BCODR Bidirectional Circular Overlap Decomposition and Recomposition v0.1: A General Methodology That Simultaneously Overlaps Outward-Expanding and Inward-Contracting Circular-Kernel Fields, Cuts Them Apart, and Traceably Recomposes the Candidate Space — Not Specific to the Lebesgue Problem

BCODR (Bidirectional Circular Overlap Decomposition and Recomposition) is one of several methodologies Neo.K personally designed for this round of research; it is a general candidate-space search and bookkeeping method, not specific to the Lebesgue problem. Its basic cycle is Generate → Overlap → Measure → Cut → Separate → Recompose → Close: within the same parameter space, it simultaneously grows an inward-to-outward-expanding candidate circular-kernel field (originating from known cores, candidate centers, and proof/shape/witness seeds) and an outward-to-inward-contracting constraint circular-kernel field (originating from known upper bounds, forbidden zones, the global domain, and counterconstraints), deliberately preserving multiple memberships rather than forcing a single Voronoi-style partition — overlap is not error, but structured information itself. The arrangement formed by the boundary circles of the two fields cuts out atomic overlap cells within which membership stays constant, which are then separated across four layers — geometry, topology (e.g. bridge cell), information (HIGH_INFORMATION/TRANSITION/LOW_INFORMATION/REDUNDANT/CONFLICT), and certificate role (PROOF_SUPPORT/COUNTER_SUPPORT/SHAPE_SUPPORT, etc.) — and recomposed, subject to preserving traceability and minimum distinguishability, into a surviving structural spine (surviving structural spine). The method's own "honest boundaries" statement clearly acknowledges that v0.1 has not been proven to converge for arbitrary search problems, has not been proven to find the global optimum for an arbitrary objective, and has not been proven to yield a unique recomposition or a unique final surviving spine — it describes itself as a closure architecture, not a universal convergence theorem. A concrete case from connecting BCODR back into the current Lebesgue research line shows how this honesty is realized in practice: in the 5-dimensional placement-parameter search of root-6217, overlap cell number 131 was initially misjudged to contain a "zero-radius contact nexus"; a later round, after cutting to a finer resolution, corrected this to a "coarse contact container"; only in the round after that was it precisely reclassified as a genuine, but narrowly scoped, "Interior Midpoint Balanced Germ" (IMBG, with equal-radius meeting instant T*≈0.0473333498919449>0). The source document linked from this page formally proves that, as long as T*>0, the IMBG meeting point is, by construction, necessarily at a positive distance from both the outward analytic seed and the inward prune seed — that is, the BCODR overlap germ ≠ the shape minimizer, by definition — and explicitly warns subsequent documents not to misread the "IMBG point" as the "shape minimizer." This pattern of correcting itself twice, and explicitly warning readers in its own conclusion against overreading the result, is a concrete demonstration of methodological discipline, not a flaw to be downplayed. The BCODR method itself was designed by Neo.K; the execution and writing of this document were carried out by Aletheia / GPT-5.6 Sol.