Self-consistency of the chiral soliton: the heat kernel constrains the Lagrangian
Abstract
We close the spectral chain initiated in Emergence Papers I–III by investigating the inverse problem: to what extent does the heat-kernel trace K(τ) of the Dirac operator in the soliton framework [Emergence Paper III] constrain the effective Lagrangian ℒeff and, through it, the hedgehog profile F(r)? Four numerical results are established. (N1) The Seeley–DeWitt asymptotic expansion of K(τ) saturates the Skyrme Lagrangian at O(p4) to better than 0.02%, confirming that UV spectral data constrain ℒeff but do not fully determine F(r). (N2) An inverse spectral reconstruction from phase-shift data recovers the self-consistent profile to 0.98% (L2), supporting a topological Borg–Marchenko conjecture within the B=1 sector. (N2+) The Fréchet Jacobian of the spectral map Φ: F → {δκ(ε)} has full rank 11/11 at the Hartree–Fock fixed point, establishing local injectivity via the implicit function theorem; the singular-value spectrum reveals a three-tier hierarchy with condition number κ ≈ 106 that quantitatively explains the reconstruction residual. (N3a) A self-consistent fixed-point scan shows that 6 out of 7 initial profiles converge to the same F* under Anderson mixing (<0.41% L2), providing a numerical substitute for the Hohenberg–Kohn uniqueness theorem. A detailed parallel with density-functional theory — sharing reduced descriptor, self-consistent cycle, and Anderson/DIIS acceleration — is developed. Taken together, these results demonstrate that the spectral chain K(τ) → ℒeff → F(r) → {εn, δκ(ε)} → F*(r) is self-consistently closed at Level A (local, K=0 channel), and identify precise routes toward a global uniqueness theorem.
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BibTeX
@misc{dechery2026selfconsistency,
title = {{Self-consistency of the chiral soliton: the heat kernel constrains the Lagrangian}},
author = {Fabien Déchery},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05787614v3},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Self-consistency of the chiral soliton: the heat kernel constrains the Lagrangian
AU - Fabien Déchery
PY - 2026
DA - 2026/10/08
PB - HAL
UR - https://hal.science/hal-05787614v3
AB - We close the spectral chain initiated in Emergence Papers I–III by investigating the inverse problem: to what extent does the heat-kernel trace K(τ) of the Dirac operator in the soliton framework [Emergence Paper III] constrain the effective Lagrangian ℒ_eff and, through it, the hedgehog profile F(r)? Four numerical results are established. (N1) The Seeley–DeWitt asymptotic expansion of K(τ) saturates the Skyrme Lagrangian at O(p^4) to better than 0.02%, confirming that UV spectral data constrain ℒ_eff but do not fully determine F(r). (N2) An inverse spectral reconstruction from phase-shift data recovers the self-consistent profile to 0.98% (L2), supporting a topological Borg–Marchenko conjecture within the B=1 sector. (N2+) The Fréchet Jacobian of the spectral map Φ: F → {δ_κ(ε)} has full rank 11/11 at the Hartree–Fock fixed point, establishing local injectivity via the implicit function theorem; the singular-value spectrum reveals a three-tier hierarchy with condition number κ ≈ 10^6 that quantitatively explains the reconstruction residual. (N3a) A self-consistent fixed-point scan shows that 6 out of 7 initial profiles converge to the same F* under Anderson mixing (<0.41% L2), providing a numerical substitute for the Hohenberg–Kohn uniqueness theorem. A detailed parallel with density-functional theory — sharing reduced descriptor, self-consistent cycle, and Anderson/DIIS acceleration — is developed. Taken together, these results demonstrate that the spectral chain K(τ) → ℒ_eff → F(r) → {ε_n, δ_κ(ε)} → F^*(r) is self-consistently closed at Level A (local, K=0 channel), and identify precise routes toward a global uniqueness theorem.
ER -