# PAPER_1394 — UQFF Resolution of the Galileo's Bijection Paradox (CLOSED — Dispatch Whitepaper)

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** B — Foundational / Cardinality (CLOSED)
**Date:** June 16, 2026
**Location:** 41.0997° N, 80.6495° W (Youngstown, OH, USA)
**Status:** CLOSED — Documented dispatch closure for `calculate_paradox({"paradox": "galileo_paradox"})`
**Calculator surface:** `calculate_paradox({"paradox": "galileo_paradox"})`
**Closure helper:** `_l96_uqff_axiom_galileo_bijection_closure()`

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## The Paradox

Galileo (1638): the natural numbers N = {1, 2, 3, ...} can be put in bijection with the perfect squares S = {1, 4, 9, ...} via n ↔ n². So there are 'as many' squares as integers, even though squares are 'a proper subset' of integers. Cantor's solution: countable cardinality |N| = |S| = ℵ₀.

---

## UQFF Closed Identity

```
|N| = |N²| via F_U = 1 ledger normalization (occupation, not classical cardinality)
```

F_U = 1 global ledger normalization at the UA-mode level: countably infinite sets share the SAME UA mode. Distinguishability is a ledger-occupancy question, not a phase-space-counting question. |N| = |N²| because both occupy the same F_U = 1 mode.

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## Physical Interpretation

Cantor's resolution via ℵ₀ is the pure-math version; UQFF's F_U = 1 is the physical version. Both say 'cardinality is not the right invariant' — UQFF specifies the right invariant is UA-mode occupancy, not set-membership counting.

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## Live Calculator Output

```python
import uqff_pure_calculator as u
r = u.calculate_paradox({"paradox": "galileo_paradox"})["value"]
```

The closure returns the structural identity above plus per-method anchors as fields in the result dict. See `_l96_uqff_axiom_galileo_bijection_closure` in `uqff_pure_calculator.py` for the full field list.

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## NOT REPLACEMENT

Per CLAUDE.md mandate, UQFF and Standard frameworks resolve this paradox via different methods. UQFF derives the closure listed above using **only canonical primitives** (or existing wired helpers — PAPER_646, PAPER_597, PAPER_1051, etc.). **Zero free parameters.**

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## Reference

- Closure location: `uqff_pure_calculator.py` → `_l96_uqff_axiom_galileo_bijection_closure`
- Dispatch keys: `galileo_paradox`, `galileo_bijection`
- Related foundational papers: PAPER_646 (F_U_Bi_i / F_U=1 ledger), PAPER_597 (negative-time dual existence), PAPER_1183 (paradox consolidation), PAPER_1156 (cosmology suite), PAPER_1376 (Banach-Tarski parallel structural closure).

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**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, dated June 16, 2026, location 41.0997° N, 80.6495° W (Youngstown, OH, USA). Subject matter: UQFF dispatch closure for the Galileo's Bijection Paradox via the identity above.
