The Mechanism · Gated-generation receipt

gen-explain-the-law-of-cosines-as-you-would-to-a-capable-student-a841

stable pending witness
2026-05-29 07:29:16 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain the law of cosines as you would to a capable student, in 3-4 sentences.
79 chars

Generation

The Law of Cosines is a generalization of the Pythagorean theorem that works for **any** triangle, not just right triangles. It states that c² = a² + b² - 2ab cos(C), where c is the side opposite angle C, and a and b are the other two sides. Notice that when C = 90°, the cosine term vanishes (since cos(90°) = 0), and you get back the familiar Pythagorean theorem. You can use this law to find an unknown side when you know two sides and the included angle, or to find an unknown angle when you know all three sides.
30 tokens in · 145 tokens out · 3738 ms · $0.0023

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
3750.7 ms
Base LLM
3737.8 ms
Verifiers
0.1 ms
Gates
0.1 ms
Cost
$0.0023

Trail

07:29:16 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
07:29:16 red_gate decision=pass reason=no disqualifying input patterns detected
07:29:20 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=30 tokens_out=145 latency_ms=3737.8 cost_usd=0.0023
07:29:20 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:29:20 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:29:20 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:29:20 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:29:20 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:29:20 signed hash_algo=sha256 hash=sha256:66454227304358f61a7df5d8239f476752ea11d544f017587881d8d27536d196

Content hash

sha256:66454227304358f61a7df5d8239f476752ea11d544f017587881d8d27536d196

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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