The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-law-of-cosines-state-any-key-fo-28fa

stable pending witness
2026-05-29 05:57:31 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the law of cosines. State any key formula.
68 chars

Generation

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem to work for any triangle, not just right triangles. The key formula is: **c² = a² + b² - 2ab cos(C)** where a, b, and c are the side lengths of the triangle, and C is the angle opposite side c. This formula allows you to find an unknown side length when you know two sides and the included angle, or to find an angle when you know all three sides.
27 tokens in · 130 tokens out · 3357 ms · $0.002

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
3364.2 ms
Base LLM
3356.8 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.002

Trail

05:57:31 received prompt_chars=68 verifiers=['scripture_anchors', 'theology_doctrine']
05:57:31 red_gate decision=pass reason=no disqualifying input patterns detected
05:57:34 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=130 latency_ms=3356.8 cost_usd=0.002
05:57:34 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:57:34 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:57:34 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:57:34 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:57:34 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:57:34 signed hash_algo=sha256 hash=sha256:f144503bd1ce594c4f7b4e4e4a165944bbdfd8e940b56bd8b9a81581e6de2c70

Content hash

sha256:f144503bd1ce594c4f7b4e4e4a165944bbdfd8e940b56bd8b9a81581e6de2c70

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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