A registered result in its source context
Original ledger classification: Identity (exact)
Recorded statement: DIAMOND AXIS CURVATURE LEMMA (2026-06-15): along the two canonical axes of the centered cross (C center, U winding, V sheet, v95) the Sheet-Diamond determinant is LINEAR on the winding axis and QUADRATIC on the sheet axis: det(C+xU)=14+6x…
Statement, assumptions and sources
Paper context: Standard Model and flavor structure
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DIAMOND.AXIS.01
Original ledger classification: Identity (exact)
Original claim text
DIAMOND AXIS CURVATURE LEMMA (2026-06-15): along the two canonical axes of the centered cross (C center, U winding, V sheet, v95) the Sheet-Diamond determinant is LINEAR on the winding axis and QUADRATIC on the sheet axis: det(C+xU)=14+6x (flat, slope 6=|R^+(A_3)|), det(C+yV)=14+14y+4y^2 (2nd difference 8=rank E_8); the anchor block satisfies det B(C+xU)=28+12x=2 det(C+xU) and det B(C+yV)=28+21y+3y^2 (2nd difference 6=|R^+(A_3)|). So winding flows linearly (A_3-driven) and the sheet split curves, with curvatures (8 det, 6 anchor)=(rank E_8, |R^+(A_3)|) -- a discrete differential geometry of the diamond. Exact identity UPDATE (2026-07-31, v566, DIAMOND.PARA.SELFCODE.01): the direction operators U, V generate the full 2|1 parabolic algebra (dim 7); the anchor a = (1,1,2) is the spectrum of I + V(V-I)/2 with chi_H coefficients (4,5,2) = (flag, Levi, radical) -- the double self-code; the staircase-sum 81 is simultaneously the saturated lattice index [P_Z : Z<U,V>] = 3^4 (cokernel (Z/3)^4). Markers unchanged.
Original classification detail
[E] det(C+xU)=14+6x linear, det(C+yV)=14+14y+4y^2 quadratic; curvatures (8,6)=(rank E_8, |R^+(A_3)|)
Dependencies as recorded
v94_sheet_diamond.py; v95_centered_diamond.py
Open the original ledger source, row 303Location recorded in the ledger
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- Standard Model and flavor structure
Sections that explicitly cite this claim
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- 2026-06-15 (the diamond axis geometry — two axes + the transfer corner)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
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- 2026-06-24 (The sheet generator [formula in source] as a binary internal compiler: v410– v414)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportStandard Model and flavor structure
- How to read this documenttfpt 1 architecture e8
- Front matter / supportStandard Model and flavor structure
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v218_diamond_axis_geometry.pyRegistered verification module · registry
Read the description recorded in the module registry
Diamond axis geometry: the Sheet Diamond (v94) is a discrete geometry with two axes around the centered cross (v95: C center, U winding, V sheet), and the missing corner F is the transfer completion. Three exact [E] lemmas + honest audit/heuristic typing, no new numbers. [E] AXIS CURVATURE: det(C+xU)=14+6x linear (winding flat, slope 6=|R⁺(A₃)|), det(C+yV)=14+14y+4y² quadratic with 2nd difference 8=rank E₈; det B(C+xU)=2 det(C+xU), anchor-block sheet 2nd difference 6=|R⁺(A₃)| — curvatures (8,6)=(rank E₈, |R⁺(A₃)|). [E] PLÜCKER TRANSFER LADDER: Pl(K)=(−1,6,4)→Pl(C)=(0,14,14)→Pl(F)=(1,22,30); C−K=(1,8,10)=(N_Φ,rank E₈,A_Λ), F−C=(1,8,16)=(N_Φ,rank E₈,dim S⁺) — decuple 10 then full generation 16 (identity [E], transfer reading [C]). [E] SPECTRAL RAMIFICATION: for Q,K,C,F the cubic discriminant=q(r)²·Disc(q); squares (1,3,4,6)=(N_Φ,N_fam,|μ₄|,|R⁺(A₃)|), kernels (13,48,65,105)=(Δ_Q,Ω_adm,g_car·Δ_Q,N_fam·g_car·7), F carries 105=3·5·7. [I] audit anchor-defect spectrum Δ_a(Q,K,R,C,L,F)=(8,16,18,21,24,26); [I] audit pair-sum atlas {12,24,28,36,40,52} total 64, staircase sum 81=N_fam⁴ — 28=2 dim G₂, 52=dim F₄ audit-only, not spine. [C] heuristic: F the transfer-completion corner, a search principle not a hard filter.
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