The Coxeter and Euler completion identity
Original ledger classification: Proven finite statement [E] (Identity, exact over Z/Q via sympy) plus Numerical Euler-product checks (mpmath, X=10^5); fence/no-go carry NO marker upgrade; NO RH claim
Recorded statement: THE COXETER-EULER COMPLETION OF E8 [E] Identity + Numerical (round r617, promoted 2026-09-02; module v1019, re-derived, no probe imports). Exact: C=-S^{-1}S^T, det(I-xC)=Phi_30, Tr C=-1; Moebius…
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Paper context: TFPT — Introduction and reading guide
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E8.COXETER.EULER.COMPLETION.01
Original ledger classification: Proven finite statement [E] (Identity, exact over Z/Q via sympy) plus Numerical Euler-product checks (mpmath, X=10^5); fence/no-go carry NO marker upgrade; NO RH claim
Original claim text
THE COXETER-EULER COMPLETION OF E8 [E] Identity + Numerical (round r617, promoted 2026-09-02; module v1019, re-derived, no probe imports). Exact: C=-S^{-1}S^T, det(I-xC)=Phi_30, Tr C=-1; Moebius Phi_30=(1-x^2)(1-x^3)(1-x^5)(1-x^30)/[(1-x)(1-x^6)(1-x^10)(1-x^15)] (classical cyclotomic Euler-product inversion); U=1 oplus C, det(I-xU)=(1-x)Phi_30=1-x^2-x^3+x^6+x^7-x^9, Tr U=0. Global Z_C(s)=zeta(2s)zeta(3s)zeta(5s)zeta(30s)/[zeta(s)zeta(6s)zeta(10s)zeta(15s)] and D_E8(s)=zeta(6s)zeta(10s)zeta(15s)/[zeta(2s)zeta(3s)zeta(5s)zeta(30s)], absolutely convergent for Re s>1/2. Numerical (mpmath, cutoff X=10^5): D_E8 residual 4.76e-4 at s=0.75, 4.16e-12 at s=1.5. CLASS: the vanishing linear term is the generic consequence of Tr C=-1 (controls: scalar -1 -> 1/zeta(2s); A1 -> 1/zeta(2s); A2 -> 1/zeta(3s); E6 (Phi_3 Phi_12) -> zeta(4s)zeta(6s)/[zeta(2s)zeta(3s)zeta(12s)]; negative control B2 (Phi_4, trace 0) -> zeta(2s)/[zeta(s)zeta(4s)], still contains 1/zeta(s)); E8 only selects the divisor set of 30. Beurling-generic: the identity holds for any set of primes Q subset (1,inf) -- no prime-specific input. FENCE (verbatim, no marker upgrade): The trace-free completion is zero-free and pole-free in Re s > 1/2. The splitting into the scalar zeta channel and the Coxeter channel is open and RH-equivalent. No RH claim. Typed no-go E8.COXETER.REGULARIZED_SPLIT.NO_GO.01: any construction controlling only the det_2/Hilbert-Schmidt part cannot isolate 1/zeta(s); the RH content sits in the analytic continuation of the trace term P(s)=Sum_k mu(k)/k log zeta(k s). The analytic identities are classical; the new statement is the E8 reading with the fence. Like the Eisenstein bridge, RH-neutral as a finite identity. NO RH CLAIM.
Original classification detail
[E] Identity: det(I-xC)=Phi_30, Tr C=-1, U=1 oplus C, Z_C and D_E8 zeta quotients (classical Moebius); Numerical Euler products at X=10^5 (residuals 4.76e-4 at s=0.75, 4.16e-12 at s=1.5); class statement + Beurling-genericity; fence verbatim (no marker upgrade); NO-GO E8.COXETER.REGULARIZED_SPLIT.NO_GO.01 on det_2; NO RH claim
Dependencies as recorded
v1019
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- TFPT — Introduction and reading guide
- Frontier and open physical interfaces
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- 4. Vorarbeiten und aktuelle Grenze der QuellenprüfungTFPT Universalraum Quellkomposition Konsolidierung 2026-09-15 v1.6.9
- The hard reduction: the live residual as typed interfacesTFPT — Introduction and reading guide
- 4. Vorarbeiten und aktuelle Grenze der QuellenprüfungTFPT Universalraum Spinlift Hauptdokument 2026-09-15 v1.6.10
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 4. Vorarbeiten und aktuelle Grenze der QuellenprüfungTFPT Universalraum Richtige Kette Konsolidierung 2026-09-15 v1.6.7
- 4. Vorarbeiten und aktuelle Grenze der QuellenprüfungTFPT Universalraum Gemeinsamer Prozess Konsolidierung 2026-09-15 v1.6.8
- Full quantum gravity — induced from the boundary kernelFrontier and open physical interfaces
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- 2026-10-18: Project separation and retained physics historyResearch changelog; TFPT — Introduction and reading guide
Registered verification modules
- v1019_coxeter_euler_completion.pyRegistered verification module · core
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E8.COXETER.EULER.COMPLETION.01 [E]+Numerical (2026-09-02, Runde r617): Coxeter-Euler-Vervollstaendigung -- neu abgeleitet. Exakt: det(I-xC)=Phi_30, Tr C=-1; Moebius Phi_30=(1-x^2)(1-x^3)(1-x^5)(1-x^30)/[(1-x)(1-x^6)(1-x^10)(1-x^15)]; U=1 oplus C, Tr U=0. Global Z_C und D_E8 als Zeta-Quotienten, abs. konv. Re s>1/2. Numerisch (X=10^5): D_E8 Residual 4.76e-4 bei s=0.75, 4.16e-12 bei s=1.5. Klasse: lineares Verschwinden generisch aus Tr C=-1; E8 waehlt nur die Teiler von 30. Beurling-generisch. ZAUN: The trace-free completion is zero-free and pole-free in Re s > 1/2. The splitting into the scalar zeta channel and the Coxeter channel is open and RH-equivalent. No RH claim. NO-GO E8.COXETER.REGULARIZED_SPLIT.NO_GO.01: det_2 isoliert nicht 1/zeta(s). Python-only / Wolfram zurueckgestellt.
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