The Gaussian code and four bit information space
Original ledger classification: Exact [E] SNF/census/semantics identities + [E, kill] three must-fail controls (exact arithmetic; Python-only per GATE.WOLFRAM.02).
Recorded statement: The Gaussian code bridge (review contract E8.GAUSSIAN.CODE.01) -- E8(Z[i])/(1+i) IS the information space of RM(1,3): the mu4 complex structure J makes E8 a rank-4 Z[i]-lattice L; reduction modulo the ramified Gaussian prime (1+i) gives…
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Paper context: Architecture, E8 and the electromagnetic fixed point
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E8.GAUSSCODE.01
Original ledger classification: Exact [E] SNF/census/semantics identities + [E, kill] three must-fail controls (exact arithmetic; Python-only per GATE.WOLFRAM.02).
Original claim text
The Gaussian code bridge (review contract E8.GAUSSIAN.CODE.01) -- E8(Z[i])/(1+i) IS the information space of RM(1,3): the mu4 complex structure J makes E8 a rank-4 Z[i]-lattice L; reduction modulo the ramified Gaussian prime (1+i) gives L/(1+i)L = F2^4 EXACTLY (one module, 26 checks, ~3 s, exact integer/Fraction arithmetic, sympy only for the Smith normal form; discovery probe gaussian_code_bridge_probe.py 26/26, verdict GAUSSIAN-CODE-BRIDGE-EXACT). THE QUOTIENT [E]: SNF exact; standard-model cross-check: basis |det| = 256 = 2^8 (doubled coordinates), SNF [1,1,1,1,2,2,2,2] -- presentation-independent. THE CENSUS [E]: 240 roots fall 15 x 16 over the nontrivial classes (the zero class PROVABLY EMPTY); each class = a union of 4 of the 60 G31 lines (60 = 15 x 4). THE SEMANTICS [E]: sigma = c^4 acts on the quotient as the RM(1,3) family permutation (3-cycle F1 -> F2 -> F3 + fixed anchor, exactly the v638 info-bit action); residual gauge = the order-18 centralizer of the family 3-cycle in GL(4, F2); the coordinate block = the sum-of-families label F1 + F2 + F3. MUST-FAIL CONTROLS [E, kill]: the naive placement keeps the 15 x 16 census (a property of (E8, J) alone) but the sigma-bit semantics becomes UNDEFINED (witness codeword lift off the lattice); a non-piJ placement has NO Z[i]-module; Z[i]^4 trivializes (16 minimal vectors, 11 classes empty). No marker move. [2026-08-03 LI: formally verified -- the algebraic core (Z[i]-module + SNF certificate with unimodular P/Q both directions, the 240-root census 15 x 16 with the zero class provably empty, mu4 stability, the sigma = order-3 family action with coordinate block = F1+F2+F3) is kernel-checked in Lean 4 (experiments/lean4-carrier-rigidity/TfptCarrier/GaussianCodeBridge.lean, lake build green, no sorry / native_decide).]
Original classification detail
GAUSSIAN-CODE-BRIDGE-EXACT: E8(Z[i])/(1+i) = F2^4 (SNF exact, presentation-independent), 240 = 15 x 16 census (zero class provably empty), 60 = 15 x 4, sigma = RM(1,3) family permutation (18 equivariant gauges), coordinate block = F1+F2+F3; all must-fail controls fire (naive placement, non-piJ placement, Z[i]^4).
Dependencies as recorded
E8.CODE.01; E8.CODESEM.01; E8.ST31.01; E8.ZIQUOTIENT.01; v626; v634; v638
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- Architecture, E8 and the electromagnetic fixed point
- Standard Model and flavor structure
- E8 audit and bootstrap
Sections that explicitly cite this claim
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- How to read this documenttfpt 1 architecture e8
- How to read this documentArchitecture, E8 and the electromagnetic fixed point
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- Front matter / supportStandard Model and flavor structure
- Front matter / supportE8 audit and bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- How to read this documenttfpt 1 architecture e8
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
Navigation through a shared verification script
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- Front matter / supportArchitecture, E8 and the electromagnetic fixed point
- Research Contract 2 — [formula in source]tfpt research contracts
- The Gaussian code bridge: [formula in source] over [formula in source] , the extended Hamming code, and a four-bit information layerE8 and the Gaussian code
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Front matter / supportE8 audit and bootstrap
- Research Contract 2 — [formula in source]tfpt research contracts
- 2026-10-18: Project separation and retained physics historyResearch changelog; TFPT — Introduction and reading guide
Registered verification modules
- v689_gaussian_code_bridge.pyRegistered verification module · core
Read the description recorded in the module registry
E8.GAUSSCODE.01: die Gauß-Codebrücke (Review-Vertrag E8.GAUSSIAN.CODE.01) — **E8(Z[i])/(1+i) ≅ F₂⁴ EXAKT (Smith-Normalform; Standardmodell: |det| = 256 = 2⁸, SNF [1⁴,2⁴] — präsentationsunabhängig): der 240 = 15×16-Census (Nullklasse BEWEISBAR leer, jede Klasse = 4 der 60 G31-Linien), σ = c⁴ wirkt auf dem Quotienten als RM(1,3)-Familienpermutation (3-Zyklus F1→F2→F3 + Fix-Anker, exakt die v638-Info-Bit-Aktion), Rest-Eichung = Zentralisator in GL(4,F₂) der Ordnung 18, Koordinatenblock = F1+F2+F3-Label**; alle must-fail-Kontrollen feuern: naive Platzierung behält den Census, verliert aber die σ-Semantik (Zeuge off-lattice); nicht-πJ-Platzierung: kein Z[i]-Modul; Z[i]⁴: trivialisiert (16 Vektoren, 11 Klassen leer); exakte Ganzzahl/Fraction-Arithmetik; kein Marker-Move (26 Checks, ~3 s)
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