The Mechanism · Gated-generation receipt

gen-briefly-explain-polygon-angle-sums-and-one-common-misconcept-457e

stable pending witness
2026-05-29 03:13:09 UTC · anthropic/claude-sonnet-4-5

Prompt

Briefly explain polygon angle sums and one common misconception about it.
73 chars

Generation

# Polygon Angle Sums ## The Rule The sum of **interior angles** in any polygon equals **(n - 2) × 180°**, where n is the number of sides. - Triangle (3 sides): (3-2) × 180° = 180° - Quadrilateral (4 sides): (4-2) × 180° = 360° - Pentagon (5 sides): (5-2) × 180° = 540° ## Common Misconception **Students often confuse interior and exterior angle sums.** - **Interior angles**: Vary by polygon (use the formula above) - **Exterior angles**: Always sum to **360°** for ANY polygon (when taking one exterior angle per vertex) Students sometimes try to apply the (n-2) × 180° formula to exterior angles, or assume exterior angles sum to different values for different polygons. In reality, exterior angles always total 360° whether you have a triangle or a dodecagon!
22 tokens in · 229 tokens out · 6512 ms · $0.0035

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
6525.9 ms
Base LLM
6512.4 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0035

Trail

03:13:09 received prompt_chars=73 verifiers=['scripture_anchors', 'theology_doctrine']
03:13:09 red_gate decision=pass reason=no disqualifying input patterns detected
03:13:15 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=22 tokens_out=229 latency_ms=6512.4 cost_usd=0.0035
03:13:15 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:13:15 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:13:15 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:13:15 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:13:15 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:13:15 signed hash_algo=sha256 hash=sha256:ee693c1740582c985b223c2afb3a94558ef54c3a345e0d2ce81efd671727f6d8

Content hash

sha256:ee693c1740582c985b223c2afb3a94558ef54c3a345e0d2ce81efd671727f6d8

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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