The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-multinomial-coefficient-state-a-4166

stable pending witness
2026-05-29 06:34:39 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the multinomial coefficient. State any key formula.
77 chars

Generation

The multinomial coefficient generalizes the binomial coefficient to situations where items are divided into more than two groups. It counts the number of ways to partition a set of n objects into k groups of specified sizes n₁, n₂, ..., nₖ. The key formula is: $$\binom{n}{n_1, n_2, \ldots, n_k} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}$$ where n = n₁ + n₂ + ... + nₖ, representing the total number of objects being distributed.
26 tokens in · 153 tokens out · 3568 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
3575.3 ms
Base LLM
3567.8 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0024

Trail

06:34:39 received prompt_chars=77 verifiers=['scripture_anchors', 'theology_doctrine']
06:34:39 red_gate decision=pass reason=no disqualifying input patterns detected
06:34:43 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=153 latency_ms=3567.8 cost_usd=0.0024
06:34:43 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:34:43 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:34:43 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:34:43 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:34:43 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:34:43 signed hash_algo=sha256 hash=sha256:27083a9e458ee52ef6dd2f649fc0cdd70f1be3645d9a08b185a3564dbc724cc8

Content hash

sha256:27083a9e458ee52ef6dd2f649fc0cdd70f1be3645d9a08b185a3564dbc724cc8

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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