The Mechanism · Gated-generation receipt

gen-briefly-explain-the-law-of-cosines-and-one-common-misconcept-0bec

stable pending witness
2026-05-29 03:44:54 UTC · anthropic/claude-sonnet-4-5

Prompt

Briefly explain the law of cosines and one common misconception about it.
73 chars

Generation

## Law of Cosines The law of cosines relates the three sides of any triangle to one of its angles: **c² = a² + b² - 2ab cos(C)** where c is the side opposite angle C, and a and b are the other two sides. This formula allows you to find an unknown side when you know two sides and the included angle, or find an angle when you know all three sides. ## Common Misconception **"The law of cosines only works for right triangles"** This is backwards! The law of cosines works for *all* triangles. In fact, when you have a right triangle (C = 90°), cos(90°) = 0, and the formula reduces to the Pythagorean theorem: c² = a² + b². The law of cosines is actually most useful for non-right triangles, where the Pythagorean theorem doesn't apply.
23 tokens in · 213 tokens out · 5207 ms · $0.0033

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
5213.5 ms
Base LLM
5206.5 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0033

Trail

03:44:54 received prompt_chars=73 verifiers=['scripture_anchors', 'theology_doctrine']
03:44:54 red_gate decision=pass reason=no disqualifying input patterns detected
03:44:59 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=23 tokens_out=213 latency_ms=5206.5 cost_usd=0.0033
03:44:59 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:44:59 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:44:59 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:44:59 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:44:59 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:44:59 signed hash_algo=sha256 hash=sha256:2b7341c3851be9f4c251794c40c1a4255a7cc6e1967e1fed9a72acc9700ad25f

Content hash

sha256:2b7341c3851be9f4c251794c40c1a4255a7cc6e1967e1fed9a72acc9700ad25f

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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