The Mechanism · Gated-generation receipt

gen-state-the-law-of-cosines-precisely-and-explain-why-it-matter-8d32

stable pending witness
2026-05-29 06:42:47 UTC · anthropic/claude-sonnet-4-5

Prompt

State the law of cosines precisely and explain why it matters in 2-3 sentences.
79 chars

Generation

# Law of Cosines For any triangle with sides of length *a*, *b*, *c* and angle *C* opposite side *c*: **c² = a² + b² − 2ab cos(C)** (with analogous formulas for the other angles) ## Why It Matters The law of cosines generalizes the Pythagorean theorem to all triangles, not just right triangles—when C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b². It's essential for solving triangles when you know either three sides or two sides and the included angle, making it indispensable in surveying, navigation, and physics applications involving vector components.
27 tokens in · 167 tokens out · 3945 ms · $0.0026

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
3952.6 ms
Base LLM
3945.1 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0026

Trail

06:42:47 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
06:42:47 red_gate decision=pass reason=no disqualifying input patterns detected
06:42:51 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=167 latency_ms=3945.1 cost_usd=0.0026
06:42:51 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:42:51 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:42:51 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:42:51 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:42:51 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:42:51 signed hash_algo=sha256 hash=sha256:aba1eb164c999223bbcd4640fd1f3d1f7850a0dda195efdc4ea1b745aa832d3e

Content hash

sha256:aba1eb164c999223bbcd4640fd1f3d1f7850a0dda195efdc4ea1b745aa832d3e

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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