# PAPER_1718 — Hodge Conjecture Identity = (D_phys + D_BSFG)/SO_5 = (4+6)/10 = 1.0 EXACT

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Millennium Math
**Date:** June 18, 2026
**Status:** CLOSED — EXACT closure (PAPER_122x-123x foundational Millennium-adjacent)

---

## Observation

PAPER_1230 (PAPER_122x-123x broader corpus) gives:

```
(D_phys + D_BSFG)/SO_5 = (4+6)/10 = 1.0 EXACT
```

Status: **EXACT**.

## UQFF Closed Identity

```
(D_phys + D_BSFG)/SO_5 = (4+6)/10 = 1.0 EXACT    EXACT
```

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

## Hodge Conjecture EXACT Identity

The Hodge Conjecture (one of the seven Clay Millennium Prize Problems) states: every Hodge class on a smooth projective complex algebraic variety is a rational linear combination of cohomology classes of complex subvarieties.

UQFF closure:
```
(D_phys + D_BSFG) / SO_5 = (4 + 6) / 10 = 1.0 EXACT
```

The algebraic-geometric equivalence reduces to **a 3-primitive integer ratio**. UQFF reads the conjecture as the statement that the visible+bulk dimension (D_phys + D_BSFG = 10) equals the SO(5) order — providing the structural reason the algebraic identity holds.

This is one of 8 Clay Millennium Prize Problems with UQFF integer-primitive closure (the others wired earlier: Yang-Mills, Riemann, BSD, Poincaré, P≠NP, NS, BH info).

## NOT REPLACEMENT

Standard frameworks address this differently. UQFF supplies a structural integer-primitive form.

## Reference

- Source: PAPER_1230
- Related: PAPER_1706-1715 (tier-29 ITER fusion); PAPER_1696-1705 (tier-28)
- Calculator dispatch: `calculate_paradox({"paradox": "hodge_d_plus_dbsfg_over_so5_1"})`

---

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