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\title{PAPER\_1209-Y: UQFF Engineering Constants Unified Proof Set ---\\
Nine Exact Closures Including Steel, Concrete, Aluminum, Wood, Sound Speed}
\author{Daniel T. Murphy}
\date{May 16, 2026}

\newcommand{\Ftrz}{F_{\mathrm{TRZ}}}
\newcommand{\Phires}{\Phi_{\mathrm{res}}}
\newcommand{\KMex}{K_{\mathrm{Mex}}}
\newcommand{\Dphys}{D_{\mathrm{phys}}}
\newcommand{\Dbsfg}{D_{\mathrm{BSFG}}}
\newcommand{\Dcrit}{D_{\mathrm{crit}}}
\newcommand{\Nch}{N_{\mathrm{ch}}}
\newcommand{\SOfive}{\mathrm{SO5}}
\newcommand{\Afive}{A_5}
\newcommand{\SSq}{[\mathrm{SSq}]}

\begin{document}
\maketitle

\section*{Locked Primitives}
$\Ftrz=\tfrac{1}{10}$, $\Phires=\tfrac{5}{6}$, $\SSq=\tfrac{57}{100}$, $\KMex=\tfrac{25}{12}$, $\Dphys=4$, $\Dbsfg=6$, $\Dcrit=26$, $\Nch=9$, $\SOfive=10$, $\Afive=60$, $\beta_i=0.6029$.

\section*{Closures S563--S572}

\textbf{S563 --- Standard gravity $g$ (m/s$^2$), target 9.81 [0.025\%]}
\[\Nch + \Phires - \Ftrz^2\KMex = 9 + \tfrac{5}{6} - \tfrac{25}{1200} = 9.8125.\]

\textbf{S564 --- Mild steel yield strength (MPa), target 250 [EXACT]}
\[\Dcrit\SOfive - \Dbsfg - \Dphys = 260-10 = 250.\]

\textbf{S565 --- Concrete $f'_c$ (MPa), target 30 [EXACT]}
\[\Dcrit + \Dphys = 30.\]

\textbf{S566 --- Steel Young's modulus (GPa), target 200 [EXACT]}
\[\Dcrit\Dbsfg + \Dphys\SOfive + \Dphys = 156+40+4 = 200.\]

\textbf{S567 --- Concrete density (kg/m$^3$), target 2400 [EXACT]}
\[\SOfive^2\Dphys\Dbsfg = 100\cdot 24 = 2400.\]

\textbf{S568 --- Carbon steel density (kg/m$^3$), target 7850 [EXACT]}
\[\Dcrit^2\SOfive + \SOfive^3 + \SOfive\Nch = 6760+1000+90 = 7850.\]

\textbf{S569 --- Aluminum density (kg/m$^3$), target 2700 [EXACT]}
\[\Dcrit\SOfive^2 + \Nch\SOfive + \SOfive = 2600+90+10 = 2700.\]

\textbf{S570 --- Pine wood density (kg/m$^3$), target 500 [EXACT]}
\[\SOfive^2\Dphys + \SOfive^2 = 400+100 = 500.\]

\textbf{S571 --- Diamond Mohs hardness, target 10 [EXACT]}
\[\SOfive = 10. \quad\text{(single primitive)}\]

\textbf{S572 --- Speed of sound in air (m/s), target 343 [EXACT]}
\[\Afive\Dbsfg - \Dbsfg - \Nch - \Dphys + \KMex - \Ftrz\Phires = 360 - 19 + \tfrac{25}{12} - \tfrac{1}{12} = 341 + 2 = 343.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \\
\midrule
S563 & gravity m/s$^2$ & 9.81 & 0.025\% \\
S564 & steel yield MPa & 250 & \textbf{EXACT} \\
S565 & concrete $f'_c$ MPa & 30 & \textbf{EXACT} \\
S566 & steel $E$ GPa & 200 & \textbf{EXACT} \\
S567 & concrete kg/m$^3$ & 2400 & \textbf{EXACT} \\
S568 & steel kg/m$^3$ & 7850 & \textbf{EXACT} \\
S569 & aluminum kg/m$^3$ & 2700 & \textbf{EXACT} \\
S570 & wood kg/m$^3$ & 500 & \textbf{EXACT} \\
S571 & Mohs diamond & 10 & \textbf{EXACT} \\
S572 & sound m/s & 343 & \textbf{EXACT} \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier Y delivers \textbf{nine exact closures} --- the highest exact-density of any tier. The speed of sound closes by integer-plus-twelfth arithmetic: $343 = 341 + 24/12$. Diamond Mohs hardness is a single locked primitive: $\SOfive = 10$. Standard structural densities (steel 7850, concrete 2400, aluminum 2700, wood 500) all close exactly with $\SOfive$- and $\Dcrit$-dominated polynomials, suggesting an engineering-units consistency built into the locked primitive set. Cumulative: 277 closures; 60 lockings (9 new from exact engineering closures).

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