Generation
# Metric Space
## Definition
A **metric space** is a set *X* together with a function *d*: *X* × *X* → ℝ (called a **metric** or **distance function**) satisfying for all *x*, *y*, *z* ∈ *X*:
1. **Non-negativity**: *d*(*x*, *y*) ≥ 0
2. **Identity**: *d*(*x*, *y*) = 0 if and only if *x* = *y*
3. **Symmetry**: *d*(*x*, *y*) = *d*(*y*, *x*)
4. **Triangle inequality**: *d*(*x*, *z*) ≤ *d*(*x*, *y*) + *d*(*y*, *z*)
## Worked Example
**Claim**: ℝ² with the Euclidean metric *d*(**p**, **q**) = √[(x₂ - x₁)² + (y₂ - y₁)²] is a metric space, where **p** = (x₁, y₁) and **q** = (x₂, y₂).
**Verification**: Let **p** = (1, 2), **q** = (4, 6), **r** = (0, 0).
1. *d*(**p**, **q**) = √[(4-1)² + (6-2)²] = √[9 + 16] = 5 ≥ 0 ✓
2. *d*(**p**, **p**) = √[(1-1)² + (2-2)²] = 0, and if *d*(**p**, **q**) = 0, then (x₂ - x₁)² + (y₂ - y₁)² = 0, forcing **p** = **q** ✓
3. *d*(**p**, **q**) = √[(4-1)² + (6-2)²] = √[(1-4)² + (2-6)²] = *d*(**q**, **p**) ✓
4. *d*(**p**, **r**) = √5 ≈ 2.24, *d*(**r**, **q**) = √52 ≈ 7.21, *d*(**p**, **q**) = 5
Indeed: 5 ≤ 2.24 + 7.21 ✓
18 tokens in · 615 tokens out · 9068 ms · $0.0093
Trail
06:33:14 received prompt_chars=56 verifiers=['scripture_anchors', 'theology_doctrine']
06:33:14 red_gate decision=pass reason=no disqualifying input patterns detected
06:33:23 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=18 tokens_out=615 latency_ms=9067.9 cost_usd=0.0093
06:33:23 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:33:23 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:33:23 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:33:23 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:33:23 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:33:23 signed hash_algo=sha256 hash=sha256:1a84230246d7e12cbf2b7020f87d96cea57803c71e178b0270c4a3fd64a58994