A registered result in its source context
Original ledger classification: Exact [E] existence realization (integrality, conjugation, saturation index all exact sympy) + typed open canonicity [O]; NO marker moves
Recorded statement: The joint embedding -- the compiler pair (U, V) is realized on the mu3-cover (one module, 15 checks, ~4 s; discovery probe joint_embedding_probe.py 15/15, verdict JOINT-EMBEDDING-LANDED). THE PAIR [E]: in the Gamma-even DEGREE-2 sector…
Statement, assumptions and sources
Paper context: E8 audit and bootstrap
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QGEO.EMBED.01
Original ledger classification: Exact [E] existence realization (integrality, conjugation, saturation index all exact sympy) + typed open canonicity [O]; NO marker moves
Original claim text
The joint embedding -- the compiler pair (U, V) is realized on the mu3-cover (one module, 15 checks, ~4 s; discovery probe joint_embedding_probe.py 15/15, verdict JOINT-EMBEDDING-LANDED). THE PAIR [E]: in the Gamma-even DEGREE-2 sector (the v599 grading functor applied to cusp-Gram products), U* = -2E1 + (G1 Gamma(G1) + Gamma(G1) G1) + 3 and V* = -E3 + (1/2)(G2 Gamma(G2) + Gamma(G2) G2) + 1 are both integral over Z[omega], Gamma-even, with char U* = x^2(x-3), char V* = x(x-1)(x-2), char(U*+V*) = (x-1)(x^2-5x+3) = char Q, and the compiler joint traces tr(U*V*) = 3, tr(U*V*^2) = 6. THE ALGEBRA EMBEDDING [E]: the seven words {1, U*, V*, U*V*, V*U*, V*^2, V*U*V*} are independent (rank 7 = the full 2|1 parabolic) and the simultaneous-conjugation system is EXACTLY solved: an explicit invertible g with g U* g^-1 = U_c and g V* g^-1 = V_c -- the v566 parabolic self-code algebra lives on the cover. THE INTEGRAL FORM [E]: the real structure R as an integer matrix on Z^6 has R^2 = 1; its fixed lattice has the explicit saturated basis (1,2,2,0,0,0), (0,1,1,-1,0,0), (0,0,1,1,-1,1) (SNF certificate (1,1,1)); U*, V* restrict to INTEGER 3x3 matrices with the same chars. THE 81 [E, the hard test]: the Z-order of the seven restricted words has Smith elementary divisors (1,1,1,3,3,3,3) -- saturation index 81 = 3^4, EXACTLY the v566 compiler self-code index with the same divisor pattern. HONEST LIMITS: must-break control passes (the v599 degree-1 pair fails the joint criterion); CANONICITY IS OPEN and typed open -- the degree-2 scan admits many joint pairs, so this is an EXISTENCE claim, not uniqueness; GATE.QGEO does not move until the canonicity round and the two-demand continuum derivation land; no marker changes. [2026-08-01 XVII: the canonicity question is answered -- v601: the J-adapted projector presentation is D4-equivariant by construction (mark choice = winding choice) and realizes the DUAL parabolic exactly; the Klein group lives integrally on the curve; see QGEO.CANON.01.] [2026-08-01 XVIII: the integer core is Lean-certified -- FORM.QGEO.04 in TfptCarrier/CoverEmbedding.lean: W = T*S, Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P, S-primitivity via unimodular minor (determinant-free, kernel decide; lake build green).]
Original classification detail
JOINT-EMBEDDING-LANDED: (U*, V*) Gamma-even degree-2, Z[omega]-integral, chars + joint traces compiler-exact, word rank 7, conjugation solved, integer on saturated Fix(R), Smith (1,1,1,3,3,3,3) index 81 = v566; canonicity open; 15/15 checks
Dependencies as recorded
QGEO.REAL.01; QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v597; v598; v599
Open the original ledger source, row 618Location recorded in the ledger
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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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- =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
- =150 2026-08-01 (XVI) (the joint embedding lands ( v600 promoted): the compiler pair [formula in source] is REALIZED on the [formula in source] -cover — a [formula in source] -even degree-2 pair, [formula in source] -integral, seven-word rank [formula in source] , simultaneous conjugation to the compiler pair solved exactly, integer restrictions on the saturated [formula in source] lattice, and saturation index [formula in source] with the exact v566 divisor pattern; canonicity typed open, the gate does not move)Research changelog; TFPT — Introduction and reading guide
- 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
Navigation through a shared verification script
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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 / supportE8 audit and bootstrap
- Front matter / supportStandard Model and flavor structure
- Research Contract 1 — [formula in source]tfpt research contracts
- How to read this documenttfpt 1 architecture e8
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Registered verification modules
- v600_joint_embedding.pyRegistered verification module · core
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QGEO.EMBED.01: die Gelenk-Einbettung — DAS COMPILER-PAAR (U, V) IST AUF DEM μ₃-COVER REALISIERT: der Γ-gerade Grad-2-Sektor liefert U* = −2E₁ + (G₁Γ(G₁)+Γ(G₁)G₁) + 3 und V* = −E₃ + ½(G₂Γ(G₂)+Γ(G₂)G₂) + 1 — beide Z[ω]-integral, alle Chars und Gelenk-Spuren exakt Compiler (tr(UV) = 3, tr(UV²) = 6, char(U+V) = charQ); die Sieben-Wort-Algebra hat Rang 7 und die Simultan-Konjugation g U* g⁻¹ = U_c, g V* g⁻¹ = V_c ist EXAKT gelöst; auf dem saturierten R-Fixgitter (SNF-Zertifikat (1,1,1)) sind die Restriktionen GANZZAHLIG, und die Z-Ordnung der sieben Wörter hat Smith-Elementarteiler (1,1,1,3,3,3,3) — SATURATIONSINDEX 81 = 3⁴, exakt der v566-Selfcode-Index; must-break-Kontrolle besteht; KANONIZITÄT OFFEN (Existenz-Claim); GATE.QGEO bewegt sich nicht (15 checks, ~4 s)
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