The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-catalan-numbers-state-any-key-formu-6cf9

stable pending witness
2026-05-29 03:50:00 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Catalan numbers. State any key formula.
65 chars

Generation

Catalan numbers are a sequence of natural numbers (1, 1, 2, 5, 14, 42, ...) that appear in various counting problems in combinatorics. They count structures like the number of ways to parenthesize expressions, the number of valid sequences of balanced parentheses, the number of binary trees with n internal nodes, and paths on a grid that don't cross the diagonal. The nth Catalan number is given by the formula: **C_n = (1/(n+1)) × (2n choose n) = (2n)! / ((n+1)! × n!)** They can also be computed recursively: C_0 = 1 and C_n = Σ(C_i × C_{n-1-i}) for i from 0 to n-1.
26 tokens in · 189 tokens out · 3837 ms · $0.0029

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
3844.3 ms
Base LLM
3837.0 ms
Verifiers
0.4 ms
Gates
0.1 ms
Cost
$0.0029

Trail

03:50:00 received prompt_chars=65 verifiers=['scripture_anchors', 'theology_doctrine']
03:50:00 red_gate decision=pass reason=no disqualifying input patterns detected
03:50:04 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=189 latency_ms=3837.0 cost_usd=0.0029
03:50:04 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:50:04 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:50:04 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:50:04 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:50:04 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:50:04 signed hash_algo=sha256 hash=sha256:8b45df35b763fe46dccab270043835d0bd36e0459e197e6d36096ac58f49ade4

Content hash

sha256:8b45df35b763fe46dccab270043835d0bd36e0459e197e6d36096ac58f49ade4

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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