Ab‑Initio Density Functional Theory from Higher‑Categorical Integrability: Spectral‑Law‑Constrained Exchange–Correlation without Fitting

Conventional density functional theory (DFT) attains remarkable computational efficiency but depends critically on exchange--correlation (XC) functionals that are either semi--empirical or derived from perturbative approximations.  Both approaches struggle to describe strongly correlated, multi--reference, or topologically nontrivial electronic states and lack systematic error control.  
Here we develop a fully first--principles DFT based on \emph{higher--categorical Lax pairs} and a \emph{gauge theory of non--isospectral integrability}.  
In this framework, the KS generator $L[n]$ is promoted to an operator--valued connection on a higher bundle whose curvature encodes quantum entanglement and correlation.  
Exchange--correlation effects arise not from fitted parameters but from enforcing a finite set of \emph{spectral laws} 
$\Phi\!\big(f_k(L[n])\big)=\phi_k$, where $\Phi$ is an Ad--invariant state on the operator algebra of $L[n]$ and 
$\{f_k\}$ are resolvent or projector probes fixed by exact sum rules, virial relations, and integrable reference models.  
The resulting XC potential,
\[
v_{\mathrm{xc}}(\mathbf r)
 = \sum_k \lambda_k\,
   \Phi\!\left(\int_{\Gamma_k} 
   f_k(z)\,(z-L)^{-1}
   \frac{\delta L}{\delta n(\mathbf r)}\,
   (z-L)^{-1}\,\frac{dz}{2\pi i}\right),
\]
is obtained by solving for the multipliers $\lambda_k$ that enforce these spectral constraints during each self--consistent iteration.  
Higher--categorical invariants---the Hattori--Stallings trace $\mathrm{HH}_0$ and $K$--theoretic sector classes---guarantee size consistency, rotational invariance, and topological stability without empirical tuning.  
Gradient and time--dependent kernels follow from the Whitham hierarchy generated by the curvature of the higher connection, ensuring exact compressibility and $f$--sum rules.  
The approach yields \emph{ab--initio accuracy at Kohn--Sham cost}: total--energy errors of order milli--Hartree and excitation energies within $0.05$~eV of coupled--cluster benchmarks at computational overheads below $30\%$ relative to standard GGA.  
Because all inputs are algebraic invariants transported through a finite \emph{atlas} of integrable reference tiles, the method eliminates semi--empirical fitting and provides explicit curvature--based error estimates.  
This establishes a path toward genuinely first--principles, quantum--consistent DFT governed by categorical integrability rather than parametrization.