Division Under the Unitary Reference Principle: Three Hidden Assumptions, the Incomplete Answer Problem, the Shared Reference Division Principle, and the Infinite Decimal as Depletion-Renewal Cycle
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| Format: | Recurso digital |
| Language: | English |
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2026
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| _version_ | 1866901914493386752 |
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| author | Brogley, Joshua |
| author_facet | Brogley, Joshua |
| contents | <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Abstract</strong></p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This paper extends the Unitary Reference Principle (URP), first proposed in Brogley (2026a), to the specific domain of division. It identifies three hidden assumptions in conventional division, proposes a structural reformulation under which division is a declaration of partition rather than a reduction operation, and develops the Shared Reference Division Principle for cases where quantities exist in incommensurable dimensional domains.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The paper demonstrates that the infinite repeating decimal is a depletion-renewal cycle — base-10 arithmetic repeatedly attempting and failing to express an exact fraction — and that the fraction is the exact answer while the decimal is the approximation. Division by zero is shown to be not undefined but unconstructable: zero has no declared reference, so there is nothing to divide into.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This second edition incorporates cross-series consistency updates applied uniformly across all published URP papers: all subscript notation standardised (R(bridge), R(mass), R(volume), R(distance), R(time)); all citations updated to Brogley (2026a) reflecting the numbered series; the series citation line updated to show all four published DOIs; and a note on surplus (n > R) added before Section 3.5, pointing to the formal treatment in Brogley (2026a) Sections 1.5a–1.5e. No structural changes were made to the framework, proofs, or demonstrations.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Contents:</strong> Chapter 1: The Problem With Division | Chapter 2: A History of Division — Four Thousand Years of Almost | Chapter 3: The URP Applied to Division (including Step 0, the Shared Reference Division Principle, and the Infinite Decimal as Depletion-Renewal Cycle) | Chapter 4: Demonstrations | Chapter 5: The Egyptian Fraction Connection | Chapter 6: Division Problems of Debate | Chapter 7: Implications and Conclusion</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Part of the URP Series:</strong></p> <ul class="[li_&]:mb-0 [li_&]:mt-1 [li_&]:gap-1 [&:not(:last-child)_ul]:pb-1 [&:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3"> <li class="whitespace-normal break-words pl-2">Paper 1 (Foundation): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19697119">https://doi.org/10.5281/zenodo.19697119</a></li> <li class="whitespace-normal break-words pl-2">Paper 2 (this paper — Division): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19733441">https://doi.org/10.5281/zenodo.19733441</a></li> <li class="whitespace-normal break-words pl-2">Paper 3 (Riemann Hypothesis): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19735713">https://doi.org/10.5281/zenodo.19735713</a></li> <li class="whitespace-normal break-words pl-2">Paper 4 (Geometry): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19847459">https://doi.org/10.5281/zenodo.19847459</a></li> </ul> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Version notes:</strong></p> <ul class="[li_&]:mb-0 [li_&]:mt-1 [li_&]:gap-1 [&:not(:last-child)_ul]:pb-1 [&:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3"> <li class="whitespace-normal break-words pl-2">v1 — April 2026: First edition</li> <li class="whitespace-normal break-words pl-2">v2 — April 28, 2026: Second edition — notation standardised, series citations added, surplus note added (Section 3.4), reference [1] updated to Third Edition</li> </ul> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Keywords</strong> division, Unitary Reference Principle, declared denominator, equal partition assumption, incomplete answer problem, Shared Reference Division Principle, depletion-renewal cycle, infinite decimal, division by zero, unconstructable, incommensurable references, bridge reference, Egyptian fractions, Erdős-Straus conjecture, surplus</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19851980 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Division Under the Unitary Reference Principle: Three Hidden Assumptions, the Incomplete Answer Problem, the Shared Reference Division Principle, and the Infinite Decimal as Depletion-Renewal Cycle Brogley, Joshua <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Abstract</strong></p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This paper extends the Unitary Reference Principle (URP), first proposed in Brogley (2026a), to the specific domain of division. It identifies three hidden assumptions in conventional division, proposes a structural reformulation under which division is a declaration of partition rather than a reduction operation, and develops the Shared Reference Division Principle for cases where quantities exist in incommensurable dimensional domains.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The paper demonstrates that the infinite repeating decimal is a depletion-renewal cycle — base-10 arithmetic repeatedly attempting and failing to express an exact fraction — and that the fraction is the exact answer while the decimal is the approximation. Division by zero is shown to be not undefined but unconstructable: zero has no declared reference, so there is nothing to divide into.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This second edition incorporates cross-series consistency updates applied uniformly across all published URP papers: all subscript notation standardised (R(bridge), R(mass), R(volume), R(distance), R(time)); all citations updated to Brogley (2026a) reflecting the numbered series; the series citation line updated to show all four published DOIs; and a note on surplus (n > R) added before Section 3.5, pointing to the formal treatment in Brogley (2026a) Sections 1.5a–1.5e. No structural changes were made to the framework, proofs, or demonstrations.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Contents:</strong> Chapter 1: The Problem With Division | Chapter 2: A History of Division — Four Thousand Years of Almost | Chapter 3: The URP Applied to Division (including Step 0, the Shared Reference Division Principle, and the Infinite Decimal as Depletion-Renewal Cycle) | Chapter 4: Demonstrations | Chapter 5: The Egyptian Fraction Connection | Chapter 6: Division Problems of Debate | Chapter 7: Implications and Conclusion</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Part of the URP Series:</strong></p> <ul class="[li_&]:mb-0 [li_&]:mt-1 [li_&]:gap-1 [&:not(:last-child)_ul]:pb-1 [&:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3"> <li class="whitespace-normal break-words pl-2">Paper 1 (Foundation): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19697119">https://doi.org/10.5281/zenodo.19697119</a></li> <li class="whitespace-normal break-words pl-2">Paper 2 (this paper — Division): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19733441">https://doi.org/10.5281/zenodo.19733441</a></li> <li class="whitespace-normal break-words pl-2">Paper 3 (Riemann Hypothesis): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19735713">https://doi.org/10.5281/zenodo.19735713</a></li> <li class="whitespace-normal break-words pl-2">Paper 4 (Geometry): <a class="underline underline underline-offset-2 decoration-1 decoration-current/40 hover:decoration-current focus:decoration-current" href="https://doi.org/10.5281/zenodo.19847459">https://doi.org/10.5281/zenodo.19847459</a></li> </ul> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Version notes:</strong></p> <ul class="[li_&]:mb-0 [li_&]:mt-1 [li_&]:gap-1 [&:not(:last-child)_ul]:pb-1 [&:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3"> <li class="whitespace-normal break-words pl-2">v1 — April 2026: First edition</li> <li class="whitespace-normal break-words pl-2">v2 — April 28, 2026: Second edition — notation standardised, series citations added, surplus note added (Section 3.4), reference [1] updated to Third Edition</li> </ul> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Keywords</strong> division, Unitary Reference Principle, declared denominator, equal partition assumption, incomplete answer problem, Shared Reference Division Principle, depletion-renewal cycle, infinite decimal, division by zero, unconstructable, incommensurable references, bridge reference, Egyptian fractions, Erdős-Straus conjecture, surplus</p> |
| title | Division Under the Unitary Reference Principle: Three Hidden Assumptions, the Incomplete Answer Problem, the Shared Reference Division Principle, and the Infinite Decimal as Depletion-Renewal Cycle |
| url | https://doi.org/10.5281/zenodo.19851980 |