← RCHM × Hodge Research Round / R24 · Corrections and exact saturation (with erratum)

RCHM × Hodge Research Round R24 · Corrections and exact saturation (with erratum)

Küchle c5: Exceptional-Twist Correction, Exact Saturation, the A₁⊕A₃ Inherited Lattice, and Failure of the Direct Integral K3^[2] Template

Corrects the exceptional normal class: the descended root block is A₂ (not 3A₂), the explicit visible-plus-root sublattice has determinant 72 and index 3 in its saturation A₁ + A₃ (determinant 8, discriminant group Z/2 + Z/4); it then rejects a direct integral primitive K3^[2] isometry on a finite-discriminant-form mismatch.

R24 corrects two values of R22 and R23, but its closing negative conclusion (direct integral primitive K3^[2] Hodge isometry = FAIL) is reversed by R25. Erratum banner: a reader of R24 alone gets a wrong negative conclusion; R25 points out that it used q_T = −q_A, valid only in an even unimodular ambient lattice. A₂, det 72 and A₁ + A₃ (det 8) are retained. The header date is 2026-09-17, the file time 2026-09-18.

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PreviousR21 · Ancestry normalization of Küchle c5 · Küchle c5 Ancestry Normalization: Primitive Integral Inheritance and the Re-Emergence of a K3^[2] Bridge Profile
NextR25 · The odd-unimodular gluing correction (reverses R24) · Odd-Unimodular Gluing Correction for Küchle c5: the Integral K3^[2] Primitive Lattice Reopens

Erratum: this paper's closing 'direct integral primitive K3^[2] Hodge isometry = FAIL' is reversed by R25 (R24 used q_T = −q_A, valid only in an even unimodular ambient lattice). The paper is kept as written; read it together with R25.

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