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Original ledger classification: Exact [E] equivariance/duality/Klein identities + [C] typed opens (exact sympy; one numeric inertia count of an exact hermitian matrix)
Recorded statement: The canonical equivariant realization (one module, 14 checks, ~12 s; discovery probe equivariant_dual_probe.py 14/14, verdict EQUIVARIANT-DUAL-LANDED). THE REPAIR: the naive cusp-Gram presentation of v600 is not D4-equivariant because the…
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Paper context: E8 audit and bootstrap
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QGEO.CANON.01
Original ledger classification: Exact [E] equivariance/duality/Klein identities + [C] typed opens (exact sympy; one numeric inertia count of an exact hermitian matrix)
Original claim text
The canonical equivariant realization (one module, 14 checks, ~12 s; discovery probe equivariant_dual_probe.py 14/14, verdict EQUIVARIANT-DUAL-LANDED). THE REPAIR: the naive cusp-Gram presentation of v600 is not D4-equivariant because the standard hermitian dagger is basis-dependent -- the Burau image preserves J; the canonical building blocks are the J-ADAPTED cusp projectors GJ_k = (1-T_k) J^-1 (1-T_k)^dagger J. THE PROJECTORS [E]: all four are idempotent rank-1 J-self-adjoint projectors (char x^2(x-1)) rotating exactly under D4 (r GJ_k r^-1 = GJ_(k+1)); opposite cusps J-orthogonal (tr(GJ1 GJ3) = 0), neighbors overlap with trace 1. THE EQUIVARIANT PAIR [E]: U_k = 3 GJ_k, V_k = 1 + GJ_(k-1) - GJ_k GJ_(k+1) -- Z[omega]-integral, compiler spectra, compiler joint data (char(U+V) = charQ, tr(UV) = 3, tr(UV^2) = 6), seven-word rank 7, and D4-EQUIVARIANT BY CONSTRUCTION (r^k-conjugation = mark shift, verified for all marks): the mark choice IS the compiler's winding choice. THE DUALITY [E]: the equivariant pair is NOT conjugate to (U_c, V_c) (the intertwiner space is 1-dimensional with identically singular solutions -- must-fail documented) but EXACTLY conjugate to the transposed pair (U_c^T, V_c^T): the equivariant presentation realizes the DUAL (plane-stabilizer) parabolic, the v600 Gamma-even pair the line-stabilizer -- the cover carries BOTH parabolic types, exchanged by duality. THE KLEIN GROUP ON THE CURVE [E]: all eight spectral-sign involutions of V_1 are Z[omega]-integral (v590 rigidity reproduced); the two v590 demands select the explicit integral S_0 (trace -1, commuting with V), and A B = S_0 closes the Klein four-group on the cover. HONEST OPENS [C]: the naive J-word Gram has inertia (3,2,2), not the corpus v572 RP pattern (4,0,3) -- the Hodge/RP comparison needs the Weil-twisted polarization (named); the seam-line geometric identification stays open. GATE.QGEO does not move; no marker changes. [2026-08-01 XVIII: the duality mechanism is identified -- v602: the J-adjoint exchanges the two parabolic types exactly ((U1, V1^J) ~ compiler pair); the dictionary completes in the J-projector language; Weil positivity holds exactly; see QGEO.DUALFORMS.01.]
Original classification detail
EQUIVARIANT-DUAL-LANDED: GJ_k idempotent equivariant projectors; pair family compiler-exact and equivariant by construction; conjugate to the TRANSPOSED compiler pair (dual parabolic); Klein group integral on the curve (8/8, S_0 explicit, A B = S_0); J-word-Gram inertia (3,2,2) typed open vs v572; 14/14 checks
Dependencies as recorded
QGEO.EMBED.01; QGEO.REAL.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v590; v597; v599; v600
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- Architecture, E8 and the electromagnetic fixed point
- Standard Model and flavor structure
- E8 audit and bootstrap
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- =150 2026-08-01 (XVIII) (the duality-and-forms round lands ( v602 promoted): the J-adjoint exchanges the two parabolic types exactly ( [formula in source] compiler pair); the dictionary completes in the J-projector language (the corpus anchor formula [formula in source] holds verbatim on the curve); Weil positivity holds exactly (the Riemann bilinear relations on the [formula in source] -sheet); the canonical bilinear form [formula in source] is symmetric with [formula in source] ; the v600 integer core is Lean-certified ( FORM.QGEO.04); the contracts paper gains the [formula in source] -cover model paragraph)Research changelog; TFPT — Introduction and reading guide
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- Group-theoretic audits (optional)E8 audit and bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- =150 2026-08-01 (XVII) (the canonicity round lands ( v601 promoted): the J-adapted cusp projectors give a manifestly [formula in source] -equivariant presentation of the pair (mark choice [formula in source] winding choice), which realizes the DUAL parabolic exactly (conjugate to the transposed compiler pair; the direct intertwiner is singular); the v590 Klein four-group lives integrally on the curve; the Weil-twisted RP comparison and the seam identification stay open)Research changelog; TFPT — Introduction and reading guide
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- How to read this documenttfpt 1 architecture e8
- How to read this documentArchitecture, E8 and the electromagnetic fixed point
- Research Contract 1 — [formula in source]Research contracts
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- Front matter / supportStandard Model and flavor structure
- Research Contract 1 — [formula in source]tfpt research contracts
- How to read this documenttfpt 1 architecture e8
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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Registered verification modules
- v601_equivariant_dual.pyRegistered verification module · core
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QGEO.CANON.01: die kanonische äquivariante Realisierung — die J-adaptierten Cusp-Projektoren GJₖ = (1−Tₖ)J⁻¹(1−Tₖ)†J sind idempotente Rang-1-Projektoren (char x²(x−1)), rotieren EXAKT unter D₄ (r GJₖ r⁻¹ = GJₖ₊₁; gegenüberliegende J-orthogonal); das Paar Uₖ = 3GJₖ, Vₖ = 1 + GJₖ₋₁ − GJₖGJₖ₊₁ ist Z[ω]-integral, Compiler-exakt (Spektren, Gelenk-Daten, Wortrang 7) und D₄-ÄQUIVARIANT PER BAUART (Mark-Wahl = Winding-Wahl); DUALITÄT: exakt konjugiert zum TRANSPONIERTEN Compiler-Paar (die duale Ebenen-Stabilisator-Parabolik; der Intertwiner zum direkten Paar ist identisch singulär — must-fail dokumentiert): der Cover trägt BEIDE Parabolik-Typen; die KLEIN-GRUPPE lebt integral auf der Kurve (alle 8 Spektral-Vorzeichen-Involutionen Z[ω]-integral, S₀ explizit mit Spur −1, A·B = S₀); ehrlich offen: naive J-Wort-Gram-Inertia (3,2,2) ≠ v572 (4,0,3) — Weil-Twist-Vergleich benannt (14 checks, ~12 s)
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