The Mechanism · Gated-generation receipt

gen-explain-the-multinomial-coefficient-as-you-would-to-a-capabl-b1ee

stable pending witness
2026-05-29 06:00:03 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain the multinomial coefficient as you would to a capable student, in 3-4 sentences.
88 chars

Generation

The multinomial coefficient tells you how many ways you can arrange n objects when some of them are identical to each other. For example, if you want to arrange the letters in "MISSISSIPPI," you can't just use n! because swapping identical letters doesn't create a new arrangement. The formula is n!/(n₁!n₂!···nₖ!), where n is the total number of objects and n₁, n₂, etc. are the counts of each type of identical object—this divides out the overcounting from rearranging identical items. It generalizes the binomial coefficient, which is just the special case when you have exactly two types of objects.
29 tokens in · 150 tokens out · 5320 ms · $0.0023

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
5328.0 ms
Base LLM
5320.0 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0023

Trail

06:00:03 received prompt_chars=88 verifiers=['scripture_anchors', 'theology_doctrine']
06:00:03 red_gate decision=pass reason=no disqualifying input patterns detected
06:00:09 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=150 latency_ms=5320.0 cost_usd=0.0023
06:00:09 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:00:09 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:00:09 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:00:09 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:00:09 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:00:09 signed hash_algo=sha256 hash=sha256:cf4b8989d98ca7ae568ce585ad7debf0c6d038f64c7ff17fde429ff577fb76f1

Content hash

sha256:cf4b8989d98ca7ae568ce585ad7debf0c6d038f64c7ff17fde429ff577fb76f1

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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