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Vacuum Spin-Torsion Energy Density and Two-Modulus Stability on S2×ℂP2

  • Edward J. Shaya
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Abstract

In Einstein--Cartan--Kaluza--Klein compactification on S2×ℂP2, elimination of the non-propagating torsion generates a quartic spin-current contact interaction whose vacuum expectation value is nonzero even at zero fermion occupation. We reduce the ten-dimensional ΓABC structure to the zero-mode sector and find that the reduction is channel-resolved: each of the three internal Cartan-plane bilinears contribute with the opposite sign to the external (axial) channel, so that the full trace is Rch = Tvac/Taxial = 1 + 3 (−1) = −2. The vacuum term is therefore twice the axial estimate and opposite in sign, and, combined with the elimination constant c10 = −κ10/32, verified by explicit Clifford computation, it is attractive and proportional to Nfam2. We obtain the separated-point correlator, which in the continuum scales as Nfam2 dR/(V6 r6), and, conditional on the physical quantum-boundary prescription, evaluate it at the minimum supported separation. The finite-density piece of the same correlator has the same sign and grows with chemical potential. Binding deepens after nucleation; the single-particle Hartree self-contraction vanishes for chiral zero modes. Using the derived 1/(a12a24) vacuum term with the Einstein-frame curvature and quantized-flux contributions, we analyze a coupled two-modulus toy potential. At fiducial couplings, the (m,n)=(3,3) sector has a finite-radius stationary point with positive Hessian in both radion directions, while the four-family (4,3) minimum lies beyond the adopted quantum-curvature boundary. The three-family sector (1,5) is also locally stable but is less bound. 3-family dominance of the thermal nucleation weight further imposes the condition κ4 ≲ 7.5 (16/B) ℓP2. These results serve as inputs to our higher-level synthesis.

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Cite

BibTeXRIS
BibTeX
@misc{shaya2026vacuum,
  title = {{Vacuum Spin-Torsion Energy Density and Two-Modulus Stability on $S^2\times\mathbbCP^2$}},
  author = {Edward J. Shaya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02909},
  eprint = {2609.02909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02909},
}
RIS
TY  - UNPB
TI  - Vacuum Spin-Torsion Energy Density and Two-Modulus Stability on S^2×ℂP^2
AU  - Edward J. Shaya
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.02909
UR  - https://arxiv.org/abs/2609.02909
AB  - In Einstein--Cartan--Kaluza--Klein compactification on S^2×ℂP^2, elimination of the non-propagating torsion generates a quartic spin-current contact interaction whose vacuum expectation value is nonzero even at zero fermion occupation. We reduce the ten-dimensional Γ_ABC structure to the zero-mode sector and find that the reduction is channel-resolved: each of the three internal Cartan-plane bilinears contribute with the opposite sign to the external (axial) channel, so that the full trace is R_ch = T_vac/T_axial = 1 + 3 (−1) = −2. The vacuum term is therefore twice the axial estimate and opposite in sign, and, combined with the elimination constant c_10 = −κ_10/32, verified by explicit Clifford computation, it is attractive and proportional to N_fam^2. We obtain the separated-point correlator, which in the continuum scales as N_fam^2 d_R/(V_6 r^6), and, conditional on the physical quantum-boundary prescription, evaluate it at the minimum supported separation. The finite-density piece of the same correlator has the same sign and grows with chemical potential. Binding deepens after nucleation; the single-particle Hartree self-contraction vanishes for chiral zero modes. Using the derived 1/(a_1^2a_2^4) vacuum term with the Einstein-frame curvature and quantized-flux contributions, we analyze a coupled two-modulus toy potential. At fiducial couplings, the (m,n)=(3,3) sector has a finite-radius stationary point with positive Hessian in both radion directions, while the four-family (4,3) minimum lies beyond the adopted quantum-curvature boundary. The three-family sector (1,5) is also locally stable but is less bound. 3-family dominance of the thermal nucleation weight further imposes the condition κ_4 ≲ 7.5 (16/B) ℓ_P^2. These results serve as inputs to our higher-level synthesis.
ER  -