The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-binomial-distribution-state-any-17a4

stable pending witness
2026-05-29 04:41:06 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the binomial distribution. State any key formula.
75 chars

Generation

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant. It's commonly used for scenarios like counting heads in coin flips or defective items in a sample. The probability of getting exactly k successes in n trials is given by: **P(X = k) = C(n,k) × p^k × (1-p)^(n-k)** where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient, p is the probability of success, and k ranges from 0 to n.
26 tokens in · 152 tokens out · 4185 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
4192.3 ms
Base LLM
4184.9 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0024

Trail

04:41:06 received prompt_chars=75 verifiers=['scripture_anchors', 'theology_doctrine']
04:41:06 red_gate decision=pass reason=no disqualifying input patterns detected
04:41:10 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=152 latency_ms=4184.9 cost_usd=0.0024
04:41:10 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:41:10 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:41:10 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:41:10 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:41:10 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:41:10 signed hash_algo=sha256 hash=sha256:3ec653bf7ab3d4f906497e86e81fd9597f54c60554b02197e8a8665cb8acb801

Content hash

sha256:3ec653bf7ab3d4f906497e86e81fd9597f54c60554b02197e8a8665cb8acb801

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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