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Formalize totally positive elements of number fields for the Lean-LMFDB bridge

Define `IsTotallyPositive (x : K)` for a number field as positivity under every real embedding, prove closure under addition and multiplication, prove the nonzero-square fact over totally real fields, and add the issue's Q(sqrt 2) examples, following the existing infinite-places API.

Why this problem

This is an explicit beginner project connecting Mathlib with the L-functions and Modular Forms Database. The predicate is elementary and the requested API and examples are enumerated, while acceptance has a concrete Mathlib/LMFDB/LeanBridge path.

Verification contract

The declarations and examples must typecheck against the pinned Mathlib dependency, pass `lake build` and the blueprint declaration checker, and use no `sorry` or new axioms.

Tracking
Difficulty
4/10
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0
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Source status
open
External validation
none
Techniques and harnesses
Lean 4number fieldsreal embeddingsordered fieldsLMFDB
Resumable campaign memory

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