A registered result in its source context
Original ledger classification: Identity [I] (character-exact multiset identities) + Physical [P] (QBL reading)
Recorded statement: SHEET-PAIRING LEMMA (the first exact representation-theoretic anchor of the QBL programme; connects red-team Target B to the sheet Z2): character-exact weight-multiset results from the D5 spinor weights (+-1/2)^5 alone - (1) NO SCALAR…
Statement, assumptions and sources
Paper context: Architecture, E8 and the electromagnetic fixed point
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CAR.PAIR.01
Original ledger classification: Identity [I] (character-exact multiset identities) + Physical [P] (QBL reading)
Original claim text
SHEET-PAIRING LEMMA (the first exact representation-theoretic anchor of the QBL programme; connects red-team Target B to the sheet Z2): character-exact weight-multiset results from the D5 spinor weights (+-1/2)^5 alone - (1) NO SCALAR WITHIN A SHEET: zero-weight multiplicity of S+xS+ (and S-xS-) is 0, because the odd slot count g=5 flips the sign parity of -w (a parity theorem of the five-slot code): there is no invariant bilinear pairing inside one chirality. (2) THE SCALAR LIVES ACROSS THE SHEETS: S+xS- has zero-weight multiplicity 16 = 1+5+10 (the Pascal triple) and equals exactly, as a weight multiset, the even-form tower Lambda^0 + Lambda^2 + Lambda^4 (256 = 1+45+210) with zero-mode grading (1,5,10) = the carrier code (Lambda^0,Lambda^1,Lambda^2)(C^5). (3) SHEET-DIAGONAL = ODD FORMS: (S+xS+) u (S-xS-) = 2Lambda^1 + 2Lambda^3 + Lambda^5 exactly (512 = 20+240+252): within-sheet pairings are odd-form-valued, never scalar. (4) the scalar-bearing tower tops at Lambda^{2K} with K=2 - bilinear seam data reach exactly the Pascal-ladder degree (CAR.LADDER.01). (5) QBL READING [P] (recorded, not claimed): the seam's single scalar two-point kernel is necessarily a SHEET PAIRING (chirality-off-diagonal, the same Z2 as GATE.METRIC.07 / FLAV.SHEET.01-03); the remaining step is to show the Calderon kernel supplies exactly this pairing and nothing beyond - then QBL => K=2 <=> g=5 closes the carrier choice
Original classification detail
Identity: scalar pairing only across sheets; even tower tops at 2K; QBL anchored
Dependencies as recorded
CAR.LADDER.01; ARCH.QUAD.01 (QBL); GATE.METRIC.07 (sheet Z2); FLAV.SHEET.01-03; CAR.PASCAL.01
Open the original ledger source, row 193Location recorded in the ledger
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- Architecture, E8 and the electromagnetic fixed point
Navigation through a shared verification script
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- How to read this documenttfpt 1 architecture e8
- How to read this documenttfpt 1 architecture e8
- 2026-06-11Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Front matter / supportArchitecture, E8 and the electromagnetic fixed point
Registered verification modules
- v109_sheet_pairing.pyRegistered verification module · registry
Read the description recorded in the module registry
SHEET-PAIRING LEMMA (the first exact rep-theoretic anchor of QBL; ties Target B to the sheet ℤ₂): character-exact weight multisets from the D₅ spinor weights (±½)⁵ alone — NO SCALAR WITHIN A SHEET (zero-weight multiplicity of S⁺⊗S⁺ = 0: the odd slot count g=5 flips the sign parity of −w, a parity theorem of the five-slot code); THE SCALAR LIVES ACROSS THE SHEETS: S⁺⊗S⁻ = Λ⁰⊕Λ²⊕Λ⁴ exactly (256 = 1+45+210) with zero-mode grading (1,5,10) = the carrier code; SHEET-DIAGONAL = ODD FORMS: (S⁺⊗S⁺)∪(S⁻⊗S⁻) = 2Λ¹+2Λ³+Λ⁵ exactly (never scalar); the scalar-bearing tower tops at Λ^{2K} with K=2 — bilinear seam data reach exactly the Pascal-ladder degree; QBL reading [P]: the seam's single scalar kernel is necessarily a SHEET PAIRING (chirality-off-diagonal — the same ℤ₂ as the glue chiralities and branch kernels); remaining step: show the Calderón kernel supplies exactly this pairing
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