The Mechanism · Gated-generation receipt

gen-explain-the-distance-formula-as-you-would-to-a-capable-stude-0f32

stable pending witness
2026-05-29 05:22:36 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain the distance formula as you would to a capable student, in 3-4 sentences.
81 chars

Generation

The distance formula calculates the straight-line distance between two points in a coordinate plane, just like measuring with a ruler. It comes directly from the Pythagorean theorem: if you imagine a right triangle connecting your two points, the horizontal and vertical legs have lengths |x₂ - x₁| and |y₂ - y₁|, so the distance (hypotenuse) is d = √[(x₂ - x₁)² + (y₂ - y₁)²]. Essentially, you're finding how far apart the points are horizontally, how far apart they are vertically, then combining these using the Pythagorean theorem to get the direct distance.
28 tokens in · 157 tokens out · 3280 ms · $0.0024

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
3288.4 ms
Base LLM
3279.8 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0024

Trail

05:22:36 received prompt_chars=81 verifiers=['scripture_anchors', 'theology_doctrine']
05:22:36 red_gate decision=pass reason=no disqualifying input patterns detected
05:22:39 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=157 latency_ms=3279.8 cost_usd=0.0024
05:22:39 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:22:39 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:22:39 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:22:39 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:22:39 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:22:39 signed hash_algo=sha256 hash=sha256:ac9337bac7f01e50372feeaac6bcdff9012d585860cec508b20446f37d1b91e2

Content hash

sha256:ac9337bac7f01e50372feeaac6bcdff9012d585860cec508b20446f37d1b91e2

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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