FDCL Part III: Resistance and Diffusion on FDCL Graphs — Exact Flows and Harmonic Reduction
- 1Independent Researcher
Abstract
This article studies prescribed electrical loads on the six-digit dyadic FDCL voxel graph and source-retaining reduction on finite weighted graphs. For the native unit-conductance graph, the uniform top-line to bottom-face energy converges to 17/16. Exact computer-assisted certificates restrict every supremum maximizer to levels 9–17 and enclose the supremum; its exact value and maximizing level remain open. The work develops constructive Poincaré and Haar-current estimates, subsequential Mosco convergence on a common address space, norm convergence of embedded resolvents and positive-time semigroups, and linear eigenvalue order for every cluster operator. It also gives a growing-dimensional recursion for boundary Kron conductance and the uniform-volume first-hit law, together with source-preserving positive compression and physical-mass current certificates. Uniform approximation results are stated at fixed positive penalty. The package includes the 121-page article, LaTeX sources, nine figures, exact potential witnesses, rational certificates, verification programs, reproducibility inputs, and a SHA-256 manifest (145 files). The verification driver provides core and full modes; full mode reconstructs the level-nine graph and requires substantially more memory and time. Finite checks certify their specified finite assertions, while all-generation conclusions rely on the stated analytic proofs. Generation-uniform unpenalized current bounds, energy nonconcentration, full-sequence uniqueness, and pointwise continuum heat-kernel identification remain unresolved. Series and status. FDCL Part III, Version 2.0, dated 23 October 2026. This substantially expanded working paper updates version 1.0 (https://doi.org/10.5281/zenodo.23227301). It is not a peer-reviewed publication. The bibliography contains 110 entries. AI assistance and author responsibility. ChatGPT and Codex (OpenAI) and Claude (Anthropic) were used substantively in developing arguments, proofs, verification code and manuscript text, computational analysis, literature comparison and revision. Bin Seol formulated the research questions, directed the mathematical and computational investigations, developed and checked proof and verification strategies, and consolidated the manuscript and supporting data. The author accepts responsibility for the accuracy, integrity and originality of the work. No external funding or competing interests are declared.
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BibTeX
@article{seol2026fdcl,
title = {{FDCL Part III: Resistance and Diffusion on FDCL Graphs — Exact Flows and Harmonic Reduction}},
author = {Bin Seol},
journal = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23227300},
url = {https://doi.org/10.5281/zenodo.23227300},
note = {CC BY},
}
RIS
TY - JOUR
TI - FDCL Part III: Resistance and Diffusion on FDCL Graphs — Exact Flows and Harmonic Reduction
AU - Bin Seol
PY - 2026
DA - 2026/10/08
JO - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23227300
UR - https://doi.org/10.5281/zenodo.23227300
AB - This article studies prescribed electrical loads on the six-digit dyadic FDCL voxel graph and source-retaining reduction on finite weighted graphs. For the native unit-conductance graph, the uniform top-line to bottom-face energy converges to 17/16. Exact computer-assisted certificates restrict every supremum maximizer to levels 9–17 and enclose the supremum; its exact value and maximizing level remain open. The work develops constructive Poincaré and Haar-current estimates, subsequential Mosco convergence on a common address space, norm convergence of embedded resolvents and positive-time semigroups, and linear eigenvalue order for every cluster operator. It also gives a growing-dimensional recursion for boundary Kron conductance and the uniform-volume first-hit law, together with source-preserving positive compression and physical-mass current certificates. Uniform approximation results are stated at fixed positive penalty. The package includes the 121-page article, LaTeX sources, nine figures, exact potential witnesses, rational certificates, verification programs, reproducibility inputs, and a SHA-256 manifest (145 files). The verification driver provides core and full modes; full mode reconstructs the level-nine graph and requires substantially more memory and time. Finite checks certify their specified finite assertions, while all-generation conclusions rely on the stated analytic proofs. Generation-uniform unpenalized current bounds, energy nonconcentration, full-sequence uniqueness, and pointwise continuum heat-kernel identification remain unresolved. Series and status. FDCL Part III, Version 2.0, dated 23 October 2026. This substantially expanded working paper updates version 1.0 (https://doi.org/10.5281/zenodo.23227301). It is not a peer-reviewed publication. The bibliography contains 110 entries. AI assistance and author responsibility. ChatGPT and Codex (OpenAI) and Claude (Anthropic) were used substantively in developing arguments, proofs, verification code and manuscript text, computational analysis, literature comparison and revision. Bin Seol formulated the research questions, directed the mathematical and computational investigations, developed and checked proof and verification strategies, and consolidated the manuscript and supporting data. The author accepts responsibility for the accuracy, integrity and originality of the work. No external funding or competing interests are declared.
ER -