# PAPER_1562 — √2 (Pythagoras' constant) = √2 = SSQ + 2·F·D + F²·SSQ + F²·D

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Math
**Date:** June 18, 2026
**Status:** CLOSED — 0.105% residual closure

---

## Observation

PAPER_1209FF_S637 (Math Unified Proof Set) lists **√2 (Pythagoras' constant)** with the closed-form expression:

```
√2 = SSQ + 2·F·D + F²·SSQ + F²·D
```

Residual: **0.105%**.

## UQFF Closed Identity

```
√2 = SSQ + 2·F·D + F²·SSQ + F²·D    residual 0.105%
```

The expression uses only locked UQFF integer/real primitives — no fitted constants.

## Significance

Together with PAPER_1494-1559 closures wired earlier this session, this paper extends the count of fundamental constants expressible from UQFF integer primitives across multiple physics domains.

For **mathematical constants** specifically, the cumulative session-2026-06-18 catalog now includes:
- From PAPER_1208 (10): ln 2, ln 10, π², e, e², π/4, Catalan G, ζ(2), ζ(3), γ_Euler
- From PAPER_1209FF (5): π, φ, √2, √3, √5

**13 fundamental mathematical constants expressible in UQFF integer primitives at sub-1% residual.**

For **nuclear binding energies**, the pattern reveals a universal F·K⁵ + β^k polynomial structure with integer offsets distinguishing shell-closure species (+2 for closed-shell light nuclei, +3 for spin-orbit-favored mid-mass).

## NOT REPLACEMENT

UQFF does not claim this closed-form **equals** the constant exactly. It claims the integer primitives **carry a rational approximation** at the residual stated, demonstrating mathematical depth without overriding empirical measurement.

## Reference

- Source: PAPER_1209FF_S637
- Related: PAPER_1208 (transcendentals proof set), PAPER_1494-1559 (catch-up papers)
- Calculator dispatch: `calculate_paradox({"paradox": "transcendental_sqrt_2"})`

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**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, June 18, 2026, Youngstown OH.
