Quantum Structural Theory of Harmony (QSTH 8.3) formulates the Geometry of Convergence of Paths as an audit geometry and defines the Condensation Hexagon as a local 2D cell of admissibility. This publication belongs to the broader QSTH 8.x sequence devoted to the condensation of information into structure. QSTH 8.1 maps the atlas of admissibility; QSTH 8.2 translates this atlas into gate protocols, null tests and PASS / FAIL / INCONCLUSIVE discipline; QSTH 8.M separates the improbable from the inadmissible through the Other Shore Protocol; and QSTH 8.3 shows that different paths do not have to meet at one point — they may converge within one geometry of admissibility. The central claim of QSTH 8.3 is deliberately status-bound and methodologically conservative: the convergence of paths is not a similarity of words, but the ability of independent paths to translate their outputs through edges, preserve the ledger between slices and meet in a center capable of record. The Condensation Hexagon is introduced not as decoration, but as an audit apparatus. A vertex carries a path, an edge carries a translation relation, a diagonal carries a deeper relation, and the center carries a minimal closure core: admissibility, readability, stability, record and ledger preservation. The sixth vertex remains status-cautious. Its strongest current CAND filling is resonance-phase admissibility: the candidate condition that information does not become structure merely because it is possible, readable and bound, but only when its regime, carrier, readability and ledger-record enter stable phase or resonance alignment. QSTH 8.3 also defines failure modes of convergence, including false convergence, decorative hexagon, analogy overload, post-hoc shore, broken ledger, no record and premature 8.4. These failure modes protect the document from aesthetic overreach and preserve the CORE / CAND / SUPPORT / FUTURE FORMALIZATION discipline. A careful SUPPORT block is included on topological protection as a possible mathematical language of admissibility. This is not used as confirmation of QSTH, but as an external theoretical resonance: stability may arise not only from force, but from compatibility with an admissible protected sector. QSTH 8.3 does not derive dimensionality, does not prove a helical scaffold, and does not claim that the Universe has a hexagonal shape. Its role is narrower and stronger: it closes a methodological layer by translating scattered paths, analogies and working motives into one audit geometry of admissibility. CORE SENTENCE The Condensation Hexagon is not a symbol of harmony; it is an audit geometry. Each vertex must carry a path, each edge must carry a translation, and the center must carry a record-capable core of admissibility. FINAL SYNTHESIS QSTH 8.3 is an audit geometry of the convergence of paths, where the hexagon is not an image, but a rule for auditing relations between relations. Short Description QSTH 8.3 defines the Geometry of Convergence of Paths as an audit geometry and introduces the Condensation Hexagon as a local 2D cell of admissibility. It connects the Other Shore Protocol, resonance-phase admissibility and Relations Between Relations into a disciplined CORE/CAND framework for testing whether independent paths can converge into a record-capable center. Related QSTH Zenodo Records This record belongs to the broader QSTH 8.x sequence and should be read in continuity with the following Zenodo publications: QSTH 1.0 — Quantum Shadow and Tension Hypothesis: Foundational Publications (1–10), DOI: 10.5281/zenodo.17455814 QSTH 8.x — Opening Note for the Horizon Set Invariants Series: The Condensation of Information into Structure, DOI: 10.5281/zenodo.19760499 QSTH 8.0 — The Horizon Set of Invariants: Toward the Condensation of Information into Structure, DOI: 10.5281/zenodo.19764819 QSTH 8.1 — The Main Mendeleev Audit Atlas: The Condensation of Information into Structure, DOI: 10.5281/zenodo.20002922 QSTH 8.2 — The Verification and Operational Window of the Mendeleev Audit Atlas, DOI: 10.5281/zenodo.20037362 QSTH 8.M — Mathematical Method Note: Improbability Field and Admissible States, DOI: 10.5281/zenodo.20047090, Status: Direct methodological predecessor of QSTH 8.3
Rostislav Stepanik (Thu,) studied this question.