# 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

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

## Closures S563--S572

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

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

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

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

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

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

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

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

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

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

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

\medskip
**Highlights.** Tier Y delivers **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).