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Original ledger classification: Numerical (finite MTC) [E]/[C]

Recorded statement: PARTICLES = DHR SUPERSELECTION SECTORS (2026-06-17): the discriminant/anyon content of the carrier and the condensation tower (v235/v237) made explicit as an abelian modular tensor category (MTC) -- the particle-spectrum layer of the…

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DHR.SECTORS.01

Original ledger classification: Numerical (finite MTC) [E]/[C]

Original claim text

PARTICLES = DHR SUPERSELECTION SECTORS (2026-06-17): the discriminant/anyon content of the carrier and the condensation tower (v235/v237) made explicit as an abelian modular tensor category (MTC) -- the particle-spectrum layer of the emergent QFT. The carrier discriminant form is the Lean-verified glue form (FORM.GLUE.01): on Z4 x Z4, q(x,y)=(5x^2+3y^2)/8 mod 1, bilinear B((x,y),(x',y'))=(5xx'+3yy')/4 mod 1, topological spin theta(a)=e^{2 pi i q(a)}. (1) [E] 16 sectors = Z4 x Z4 (= det K_carrier = dim S^+ = 16, v235); fusion = abelian group law (closed, associative, vacuum (0,0), inverses). (2) [E] spin/condensability: bosonic <=> theta=1 <=> q in Z <=> 5x^2+3y^2=0 mod 8 -- exactly the six isotropic elements {(0,0),(1,1),(1,3),(2,2),(3,1),(3,3)}. (3) [E] condensation tower = v235/v237: condensing the Lagrangian H1=<(1,1)> (order 4, self-dual H1^perp=H1) leaves |H^perp/H|=1 sector = (E8)_1 (det 1, holomorphic); the order-2 isotropic <(2,2)> leaves 4 sectors = SO(16)_1 = D8 (det 4); so carrier 16 -> D8 4 -> E8 1. (4) [E] NEG: a non-isotropic anyon (1,0), q=5/8, theta != 1, is confined; <(1,0)> contains non-isotropic elements so cannot be a condensation algebra -- the tower is forced by isotropy. (5) [C] identifying sectors / deconfined anyons with SM particle multiplets is the interpretation; the MTC data (fusion + spins + tower) is exact

Original classification detail

[E] 16-sector abelian MTC from the glue form, fusion = group law, condensation tower 16->4->1 = v235/v237 (Lagrangian -> E8, order-2 -> D8); [C] sector<->SM-multiplet identification -- particles = stable defect sectors of the seam code, made precise

Dependencies as recorded

GATE.HOLO.02; GATE.HOLO.03; FORM.GLUE.01

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  • v241_dhr_sectors.pyRegistered verification module · registry
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    DHR.SECTORS.01 — particles = DHR superselection sectors: the carrier discriminant/anyon content and the v235/v237 condensation tower as an abelian MTC. Carrier form (Lean-verified, FORM.GLUE.01): on Z4×Z4, q(x,y)=(5x²+3y²)/8 mod 1, θ(a)=e^{2πi q(a)}. [E] 16 sectors = Z4×Z4 (= det K_carrier = dim S⁺), fusion = abelian group law. [E] bosonic/condensable ⟺ θ=1 ⟺ isotropic ⟺ 5x²+3y²≡0 mod 8 — the six elements {(0,0),(1,1),(1,3),(2,2),(3,1),(3,3)}. [E] condensation tower = v235/v237: Lagrangian ⟨(1,1)⟩ (order 4, self-dual) → 1 sector = (E8)₁; order-2 ⟨(2,2)⟩ → 4 = D8; carrier 16→D8 4→E8 1. [E] NEG: a non-isotropic anyon (1,0) is confined. [C] sector↔SM-multiplet map is the interpretation; the MTC is exact. 'Particles = stable defect sectors of the seam code' made precise.

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