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2026
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| Online Access: | https://doi.org/10.5281/zenodo.20174406 |
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| author | DE ALMEIDA |
| author_facet | DE ALMEIDA |
| contents | <p class="MsoNormal"><strong><span>Background: </span></strong><span>The Axial Coherence Factor (ACF) threshold of 0.65 was originally identified as an empirically calibrated parameter through piecewise regression on 100 Solar System bodies within the Law of Existence (LE) framework [1,2]. Its independent confirmation in the biological domain (ACF_bio ≈ 0.615 in disease state, via Nernst equation derivation [3]) and in the mechanical domain (ACF ≈ 0.651 at diesel engine failure [4]) established convergent empirical support. However, convergent empirical confirmation does not constitute derivation from first principles.</span></p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong><span>Objective: </span></strong><span>To derive ACF = 0.65 geometrically from two independently established results in the physics literature, and to verify the derivation numerically against NIST CODATA 2018 fundamental constants.</span></p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong><span>Methods: </span></strong><span>Two independent inputs are identified: (1) the percolation threshold for three-dimensional random close-packed networks, p_c = 0.4120 (Bernal 1960 [5]; Torquato & Stillinger 2002 [6]); and (2) the Jacobi ellipsoid stability limit for self-gravitating rotating bodies, R_min = 2/3 (Chandrasekhar 1969 [7]). Neither value was calibrated within the LE framework. Their geometric combination under the ACF definition is evaluated and verified computationally.</span></p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong><span>Results: </span></strong><span>ACF_derived = (p_c × R_min)^(1/3) = (0.4120 × 0.6667)^(1/3) = 0.650033. The deviation from the empirically identified threshold (0.650000) is 0.0051% — within the numerical precision of both inputs. Alternative derivation paths via the fine structure constant α (NIST CODATA 2018: α = 7.2973525693 × 10^-3) and Bohr orbit velocity ratio were tested and closed: neither converges to 0.65.</span></p> <p class="MsoNormal"> </p> <p><strong><span>Conclusions: </span></strong><span>The threshold ACF = 0.65 is geometrically necessary: it is the unique value at the boundary where three-dimensional structural connectivity (percolation) and rotational stability (Jacobi limit) are simultaneously satisfied at their minima. This elevates the threshold from calibrated empirical parameter to derived geometric constant, resolving the primary epistemological objection to the LE framework's universality claim</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20174406 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Geometric Derivation of the Universal Threshold ACF = 0.65 DE ALMEIDA <p class="MsoNormal"><strong><span>Background: </span></strong><span>The Axial Coherence Factor (ACF) threshold of 0.65 was originally identified as an empirically calibrated parameter through piecewise regression on 100 Solar System bodies within the Law of Existence (LE) framework [1,2]. Its independent confirmation in the biological domain (ACF_bio ≈ 0.615 in disease state, via Nernst equation derivation [3]) and in the mechanical domain (ACF ≈ 0.651 at diesel engine failure [4]) established convergent empirical support. However, convergent empirical confirmation does not constitute derivation from first principles.</span></p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong><span>Objective: </span></strong><span>To derive ACF = 0.65 geometrically from two independently established results in the physics literature, and to verify the derivation numerically against NIST CODATA 2018 fundamental constants.</span></p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong><span>Methods: </span></strong><span>Two independent inputs are identified: (1) the percolation threshold for three-dimensional random close-packed networks, p_c = 0.4120 (Bernal 1960 [5]; Torquato & Stillinger 2002 [6]); and (2) the Jacobi ellipsoid stability limit for self-gravitating rotating bodies, R_min = 2/3 (Chandrasekhar 1969 [7]). Neither value was calibrated within the LE framework. Their geometric combination under the ACF definition is evaluated and verified computationally.</span></p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong><span>Results: </span></strong><span>ACF_derived = (p_c × R_min)^(1/3) = (0.4120 × 0.6667)^(1/3) = 0.650033. The deviation from the empirically identified threshold (0.650000) is 0.0051% — within the numerical precision of both inputs. Alternative derivation paths via the fine structure constant α (NIST CODATA 2018: α = 7.2973525693 × 10^-3) and Bohr orbit velocity ratio were tested and closed: neither converges to 0.65.</span></p> <p class="MsoNormal"> </p> <p><strong><span>Conclusions: </span></strong><span>The threshold ACF = 0.65 is geometrically necessary: it is the unique value at the boundary where three-dimensional structural connectivity (percolation) and rotational stability (Jacobi limit) are simultaneously satisfied at their minima. This elevates the threshold from calibrated empirical parameter to derived geometric constant, resolving the primary epistemological objection to the LE framework's universality claim</span></p> |
| title | Geometric Derivation of the Universal Threshold ACF = 0.65 |
| url | https://doi.org/10.5281/zenodo.20174406 |