A registered result in its source context
Original ledger classification: Exact [E] reality/uniqueness/must-fail structure + [E-float] PSD Grams (N-stable) + [C] typed continuum remainder (Python-only per GATE.WOLFRAM.02).
Recorded statement: Parafermionic Klein-twist reflection positivity of the Z3 twist sector (one module, 17 checks, ~30 s; discovery probe parafermion_klein_rp_probe.py 17/17, verdict KLEIN-RP-POSITIVE). The named open item of QGEO.TWISTOPE.01 (the naive…
Statement, assumptions and sources
Paper context: input 5 tfpt 3 e8 audit bootstrap
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QGEO.KLEINRP.01
Original ledger classification: Exact [E] reality/uniqueness/must-fail structure + [E-float] PSD Grams (N-stable) + [C] typed continuum remainder (Python-only per GATE.WOLFRAM.02).
Original claim text
Parafermionic Klein-twist reflection positivity of the Z3 twist sector (one module, 17 checks, ~30 s; discovery probe parafermion_klein_rp_probe.py 17/17, verdict KLEIN-RP-POSITIVE). The named open item of QGEO.TWISTOPE.01 (the naive antiunitary Z3 pairing fails positivity N-stably at -4.5e-3) is RESOLVED at the lattice OS level: the positivity-carrying pairing composes the mirror with CHARGE CONJUGATION -- Theta_K(sigma_k(x)) = eta_k sigma_{3-k}(r(x)) with omega^{k q} zero-mode phases (q = half-position mod 3). MACHINERY [E]: the dressed-string reality identity (particle-hole I - C = DCD at half filling) makes every Klein Gram entry EXACTLY real (1.4e-15 over 40 mixed configs); for lam = pi the naive and Klein pairings coincide identically (the v639 Z2 control could not distinguish them); the naive glue pins a spurious 2 lam = 4 pi/3 defect on the axis bond, the Klein glue is defect-free. POSITIVITY [E-float, central]: every charged Klein block is real symmetric AND PSD (worst min/max 2.9e-12), k = 1, 2, BOTH diameter axes, N = 48/96/192/384 (N-stable, no drift); charge blocks coincide exactly. UNIQUENESS [E]: over the 36 pairs (eta_1, eta_2) in {+-1, +-w, +-w^2}^2 exactly ONE satisfies consistency + Hermiticity + PSD: eta = (1, 1); no phase rescues the naive pairing (the fix is structural). MUST-FAIL [E]: the naive pairing reproduces the v639 violation (-4.4e-3..-4.5e-3 N-stable); eta = -1 loses PSD, eta = omega breaks Hermiticity. THE LATTICE OS STATEMENT [E-float]: the FULL mixed Gram over {1, sigma_1, sigma_2, dipoles} with NO sector projection is real symmetric and PSD at N = 48/96 (min/max -5e-17), stable at N = 192. HONEST TYPING [C]: continuum OS reconstruction, R/fermion-parity sectors (deck^3 = -1), modular invariance, chiral factorization, non-abelian content stay NAMED, not claimed; GATE.QGEO does not move. [2026-08-02 XLV: the modularity/assembly successors land -- v650 (S measured with a clean N^{-4} rate, T exact), v651 (the seam assembly is B, matching this row's Klein phase eta = (1,1)), v652 (Arf hardening; the half-twist tau sector is Klein-PSD too), v653 (bond defect = lattice theorem); the orbifold front is complete at the lattice level; GATE.QGEO does not move.]
Original classification detail
KLEIN-RP-POSITIVE: the Z3 twist sector is reflection positive on the seam-double lattice under the parafermionic Klein-twist pairing (mirror x charge conjugation, omega^{k q} phases, eta = (1,1) unique among 36), N-stable 48..384, both axes, including the full mixed OS Gram; the naive pairing keeps its N-stable violation (must-fail); continuum OS/modularity typed open; GATE.QGEO does not move.
Dependencies as recorded
QGEO.TWISTOPE.01; QGEO.ORBCAS.01; QGEO.COVERSEAM.01; v519; v622; v623; v628; v639
Open the original ledger source, row 649Location recorded in the ledger
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- E8 audit and bootstrap
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- Front matter / supportE8 audit and bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
Navigation through a shared verification script
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- Front matter / supportE8 audit and bootstrap
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-10-18: Project separation and retained physics historyResearch changelog; TFPT — Introduction and reading guide
Registered verification modules
- v645_klein_rp.pyRegistered verification module · core
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QGEO.KLEINRP.01: der parafermionische Klein-Twist — **DER ℤ₃-TWIST-SEKTOR IST REFLEXIONSPOSITIV auf dem Naht-Doppel-Gitter unter der Klein-Paarung (Spiegelung ∘ LADUNGSKONJUGATION, ω^{k·q}-Nullmoden-Phasen, η = (1,1) EINDEUTIG aus 36)** — das benannte Offen-Item von QGEO.TWISTOPE.01 ist auf Gitter-OS-Ebene GELÖST: alle geladenen Klein-Blöcke exakt reell-symmetrisch UND PSD (worst 2.9e−12), N-stabil 48…384, beide Achsen, **volle gemischte OS-Gram {1, σ₁, σ₂, Dipole} ohne Sektor-Projektion PSD**; die naive Paarung reproduziert ihre N-stabile Verletzung (−4.5e−3, must-fail); Kontinuums-OS/Modularität getypt offen; GATE.QGEO bewegt sich nicht (17 checks, ~30 s)
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