Boundary-Enabled Directional Transport: Nonzero-Displacement Cycles, Activation Thresholds, and Minimum-Cost Restoration (BIT)

Abstract

ENGLISH This research report develops a conditional mathematical framework, motivated by Boundary Information Theory (BIT), for the opening, suppression, and restoration of directional transport. The selected model is a finite connected reversible continuous-time transition network with positive storage weights, paired reverse transitions, and displacement labels. These are explicit modeling assumptions, not a unique derivation from BIT’s proposed stress–causal-information pair. Within this model, the report presents analytic proofs of a variational formula for long-time diffusivity; a necessary and sufficient cycle criterion for positive transport in a projected direction and the associated transport rank; constructive two-sided Arrhenius bounds whose activation exponent is the minimum bottleneck barrier over cycles with nonzero projected displacement; and the minimum cost of lowering that threshold by independent barrier reductions. A conditional linear-response proposition is also included. Counterexamples separate strong anisotropy from structural loss of a transport direction, local motion from unbounded transport capability, and macroscopic transport measurements from unique microscopic identification. The preserved R01 archive records 263 passed finite checks, zero failed checks, 24 synthetic networks, and 782 integer interventions enumerated in cost checks. These are finite internal computations, not proof-assistant verification or independent replication. All computational test data are synthetic; no experimental curves were fitted. The archive preserves the initial failed execution, the one-line implementation patch, successful logs, source code, environment specifications, claim and gap ledgers, and checksums. Classical random-walk, corrector, homogenization, and hopping methods are acknowledged. Novelty of the combined result package remains unverified. A quantitative bridge from BIT ontology to the network parameters, empirical validation, a finite-time control certificate, and a full coherent quantum model are not established. The release invites independent review of the proofs, assumptions, prior art, and reproducibility. New results, corrections, and physical extensions should be recorded separately from the frozen R01 artifacts. TIẾNG VIỆT Báo cáo xây dựng một lớp mô hình toán học có điều kiện, lấy câu hỏi bản thể BIT về ranh giới và hiệu dụng làm định hướng. Trong mạng chuyển tiếp thuận nghịch hữu hạn có nhãn độ dời, tài liệu trình bày bốn định lý và một mệnh đề về tensor khuếch tán dài hạn, tiêu chuẩn chu trình có độ dời ròng, ngưỡng kích hoạt, chi phí hạ rào cản và đáp ứng tuyến tính có điều kiện. Bộ nguồn R01 lưu 263 kiểm hữu hạn đạt, 24 mạng tổng hợp và 782 phương án can thiệp nguyên được duyệt; không có dữ liệu thực nghiệm được khớp. Chứng minh, mã, dữ liệu, lần chạy thất bại, bản vá, nhật ký và các giới hạn đều được giữ lại để cộng đồng kiểm tra. Tính mới chưa được xác minh; chưa có kiểm kernel, kiểm độc lập, hay xác thực vật lý đối với mô hình BIT. Đợt phát hành này chỉ bổ sung hồ sơ công bố, không sửa hoặc mở lại nghiên cứu R01. AI assistance and provenance The source report credits GPT + BIT assistance for model construction, derivations, coding, and internal checks. No independent review or proof-assistant kernel verification is claimed. Public review invitation Readers are invited to inspect the assumptions and proofs, reproduce the archived finite checks, examine prior art, and report precise corrections or counterexamples with the affected theorem ID, assumptions, and evidence. The frozen R01 archive should be preserved; later findings should be documented in a separately identified review or revision. Rights and attribution CC BY 4.0 applies to the report, explanatory documentation and author-controlled synthetic data. The original R01 archive contains no explicit software license; no MIT, Apache or other software license is added. Code is retained as research evidence for inspection and verification. No additional software reuse permission should be inferred solely from the documentation license. See RIGHTS_AND_ATTRIBUTION.md. Existing third-party rights remain unchanged.