Minkowski–Euclidean Duality and Gravitational Coupling from Block-Alternating Infinite Products
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2026
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| author | Fujii, Masanori |
| author_facet | Fujii, Masanori |
| contents | <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">We study the physical content of two block-alternating infinite products A(K,d) and B(K,d), identifying the parameter K as a distance scale. Three results emerge from this identification, without any relativistic or field-theoretic assumption.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">First, the logarithmic K-derivative of A satisfies d/dK log A = (d/4) K^{-2} + O(K^{-3}), establishing a Newtonian gravitational potential with coupling GM = d/4. The null point K = 1/2 coincides exactly with the Schwarzschild radius r_S = d/2, and the determinant identity det M_A + det M_B = 0 corresponds to the on-shell geodesic condition v^2 = GM/r for circular orbits.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Second, extending K to the complex plane via K = 1/2 + r exp(i<em>theta), the matrix M_B satisfies the exact identity det M_B(1/2 + r exp(i</em>theta)) = -4r^2 exp(2i*theta). This provides a structure analogous to the Wick rotation: the Minkowski and Euclidean regimes appear as two faces of a single analytic expression, with K = 1/2 as the rotation axis.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Third, the product family A^a × B^b carries gravitational coupling GM = (a+2b)d/4, forming a discrete spectrum with quantum number n = a+2b. Setting G = 1/4 gives M = d: the mass of the gravitational source equals the block length d, which is a positive integer by construction. This is a form of mass quantization arising purely from the combinatorial structure of the sign alternation.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19895132 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Minkowski–Euclidean Duality and Gravitational Coupling from Block-Alternating Infinite Products Fujii, Masanori block-alternating infinite products Newtonian gravity Schwarzschild geometry Wick rotation mass quantization digamma function Ramanujan summation <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">We study the physical content of two block-alternating infinite products A(K,d) and B(K,d), identifying the parameter K as a distance scale. Three results emerge from this identification, without any relativistic or field-theoretic assumption.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">First, the logarithmic K-derivative of A satisfies d/dK log A = (d/4) K^{-2} + O(K^{-3}), establishing a Newtonian gravitational potential with coupling GM = d/4. The null point K = 1/2 coincides exactly with the Schwarzschild radius r_S = d/2, and the determinant identity det M_A + det M_B = 0 corresponds to the on-shell geodesic condition v^2 = GM/r for circular orbits.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Second, extending K to the complex plane via K = 1/2 + r exp(i<em>theta), the matrix M_B satisfies the exact identity det M_B(1/2 + r exp(i</em>theta)) = -4r^2 exp(2i*theta). This provides a structure analogous to the Wick rotation: the Minkowski and Euclidean regimes appear as two faces of a single analytic expression, with K = 1/2 as the rotation axis.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Third, the product family A^a × B^b carries gravitational coupling GM = (a+2b)d/4, forming a discrete spectrum with quantum number n = a+2b. Setting G = 1/4 gives M = d: the mass of the gravitational source equals the block length d, which is a positive integer by construction. This is a form of mass quantization arising purely from the combinatorial structure of the sign alternation.</p> |
| title | Minkowski–Euclidean Duality and Gravitational Coupling from Block-Alternating Infinite Products |
| topic | block-alternating infinite products Newtonian gravity Schwarzschild geometry Wick rotation mass quantization digamma function Ramanujan summation |
| url | https://doi.org/10.5281/zenodo.19895132 |