The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-stirling-numbers-state-any-key-form-710f

stable pending witness
2026-05-29 03:12:53 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Stirling numbers. State any key formula.
66 chars

Generation

# Stirling Numbers **Stirling numbers of the first kind** s(n,k) count the number of permutations of n elements with exactly k cycles, while **Stirling numbers of the second kind** S(n,k) count the number of ways to partition n elements into k non-empty subsets. **Key formula:** Stirling numbers of the second kind satisfy the recurrence relation: $$S(n,k) = k \cdot S(n-1,k) + S(n-1,k-1)$$ This reflects either placing element n into one of k existing subsets, or starting a new subset with element n alone.
26 tokens in · 151 tokens out · 4339 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
4345.7 ms
Base LLM
4338.6 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0023

Trail

03:12:53 received prompt_chars=66 verifiers=['scripture_anchors', 'theology_doctrine']
03:12:53 red_gate decision=pass reason=no disqualifying input patterns detected
03:12:57 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=151 latency_ms=4338.6 cost_usd=0.0023
03:12:57 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:12:57 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:12:57 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:12:57 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:12:57 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:12:57 signed hash_algo=sha256 hash=sha256:8b1640d4050537f4dd31af1e822482fde82d8a0fd1a6dc96be92f4d4b73c5674

Content hash

sha256:8b1640d4050537f4dd31af1e822482fde82d8a0fd1a6dc96be92f4d4b73c5674

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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