BCODR-LUC: the bidirectional circular-overlap method applied to root 6217
BCODR-LUC-07 · Candidate/germ alignment and mirror covariance
Neo.K / Aletheia
Root 131 Candidate/Germ Alignment and Mirror-Covariance Reduction: from the Non-Identity of the IMBG Midpoint to Planar Mirror Orbits — Separating the Candidate Shape, the Certificate Germ and the Chirality Obligation
Separates three things — the candidate shape, the certificate germ, and chirality: it proves an IMBG midpoint-separation theorem (T* > 0 ⇒ the germ point z* of root 131 differs from the nearest point p*, so an IMBG is not automatically the minimizer); under planar mirrors, single-axis symmetry rules out ordinary planar chirality (S_yK = K ⇒ S_xK = R_πK), the zero-rotation barrier reconstructed in Paper 07 rests on an exactly symmetric contact state, and the feasibility and area of the Lebesgue problem are mirror-covariant; it lists what is "closed" (A to E) and what "remains open" (the Cycle-17 candidate parameter vector, alignment of the candidate point with the IMBG, marked witness chirality, individual-shape chirality at nonzero ρ, the global extremal shape, attainment, uniqueness), and eight local certificate obligations CERT-07.1 to 07.8.
The paper lists as proved that an IMBG is not automatically the minimizer, that single-axis symmetry rules out ordinary two-dimensional chirality, and that feasibility and area are mirror-covariant; it explicitly lists alignment of the candidate with the germ, marked chirality, the global extremal shape, attainment and uniqueness as still open.
Layer two (RUN-068) reproduced the midpoint-separation theorem, T* and κ₁₃₁, the mirror algebra and the support-function covariance, and the up/down symmetry of Paper 07's barrier (CERT-07.4, replayed body by body and at 50 digits), hence the barrier is achiral; not as stated is that Cycle 17's candidate is recorded by value only (§6, §20) — the candidate vector is in Cycle 17's diagnostic, 7.7×10⁻⁴ from the ray and 0.109 from the IMBG point, and it is not 131's lowest point: 131's corner holds the barrier itself, 1.38×10⁻⁷ lower — and the positions of the B5 slivers (§22). Noted: the printed p*, q* and z* are LUC-03's optimizer values, 5×10⁻⁷ from the exact pair; the schema and verifier that §23 describes are not in the archive. This paper is a single md with no package, so there is no verifier script to re-run.