The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-the-multinomial-coefficient-answ-c1d1

stable pending witness
2026-05-29 02:56:27 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of the multinomial coefficient? Answer concisely in 2-3 sentences.
91 chars

Generation

The multinomial coefficient counts the number of ways to divide n distinct objects into k groups of specified sizes, generalizing the binomial coefficient. It appears in the multinomial theorem, which extends the binomial theorem to expressions with multiple terms, giving the coefficient of each term in the expansion of (x₁ + x₂ + ... + xₖ)ⁿ. The formula is n!/(n₁!n₂!...nₖ!) where n₁ + n₂ + ... + nₖ = n.
29 tokens in · 125 tokens out · 2672 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
2679.9 ms
Base LLM
2672.4 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.002

Trail

02:56:27 received prompt_chars=91 verifiers=['scripture_anchors', 'theology_doctrine']
02:56:27 red_gate decision=pass reason=no disqualifying input patterns detected
02:56:30 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=125 latency_ms=2672.4 cost_usd=0.002
02:56:30 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:56:30 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:56:30 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:56:30 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:56:30 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:56:30 signed hash_algo=sha256 hash=sha256:87f56854789d5b71edc88bffea9984826f5528c39c80a3e576fa50cf9780eb86

Content hash

sha256:87f56854789d5b71edc88bffea9984826f5528c39c80a3e576fa50cf9780eb86

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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