BCODR-LUC: the bidirectional circular-overlap method applied to root 6217
BCODR-LUC-05 · Balanced overlap-germ taxonomy
Neo.K / Aletheia
Balanced Overlap Germ Taxonomy: a Unified Classification of Face-Constrained and Interior-Midpoint First Contacts — Root 127 vs Root 131: Different Tangent Dimensions, the Same Second-Order Isolation Mechanism
Puts root 127 (a face-constrained balanced germ, FCBG, tangent dimension 3) and root 131 (an interior-midpoint balanced germ, IMBG, tangent dimension 4) into one classification: 131's true radius T* ≈ 0.0473333 (the nearest ray–prune pair at ρ ≈ 0.00254), a midpoint theorem, a rank-1 first-order structure, and κ₁₃₁ ≈ 0.45465, within a tenth of a percent of κ₁₂₇ ≈ 0.45429, on which it claims that different first-order contact topologies collapse to the same second-order isolation class; it closes with a general birth-classification rule by the sign of κ and proposes as the next node the frontier-spine closure against the A_ref tail (that is, LUC-06).
Under the paper's definition of combined stiffness both germs are isolated at second order (birth dimension 0); the paper presents this as the start of a reusable germ taxonomy, not a conclusion about a single root.
Layer two (RUN-067) reproduced, at 50 digits, 131's nearest pair, T*, the midpoint and its place inside 131, the rank-1 first order and the four-dimensional critical space, H±, κ₁₃₁ and the isolation of both germs, and found that at 131 the straight-line stiffness is indeed the right one; not as stated are the comparison of the two germs (§13–14) and §22's rule: the "same second-order mechanism" ratio of 1.0008 is 1.92 for the overlaps' own constants, and the classification rule can neither reach the unstable class it defines nor certify isolation. Noted: the midpoint theorem needs the nearest pair of all. The package's verifier was re-run here and passes (it only checks the type, the dimension and κ > 0.4 recorded in the ledger).