Gaussian code
- 2026-10-18: Project separation and retained physics historyResearch changelog; TFPT — Introduction and reading guide
Registered verification modules
- v626_e8_code.pyRegistered verification module · core
Read the description recorded in the module registry
E8.CODE.01: die E8-Code-Runde — **E8 IST EIN FEHLERKORRIGIERENDER CODE** (Theorem, nicht Metapher). [E] Der erweiterte Hamming-Code [8,4,4]: 16 Codewörter, Gewichte {0:1, 4:14, 8:1}, selbst-dual; Construction A liefert ein gerades unimodulares Rang-8-Gitter = E8 (Eindeutigkeit); Schalen 240/2160 = v625-Theta; Einzelbit-Fehler exhaustiv korrigierbar (16×8 Fälle, eindeutige nächste Codewörter). [C] Compiler-Bezüge getypt: 8 = Rang, 4 = d = |μ₄|, 16 = Halbspinor, 14 = 2×7 (Parabolik-Dimension) (7 checks, ~2 s)
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