# UQFF Lagrangian Bridge: 11 Primitives \& 172 Closures from $L_{\mathrm{UQFF

**Author:** Star-Magic UQFF Program (PAPER\_1210)

**Date:** May 2026

---

## Claim
Every algebraic closure produced by the UQFF program to date (S261--S682,
172 numeric verifications captured in `master_closures.csv`) is a
descendant of the unified action
$$
S_{\mathrm{UQFF}}=\int d^{D_c}x\sqrt{-g}\sum_{a=1}^{N_{ch}} \mathcal{L}_a,
\quad N_{ch}=9,\ D_c=26,
$$
through stationarity $\delta S_{\mathrm{UQFF}}/\delta\phi_I=0$ in one of the
nine sectors $\mathcal{L}_a\in{\mathrm{L_{EH},L_{YM},L_{Dirac},L_{\phi},
L_{mag},L_{buoy},L_{aether},L_{LENR},L_{KK}}}$
(`uqff_lagrangian_derivation.py`,
sectors~1--9).

## Primitive Origin Map
\begin{center}
\begin{tabular}{lll}
\toprule
Primitive & Value & Originating sector \midrule
$\Ftrz$         & $1/10$       & $\mathcal{L}_{\mathrm{aether}}$ (7) $\Phires$       & $5/6$        & $\mathcal{L}_{\mathrm{KK}}$     (9) $\SSq$          & $57/100$     & $\mathcal{L}_{\phi}$            (4) $\KMex$         & $25/12$      & $\mathcal{L}_{\phi}$            (4) $D_\phi$        & $4$          & $\mathcal{L}_{\mathrm{EH}}$     (1) $D_B$           & $6$          & $\mathcal{L}_{\mathrm{buoy}}$   (6) $D_c$           & $26$         & $\mathcal{L}_{\mathrm{KK}}$     (9) $N_{ch}$        & $9$          & master (sector count) $\mathrm{SO}(5)$& $10$         & $\mathcal{L}_{\mathrm{YM}}$     (2) $A_5$           & $60$         & $\mathcal{L}_{\mathrm{YM}}$     (2) $\beta_i$       & $6029/10000$ & $\mathcal{L}_{\mathrm{buoy}}$   (6) \bottomrule
\end{tabular}
\end{center}

Each entry is verified by S683 (`_session683_lagrangian_primitives.py`)
which prints the stationarity condition that fixes the primitive.

## Closure-to-Sector Bridge (S684)
$_session684_closure_to_sector_bridge.py$ scans
`master_closures.csv` and parses each S533--S682 expression for
primitive tokens, mapping each closure to the union of sectors needed.
Result:
\begin{center}
\begin{tabular}{lr}
\toprule
Sector & Closures invoking it \midrule
$\mathcal{L}_{\mathrm{aether}}$ (7) & 135 $\mathcal{L}_{\phi}$            (4) & 109 $\mathcal{L}_{\mathrm{EH}}$     (1) &  96 $\mathcal{L}_{\mathrm{buoy}}$   (6) &  93 $\mathcal{L}_{\mathrm{KK}}$     (9) &  88 $\mathcal{L}_{\mathrm{YM}}$     (2) &  85 master (channel count)              &  56 \bottomrule
\end{tabular}
\end{center}
Total: **172/172 closures bridged** to one or more $L_{\mathrm{UQFF}}$ sectors.

## Sigma Census
From `sigma_table.csv`:
\begin{center}
\begin{tabular}{lr}
\toprule
Accuracy band & Count \midrule
EXACT (= 0%)        & 47 $< 0.01%$           & 17 $0.01$--$0.1%$      & 60 $0.1$--$1%$         & 45 $\geq 1%$           &  3 \bottomrule
\end{tabular}
\end{center}

## What This Closes
\begin{itemize}
\item **Priority #2 (Lagrangian re-run on S533--S682).**  Every closure
      now carries an explicit sector tag rather than living as a
      free-floating algebraic identity.
\item **Priority #3 (sigma falsifiability table).**  `sigma_table.csv`
      bins all 172 closures by accuracy; the 47 EXACT entries are unfalsifiable
      identities, the 64 sub-$0.01%$ entries are predictions that fail
      under any non-trivial perturbation of the 11 primitives.
\item **Priority #6 (first-principle primitive derivation).**  All 11
      frozen primitives are now traced back to the Euler--Lagrange
      stationarity of a specific sector of $L_{\mathrm{UQFF}}$.
\end{itemize}

## Reproducibility
\begin{verbatim}
.\.venv\Scripts\python.exe _uqff_program.py --audit
.\.venv\Scripts\python.exe _uqff_program.py --sigma
.\.venv\Scripts\python.exe _session683_lagrangian_primitives.py
.\.venv\Scripts\python.exe _session684_closure_to_sector_bridge.py
\end{verbatim}