A registered result in its source context
Original ledger classification: Identity [I] (explicit construction + exact grading + arithmetic) + Physical [P] (anchors)
Recorded statement: THE GRADED HULL + THE ZERO-MODE MULTIPLICITY (both named computations gain their structural anchors): R2 HALF - (1) EXPLICIT COSET CONSTRUCTION [I]: the 240 E8 roots, built as explicit norm-2 vectors in D5 (+) A3 + glue coordinates,…
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E8.GRADE.01
Original ledger classification: Identity [I] (explicit construction + exact grading + arithmetic) + Physical [P] (anchors)
Original claim text
THE GRADED HULL + THE ZERO-MODE MULTIPLICITY (both named computations gain their structural anchors): R2 HALF - (1) EXPLICIT COSET CONSTRUCTION [I]: the 240 E8 roots, built as explicit norm-2 vectors in D5 (+) A3 + glue coordinates, decompose over the glue group Z4 as 240 = 52 + 64 + 60 + 64 (C0 = carrier roots 40+12; C1 = (16, 4bar) from norm 5/4 + 3/4; C2 = (10, 6) from 1 + 1; C3 = (16bar, 4)) - exactly the GATE.METRIC.06 branching, now as vectors. (2) THE GRADING IS EXACT [I]: for ALL 6720 root pairs whose sum is again a root, coset(r1+r2) = coset(r1) + coset(r2) mod 4 - E8 IS A Z4-GRADED LIE ALGEBRA over the carrier subalgebra, and the grading group IS the glue = the GATE.METRIC.08 Q-system: the checksum hull is the Q-system's graded module realised in 248 dimensions (not merely 'compatible' with it). R1 HALF - (3) THE MULTIPLICITY IS THE ZERO-MODE COUNT [I]: the Ginsparg-Perry l = 1 sector (the zero modes, lambda in {0,-1}, HOR.CLOCK.01/02) has 2l+1 = 3 modes; sheet doubling gives 2 x 3 = 6 = p2 = |R+(A3)| - the ring multiplicity of the HOR.CLOCK.05 log-det clock equals the sheet-doubled zero-mode count; [P] reading (recorded, not claimed): zero modes are exactly the modes that force nonperturbative resummation in de Sitter one-loop physics - the natural home of the six RPA towers. (4) RESIDUE [P]: R1 = compute the RPA determinant on the six sheet-doubled l=1 zero modes; R2 = identify the seam-Calderon inclusion with the Z4-graded structure now exhibited inside E8 itself
Original classification detail
Identity: E8 = Z4-graded Lie algebra over the carrier (grading = glue Q-system; 6720 pair checks); ring multiplicity 6 = sheet-doubled GP zero modes
Dependencies as recorded
GATE.METRIC.08; GATE.METRIC.06; HOR.CLOCK.05; HOR.CLOCK.01; E8.GLU.01
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- 2026-07-20 (V) ( v492 [formula in source] the new research contract CELEST.SEAM.01 — the celestial-holographic route: the [formula in source] [formula in source] -glue is the flat [formula in source] monodromy of the equivariant celestial chiral algebra on the [formula in source] ALE space, with the exact correction clock [formula in source] deck (spin bridge [formula in source] , [formula in source] ); verdict B, no gate closes, no marker moves, SEAM.EQUIV.01 stays )Research changelog; TFPT — Introduction and reading guide
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- Research Contract — CELEST.SEAM.01 (the celestial and twistor continuum route)tfpt research contracts
- Front matter / supportStandard Model and flavor structure
- Research Contract — CELEST.SEAM.01 (the celestial and twistor continuum route)tfpt research contracts
- The celestial and twistor continuum routeinput 5 tfpt 3 e8 audit bootstrap
- Research Contract 2 — [formula in source]tfpt research contracts
- 2026-06-12Research changelog; TFPT — Introduction and reading guide
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Registered verification modules
- v128_graded_hull.pyRegistered verification module · registry
Read the description recorded in the module registry
THE GRADED HULL + THE ZERO-MODE MULTIPLICITY (both named computations R1/R2 gain their structural anchors): R2 HALF — EXPLICIT COSET CONSTRUCTION [I]: the 240 E8 roots built as explicit norm-2 vectors in D₅⊕A₃+glue coordinates decompose over the glue ℤ₄ as 240 = 52 + 64 + 60 + 64 (C₀ = carrier 40+12, C₁ = (16,4̄), C₂ = (10,6), C₃ = (16̄,4)) — the v89 branching as vectors; THE GRADING IS EXACT [I]: for ALL 6720 root pairs whose sum is again a root, coset(r₁+r₂) = coset(r₁)+coset(r₂) mod 4 — E8 IS A ℤ₄-GRADED LIE ALGEBRA over the carrier, and the grading group IS the glue = the v125 Q-system: the checksum hull is the Q-system's graded module realised in 248 dimensions; R1 HALF — THE MULTIPLICITY IS THE ZERO-MODE COUNT [I]: the Ginsparg–Perry l=1 sector (the zero modes λ ∈ {0,−1}) has 2l+1 = 3 modes, sheet doubling gives 2×3 = 6 = p₂ = |R⁺(A₃)| — the ring multiplicity of the v127 clock equals the sheet-doubled zero-mode count ([P]: zero modes are exactly the modes forcing nonperturbative resummation in dS one-loop physics — the natural home of the six RPA towers); residue [P]: R1 = compute the RPA determinant on the six sheet-doubled zero modes, R2 = identify the seam-Calderón inclusion with the ℤ₄-graded structure now exhibited inside E8 itself
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