Contact Geometry, Exact Counts and Tube Profiles: Collected Studies with Planar and Graph Companions
- 1Independent Researcher
Abstract
Version 2.0 of the registered FDCL Part I working paper, dated 25 October 2026, under the title “Contact Geometry, Exact Counts and Tube Profiles: Collected Studies with Planar and Graph Companions.” This work studies first-level contacts of digit-defined self-similar sets, exact native floor counts, geometric dimensions and Euclidean parallel volumes. For the standard six-map dyadic contact, the native count is N_n=(n+3)2^n+5 for n≥1, its voxel coordinate distortion is 2^n, and the tube volume is π ε² log₂(1/ε)+O(ε²). A reversed-address automaton certificate gives 22 native count formulas for all 255 nonempty binary cube alphabets, valid for n≥2, with a common minimal tail recurrence of order six. The fixed coordinate and endpoint conventions are distinguished from closed-cell counts and geometric dimensions. Specified q-adic families have compact-state tube identities, with horizontal and typed diagonal overlaps treated separately. Exact unit formulas, whole-union asymptotics, log-periodic phases and zeta conclusions retain their stated hypotheses and domains. Critical whole-union remainders have absolute bounds without an asserted sign. Independent companions classify sixteen deterministic planar core–arm systems and give bounds for the specified nodewise random-activation law. The core-free model is dyadic fractal percolation. For the persistent-core model, exact intermediate dimensions and the existence of connected realizations with positive probability at intermediate parameters remain open. A separate four-arc graph study combines written coverage, locality and descent arguments with complete finite certificates; it does not identify that graph with the contact union or assert a spectral limit. Files: a collected reference (90 pages), Binary Cube Contacts, Exact Native Counts and Voxel Distortion (25 pages), Tube Profiles of q-Adic Contacts with Typed Overlaps and Carries (42 pages), Planar Core–Arm Systems and Independent Random Activation (15 pages), A Four-Arc Recursive Graph: Connectivity, Cores and Finite Certificates (13 pages), and a 170-file package containing all five PDFs, editable LaTeX sources, figures, portable finite proof data, supported replay/build scripts, reference outputs and a SHA-256 manifest. The guide distinguishes exact finite proof premises, implementation comparisons and floating-point geometric regression. Automated replays and visual checks are not external peer review or proof-assistant certification. GPT systems (OpenAI) and Claude systems (Anthropic) were used extensively for mathematical arguments, proofs, verification code, manuscript text, literature research and revision. Codex (OpenAI) assisted source preparation and checks. The author takes responsibility for the accuracy, integrity and originality of the work. Original Part I registration (version 1.0): https://doi.org/10.5281/zenodo.23227281
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BibTeX
@article{seol2026contact,
title = {{Contact Geometry, Exact Counts and Tube Profiles: Collected Studies with Planar and Graph Companions}},
author = {Bin Seol},
journal = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23227280},
url = {https://doi.org/10.5281/zenodo.23227280},
note = {CC BY},
}
RIS
TY - JOUR
TI - Contact Geometry, Exact Counts and Tube Profiles: Collected Studies with Planar and Graph Companions
AU - Bin Seol
PY - 2026
DA - 2026/10/08
JO - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23227280
UR - https://doi.org/10.5281/zenodo.23227280
AB - Version 2.0 of the registered FDCL Part I working paper, dated 25 October 2026, under the title “Contact Geometry, Exact Counts and Tube Profiles: Collected Studies with Planar and Graph Companions.” This work studies first-level contacts of digit-defined self-similar sets, exact native floor counts, geometric dimensions and Euclidean parallel volumes. For the standard six-map dyadic contact, the native count is N_n=(n+3)2^n+5 for n≥1, its voxel coordinate distortion is 2^n, and the tube volume is π ε² log₂(1/ε)+O(ε²). A reversed-address automaton certificate gives 22 native count formulas for all 255 nonempty binary cube alphabets, valid for n≥2, with a common minimal tail recurrence of order six. The fixed coordinate and endpoint conventions are distinguished from closed-cell counts and geometric dimensions. Specified q-adic families have compact-state tube identities, with horizontal and typed diagonal overlaps treated separately. Exact unit formulas, whole-union asymptotics, log-periodic phases and zeta conclusions retain their stated hypotheses and domains. Critical whole-union remainders have absolute bounds without an asserted sign. Independent companions classify sixteen deterministic planar core–arm systems and give bounds for the specified nodewise random-activation law. The core-free model is dyadic fractal percolation. For the persistent-core model, exact intermediate dimensions and the existence of connected realizations with positive probability at intermediate parameters remain open. A separate four-arc graph study combines written coverage, locality and descent arguments with complete finite certificates; it does not identify that graph with the contact union or assert a spectral limit. Files: a collected reference (90 pages), Binary Cube Contacts, Exact Native Counts and Voxel Distortion (25 pages), Tube Profiles of q-Adic Contacts with Typed Overlaps and Carries (42 pages), Planar Core–Arm Systems and Independent Random Activation (15 pages), A Four-Arc Recursive Graph: Connectivity, Cores and Finite Certificates (13 pages), and a 170-file package containing all five PDFs, editable LaTeX sources, figures, portable finite proof data, supported replay/build scripts, reference outputs and a SHA-256 manifest. The guide distinguishes exact finite proof premises, implementation comparisons and floating-point geometric regression. Automated replays and visual checks are not external peer review or proof-assistant certification. GPT systems (OpenAI) and Claude systems (Anthropic) were used extensively for mathematical arguments, proofs, verification code, manuscript text, literature research and revision. Codex (OpenAI) assisted source preparation and checks. The author takes responsibility for the accuracy, integrity and originality of the work. Original Part I registration (version 1.0): https://doi.org/10.5281/zenodo.23227281
ER -