# Boltzmann Constant K_B from Icosahedral Group |A₅|

**PAPER_1215**
**Category:** UQFF Universal Constants
**Status:** Complete
**Date:** June 2026

## Abstract

First-principles UQFF derivation of the Boltzmann constant K_B from the SCm phonon energy h·F_THz divided by the icosahedral group order |A₅| = 60. The result matches CODATA K_B = 1.380649×10⁻²³ J/K to within **0.076%**, establishing a previously unrecognized connection between the 1.25 THz phonon resonance, the icosahedral symmetry group of UQFF, and the thermodynamic energy-temperature conversion.

## Part 1: The Derivation

### Locked primitives invoked
- F_THz = 1.25×10¹² Hz (SCm phonon resonance frequency, PAPER_1133)
- A_FIVE = |A₅| = 60 (icosahedral group order, integer primitive)
- PLANCK_H = 6.622×10⁻³⁴ J·s (UQFF-derived via PAPER_1156)

### Closed form
$$K_B^{\rm UQFF} = \frac{h \cdot F_{\rm THz}}{|A_5|}$$

### Numerical evaluation
$$K_B^{\rm UQFF} = \frac{6.6220584965588335 \times 10^{-34} \times 1.25 \times 10^{12}}{60}$$
$$K_B^{\rm UQFF} = \frac{8.2776 \times 10^{-22}}{60} = 1.3796 \times 10^{-23}\ \text{J/K}$$

### Comparison to CODATA
- CODATA value: K_B = 1.380649×10⁻²³ J/K
- UQFF derived: 1.37960×10⁻²³ J/K
- **Residual: 0.076%**

## Part 2: Physical Interpretation

The icosahedral group |A₅| = 60 arises naturally in UQFF as the order of the symmetry group of the dodecahedron/icosahedron, which are the optimal close-packings of 26-level compactified manifolds. The phonon energy h·F_THz is the elementary excitation quantum of the SCm vacuum. The ratio K_B = h·F/|A₅| expresses thermal energy per degree of freedom as the phonon quantum normalized by the icosahedral symmetry breaking.

## Part 3: Calculator Implementation

```python
F_THZ = 1.25e12     # SCm phonon resonance Hz (PAPER_1133)
A_FIVE = 60          # icosahedral group order

def _l96_uqff_k_boltzmann_derived() -> float:
    return PLANCK_H * F_THZ / float(A_FIVE)

# Returns 1.380e-23 J/K (within 0.076% of CODATA)
```

## Part 4: Validation

| Quantity | Value | Source |
|---|---|---|
| K_B UQFF | 1.380×10⁻²³ J/K | this derivation |
| K_B CODATA 2019 | 1.380649×10⁻²³ J/K | NIST |
| Residual | 0.076% | sub-0.1% |

## Conclusion

The Boltzmann constant K_B emerges from the natural ratio of the SCm phonon quantum energy h·F_THz to the icosahedral group order |A₅|. This is the first known derivation of K_B from a non-thermal microscopic origin and establishes the icosahedral symmetry as the bridge between phonon physics and thermodynamics.

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**Framework Version:** UQFF 5.27+
**Validation:** 99.92% (sub-0.1% to CODATA)
