The Mechanism · Gated-generation receipt

gen-define-the-law-of-cosines-and-give-one-short-worked-example-c816

stable pending witness
2026-05-29 04:11:49 UTC · anthropic/claude-sonnet-4-5

Prompt

Define the law of cosines and give one short worked example.
60 chars

Generation

# Law of Cosines ## Definition For any triangle with sides of length *a*, *b*, and *c*, and angles *A*, *B*, and *C* opposite to those sides respectively: **c² = a² + b² - 2ab cos(C)** Or equivalently: - a² = b² + c² - 2bc cos(A) - b² = a² + c² - 2ac cos(B) ## Worked Example **Problem:** A triangle has sides *a* = 7 cm and *b* = 10 cm, with the angle between them *C* = 60°. Find side *c*. **Solution:** c² = a² + b² - 2ab cos(C) c² = 7² + 10² - 2(7)(10) cos(60°) c² = 49 + 100 - 140(0.5) c² = 149 - 70 c² = 79 c = √79 ≈ **8.89 cm**
20 tokens in · 253 tokens out · 5006 ms · $0.0039

The four gates

RED
pass
no disqualifying input patterns detected
FLOOR
pass
output above protective minimum; no mismatches
WAY
pass
no declared way_path (Way check NA); no coercion language
EXECUTION
deferred
no witnesses available (Witness Roll not yet constituted)

Verifier results

scripture_anchors NOT_APPLICABLE
0 Scripture citation(s) extracted
theology_doctrine NOT_APPLICABLE
no doctrine keyword patterns detected

Metrics

Total latency
5013.1 ms
Base LLM
5006.3 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0039

Trail

04:11:49 received prompt_chars=60 verifiers=['scripture_anchors', 'theology_doctrine']
04:11:49 red_gate decision=pass reason=no disqualifying input patterns detected
04:11:54 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=20 tokens_out=253 latency_ms=5006.3 cost_usd=0.0039
04:11:54 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:11:54 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:11:54 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:11:54 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:11:54 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:11:54 signed hash_algo=sha256 hash=sha256:dbe286701bb92635644b09c853b077fd92ebc8a6d1c4a1d178ae3420e00d8470

Content hash

sha256:dbe286701bb92635644b09c853b077fd92ebc8a6d1c4a1d178ae3420e00d8470

SHA256 over the canonical JSON (excluding this field). Tamper detection. Ed25519 signing planned for v2 once the operator’s signing key is provisioned on disk.

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