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| author | Szoke, Barna |
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| contents | <h2><a name="Xd0653d366dcd5678b5f20bc76c070b7c524754d"></a>Part III of Mass Emergence from Two-Twistor Geometry</h2> <p><span><strong>Date:</strong> March 2026</span></p> <div> </div> <h2><a name="introduction-the-single-cone-framework"></a>1. Introduction: The Single-Cone Framework</h2> <p><span>Parts I and II of this series derived particle masses and coupling constants from the F2 axiom using two counter-rotating cones. The algebraic motivation, rooted in Penrose two-twistor theory, required two separate cones to carry the internal degrees of freedom. The physical picture, however, is simpler: a single cone with two counter-rotating hemispheres (upper and lower).</span></p> <p><span>The Pin_{+,-,-} algebra forces the (2,3) signature. The constraint</span></p> <p><span><span>c1 / t2 = c2 / t1 = A</span></span></p> <p><span>ensures causality: the product sigma = t1 * t2 is monotonically increasing, which excludes closed timelike curves. The two time coordinates are not independent but locked through the shared constant A.</span></p> <p><span>This paper derives the Poinsot geometry of the single-cone configuration and extracts gravitational wave predictions. The results include a geometric tensor-to-scalar ratio, black hole ringdown eigenfrequencies, and a falsifiable prediction for the LIGO IR1 data release.</span></p> <div> </div> <h2><a name="poinsot-ellipsoid-evolution"></a>2. Poinsot Ellipsoid Evolution</h2> <p><span>Consider a solid cone of half-angle theta, constant volume V, and homogeneous density. The principal moments of inertia are</span></p> <p><span><span>I3 = (3/10) m r^2<span> </span>(spin axis)</span><br><span>I1 = (3m/20)(r^2 + 4h^2) (perpendicular axes)</span></span></p> <p><span>where r is the base radius and h the height, related by r = h tan(theta).</span></p> <p><span>Three geometric thresholds exist where the ratio I1/I3 takes exact rational values.</span></p> <table style="border-collapse: collapse;"> <thead> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>theta (deg)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>I1 / I3</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r vs h</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>I1</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>I3</span></p> </td> </tr> </thead> <tbody> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>60</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>7/6 (exact)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r = h sqrt(3)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>21 h^2 / 20</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>9 h^2 / 10</span></p> </td> </tr> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>45</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>5/2 (exact)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r = h</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>3 h^2 / 4</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>3 h^2 / 10</span></p> </td> </tr> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>30</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>13/2 (exact)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r = h / sqrt(3)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>13 h^2 / 20</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>h^2 / 10</span></p> </td> </tr> </tbody> </table> <p><span>At theta = arctan(2) = 63.43 deg, the cone becomes a spherical top with I1 = I3 and surface-to-volume ratio S/V = 3.0.</span></p> <p><span>The primes 7, 5, and 13 appearing in these ratios are not inputs to the framework. They are geometric consequences of the cone geometry at distinguished angles. The 30-degree configuration, where the axial cross-section forms an equilateral triangle (h = r sqrt(3)), is the stabilization point of the evolution.</span></p> <div> </div> <h2><a name="braking-and-torsion"></a>3. Braking and Torsion</h2> <p><span>The two hemispheres of the single cone counter-rotate about the cone axis. In the idealized case of homogeneous density rho, no precession would occur: the angular momentum vectors cancel exactly and the system remains axially symmetric.</span></p> <p><span>However, the probability of identical density in both hemispheres is strictly zero:</span></p> <p><span><span>P(identical rho in both hemispheres) = 0</span></span></p> <p><span>This follows from Bose-Einstein statistics. Any finite system drawn from a thermal ensemble has a vanishing probability of exact microstate duplication across a macroscopic partition.</span></p> <p><span>The density asymmetry delta_rho causes the spin axis to precess. The own-axis rotation brakes, and the braking energy converts to torsion. Torsion is the gravitational field. The curvature-torsion relation is</span></p> <p><span><span>R(omega) = -D(K) - K wedge K</span></span></p> <p><span>so that curvature equals torsion squared.</span></p> <p><span>The graviton energy per cone is</span></p> <p><span><span>delta_rho * E_cone = (3.78 / 250) * 250 = 3.78 GeV</span></span></p> <p><span>The total graviton energy from both hemispheres is 7.56 GeV, distributed among 8 gravitons at approximately 1 GeV each.</span></p> <p><span>The final state of this braking process is complete cessation of own-axis spin. Only precession remains. Precession is mass. The particle spectrum emerges from the precession modes of the Poinsot ellipsoid.</span></p> <div> </div> <h2><a name="Xbd7f00a742c61ff1c287babeb9ecb23ce9a959e"></a>4. Symmetry Breaking from Bose-Einstein Statistics</h2> <p><span>The symmetry breaking event is not a choice imposed on the framework but a thermodynamic necessity. The probability of perfect symmetry in any finite system is</span></p> <p><span><span>P(perfect symmetry) = 1 / Omega -> 0</span></span></p> <p><span>for any finite number of microstates Omega.</span></p> <p><span>The Fano plane represents the configuration of maximum combinatorial order, corresponding to a local minimum of entropy S. The birth of structure (particles, fields, spacetime curvature) is a local entropy decrease, which is mandatory for structure formation in any thermodynamic framework.</span></p> <p><span>The symmetry breaks in entropy, not in space or energy. The geometric process of theta decreasing from 90 degrees is shifted from the ideal spherical-top angle of 63.43 degrees to the rational-ratio angles 60 and 30 degrees by the Bose-Einstein asymmetry.</span></p> <div> </div> <h2>5. c1/c2 Modulation and Polarization</h2> <p>The two light speeds are related to the two time coordinates by</p> <p><span>c1 = A * t2,<span> </span>c2 = A * t1</span></p> <p>so that</p> <p><span>c1 / c2 = t2 / t1 = exp(-phi)</span></p> <p>where phi = arctan(sigma / sigma_0). The ratio c1/c2 therefore evolves with cosmological epoch. The product is invariant:</p> <p><span>c1 * c2 = A^2 * sigma = c^2</span></p> <p>Before the axis break, the two counter-rotating waves cancel. The residual amplitude is</p> <p><span>2 * delta_rho = 0.030</span></p> <p>After the axis break, two polarization modes emerge:</p> <ul> <li><strong>E-mode</strong> (visible time modulation): gradient-like, curl-free. Compressed sinusoid with maximum amplitude less than unity.</li> <li><strong>B-mode</strong> (departing time modulation): curl-like, divergence-free. Amplitude given by</li> </ul> <p><span><span><span> </span><span> </span></span></span><span>B = sin(30) * precession rate = sin(30) * 2/13</span></p> <p>E is perpendicular to B at all times. Both E-mode and B-mode are normalized to c^2 (the common invariant), not to each other.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19164841 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Mass Emergence from Two-Twistor Geometry Szoke, Barna <h2><a name="Xd0653d366dcd5678b5f20bc76c070b7c524754d"></a>Part III of Mass Emergence from Two-Twistor Geometry</h2> <p><span><strong>Date:</strong> March 2026</span></p> <div> </div> <h2><a name="introduction-the-single-cone-framework"></a>1. Introduction: The Single-Cone Framework</h2> <p><span>Parts I and II of this series derived particle masses and coupling constants from the F2 axiom using two counter-rotating cones. The algebraic motivation, rooted in Penrose two-twistor theory, required two separate cones to carry the internal degrees of freedom. The physical picture, however, is simpler: a single cone with two counter-rotating hemispheres (upper and lower).</span></p> <p><span>The Pin_{+,-,-} algebra forces the (2,3) signature. The constraint</span></p> <p><span><span>c1 / t2 = c2 / t1 = A</span></span></p> <p><span>ensures causality: the product sigma = t1 * t2 is monotonically increasing, which excludes closed timelike curves. The two time coordinates are not independent but locked through the shared constant A.</span></p> <p><span>This paper derives the Poinsot geometry of the single-cone configuration and extracts gravitational wave predictions. The results include a geometric tensor-to-scalar ratio, black hole ringdown eigenfrequencies, and a falsifiable prediction for the LIGO IR1 data release.</span></p> <div> </div> <h2><a name="poinsot-ellipsoid-evolution"></a>2. Poinsot Ellipsoid Evolution</h2> <p><span>Consider a solid cone of half-angle theta, constant volume V, and homogeneous density. The principal moments of inertia are</span></p> <p><span><span>I3 = (3/10) m r^2<span> </span>(spin axis)</span><br><span>I1 = (3m/20)(r^2 + 4h^2) (perpendicular axes)</span></span></p> <p><span>where r is the base radius and h the height, related by r = h tan(theta).</span></p> <p><span>Three geometric thresholds exist where the ratio I1/I3 takes exact rational values.</span></p> <table style="border-collapse: collapse;"> <thead> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>theta (deg)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>I1 / I3</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r vs h</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>I1</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>I3</span></p> </td> </tr> </thead> <tbody> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>60</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>7/6 (exact)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r = h sqrt(3)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>21 h^2 / 20</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>9 h^2 / 10</span></p> </td> </tr> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>45</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>5/2 (exact)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r = h</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>3 h^2 / 4</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>3 h^2 / 10</span></p> </td> </tr> <tr style=""> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>30</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>13/2 (exact)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>r = h / sqrt(3)</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>13 h^2 / 20</span></p> </td> <td style="padding: 0in 5.4pt 0in 5.4pt;"> <p><span>h^2 / 10</span></p> </td> </tr> </tbody> </table> <p><span>At theta = arctan(2) = 63.43 deg, the cone becomes a spherical top with I1 = I3 and surface-to-volume ratio S/V = 3.0.</span></p> <p><span>The primes 7, 5, and 13 appearing in these ratios are not inputs to the framework. They are geometric consequences of the cone geometry at distinguished angles. The 30-degree configuration, where the axial cross-section forms an equilateral triangle (h = r sqrt(3)), is the stabilization point of the evolution.</span></p> <div> </div> <h2><a name="braking-and-torsion"></a>3. Braking and Torsion</h2> <p><span>The two hemispheres of the single cone counter-rotate about the cone axis. In the idealized case of homogeneous density rho, no precession would occur: the angular momentum vectors cancel exactly and the system remains axially symmetric.</span></p> <p><span>However, the probability of identical density in both hemispheres is strictly zero:</span></p> <p><span><span>P(identical rho in both hemispheres) = 0</span></span></p> <p><span>This follows from Bose-Einstein statistics. Any finite system drawn from a thermal ensemble has a vanishing probability of exact microstate duplication across a macroscopic partition.</span></p> <p><span>The density asymmetry delta_rho causes the spin axis to precess. The own-axis rotation brakes, and the braking energy converts to torsion. Torsion is the gravitational field. The curvature-torsion relation is</span></p> <p><span><span>R(omega) = -D(K) - K wedge K</span></span></p> <p><span>so that curvature equals torsion squared.</span></p> <p><span>The graviton energy per cone is</span></p> <p><span><span>delta_rho * E_cone = (3.78 / 250) * 250 = 3.78 GeV</span></span></p> <p><span>The total graviton energy from both hemispheres is 7.56 GeV, distributed among 8 gravitons at approximately 1 GeV each.</span></p> <p><span>The final state of this braking process is complete cessation of own-axis spin. Only precession remains. Precession is mass. The particle spectrum emerges from the precession modes of the Poinsot ellipsoid.</span></p> <div> </div> <h2><a name="Xbd7f00a742c61ff1c287babeb9ecb23ce9a959e"></a>4. Symmetry Breaking from Bose-Einstein Statistics</h2> <p><span>The symmetry breaking event is not a choice imposed on the framework but a thermodynamic necessity. The probability of perfect symmetry in any finite system is</span></p> <p><span><span>P(perfect symmetry) = 1 / Omega -> 0</span></span></p> <p><span>for any finite number of microstates Omega.</span></p> <p><span>The Fano plane represents the configuration of maximum combinatorial order, corresponding to a local minimum of entropy S. The birth of structure (particles, fields, spacetime curvature) is a local entropy decrease, which is mandatory for structure formation in any thermodynamic framework.</span></p> <p><span>The symmetry breaks in entropy, not in space or energy. The geometric process of theta decreasing from 90 degrees is shifted from the ideal spherical-top angle of 63.43 degrees to the rational-ratio angles 60 and 30 degrees by the Bose-Einstein asymmetry.</span></p> <div> </div> <h2>5. c1/c2 Modulation and Polarization</h2> <p>The two light speeds are related to the two time coordinates by</p> <p><span>c1 = A * t2,<span> </span>c2 = A * t1</span></p> <p>so that</p> <p><span>c1 / c2 = t2 / t1 = exp(-phi)</span></p> <p>where phi = arctan(sigma / sigma_0). The ratio c1/c2 therefore evolves with cosmological epoch. The product is invariant:</p> <p><span>c1 * c2 = A^2 * sigma = c^2</span></p> <p>Before the axis break, the two counter-rotating waves cancel. The residual amplitude is</p> <p><span>2 * delta_rho = 0.030</span></p> <p>After the axis break, two polarization modes emerge:</p> <ul> <li><strong>E-mode</strong> (visible time modulation): gradient-like, curl-free. Compressed sinusoid with maximum amplitude less than unity.</li> <li><strong>B-mode</strong> (departing time modulation): curl-like, divergence-free. Amplitude given by</li> </ul> <p><span><span><span> </span><span> </span></span></span><span>B = sin(30) * precession rate = sin(30) * 2/13</span></p> <p>E is perpendicular to B at all times. Both E-mode and B-mode are normalized to c^2 (the common invariant), not to each other.</p> |
| title | Mass Emergence from Two-Twistor Geometry |
| url | https://doi.org/10.5281/zenodo.19164841 |