The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-catalan-numbers-answer-concisely-fd00

stable pending witness
2026-05-29 04:38:33 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of Catalan numbers? Answer concisely in 2-3 sentences.
79 chars

Generation

Catalan numbers are a sequence of natural numbers that appear in numerous combinatorial counting problems, such as counting the number of valid parentheses expressions, binary trees, and paths in a grid. They are significant because they unexpectedly emerge as the solution to many seemingly unrelated mathematical problems across discrete mathematics, computer science, and algebra. The nth Catalan number is given by the formula C_n = (1/(n+1)) × (2n choose n).
29 tokens in · 101 tokens out · 3496 ms · $0.0016

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
3502.8 ms
Base LLM
3495.8 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0016

Trail

04:38:33 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
04:38:33 red_gate decision=pass reason=no disqualifying input patterns detected
04:38:36 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=101 latency_ms=3495.8 cost_usd=0.0016
04:38:36 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:38:36 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:38:36 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:38:36 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:38:36 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:38:36 signed hash_algo=sha256 hash=sha256:f6cce471dccca292689cdd7ee033187eaf2aee2cd2beefc53e33e679d00104bf

Content hash

sha256:f6cce471dccca292689cdd7ee033187eaf2aee2cd2beefc53e33e679d00104bf

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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