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A registered result in its source context

Original ledger classification: Identity/Lattice [I]/[L] + audit [P]

Recorded statement: The mass-transport pencil P(x)=K+xQ as a projective line with two anchor-block singularities, and the DOUBLE COVER branched over them (validated review block; all in admissible classes). CENTRAL [I/L]: det B(K+xQ)=(3x+2)(3x+5) is quadratic…

Statement, assumptions and sources

Paper context: Standard Model and flavor structure

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FLAV.PENCIL.03

Original ledger classification: Identity/Lattice [I]/[L] + audit [P]

Original claim text

The mass-transport pencil P(x)=K+xQ as a projective line with two anchor-block singularities, and the DOUBLE COVER branched over them (validated review block; all in admissible classes). CENTRAL [I/L]: det B(K+xQ)=(3x+2)(3x+5) is quadratic => y^2=det B(K+xQ) is a 2:1 cover ramified exactly over its zeros, so the Koide/gap point x=-|Z2|/N_fam=-2/3 and the carrier point x=-g_car/N_fam=-5/3 are THE TWO BRANCH POINTS of one double cover; deck involution y->-y of degree 2=|Z2|=the sheet (line-bundle reflection of the Spin(10)->SO(10) half-spinor cover the carrier lives on). Branch locus = -(spine ends 2,5)/N_fam; discriminant = 81 = N_fam^4 (perfect square => rational/split, sqrt=9=N_fam^2=det B_Q); separation = 1 = one transport period (deck-translation x->x+1 maps carrier->Koide). So 2/3 is not a recurring quotient but ONE ramification point of the mass<->transport double cover; sharper open question: why does the leptonic source-to-pole transfer renormalise onto a branch point and which sheet (y->+-) does parity select. DEEPER: the branch divisor symmetric functions ARE the core atoms - sum of branch points = -scalaron/N_fam = -7/3, product = A_Lambda/N_fam^2 = 10/9, integer labels {|Z2|,g_car}={2,5} have sum=7=scalaron, product=10=A_Lambda, diff=3=N_fam, so the scalaron is the TRACE of the branch divisor and all earlier scalaron readouts collapse to this one geometric origin; the fiber y^2=det B is A_Lambda at K and 4A_Lambda at L (transport K->L multiplies the fiber by |Z2|), and the Z2 sheet endpoint x=-1 lies between the branch points where y^2=-|Z2|<0 (the sheet lives on the imaginary cut joining Koide and carrier); the clean cover is a property of the anchor 2-plane <1,a> (the full-3-space cubic det(K+xQ) has disc -1132 with an irrational real root), so the anchor projection regularises the geometry (anchor-first). CLEARING MATRICES [I/L]: Koide C_{2/3}=3K-2Q reproduces D5+A3 glue (tr=15=dim A3, sum=45=dim D5, det=60=D_start), rank-one (forced) anchor block B(C23)=(g_car,7)^T(N_fam^2,11)=(5,7)^T(9,11), sum=240=|R(E8)|; carrier C_{5/3}=3K-5Q charge-neutral (sum=0), chi=(lam+|Z2|)(lam^2+lam+|R+A3|)=(lam+2)(lam^2+lam+6). Scalaron 7 = mixed anchor discriminant/N_fam = (det B_L - det B_K - det B_Q)/N_fam = (40-10-9)/3 (B_L=B_K+B_Q linearity; 7 is an interference area, not a leftover) [I]. c_u/c_d=55/117 diagnostic readout at sheet endpoint K-Q (Plucker v37 canonical) [I]

Original classification detail

Identity/Lattice [I]/[L] (double cover / branch locus, clearing matrices, mixed-disc scalaron) + [P] (per-entry atlas reading of clearing matrices)

Dependencies as recorded

FLAV.PENCIL.01; FLAV.PENCIL.02; v37; v52; v74; v80

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Registered verification modules

  • v81_singular_pencil_matrices.pyRegistered verification module · frontier
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    double cover y²=det B(K+xQ): Koide −2/3 & carrier −5/3 are the two branch points (deck 2=|ℤ₂|, disc=81=N_fam⁴); clearing matrices 3K−2Q (D₅⊕A₃ glue, sum=240) & 3K−5Q (charge-neutral)

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