The Mechanism · Gated-generation receipt

gen-explain-catalan-numbers-as-you-would-to-a-capable-student-in-90f3

stable pending witness
2026-05-29 04:49:59 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain Catalan numbers as you would to a capable student, in 3-4 sentences.
76 chars

Generation

Catalan numbers are a sequence (1, 1, 2, 5, 14, 42, ...) that count many different combinatorial structures of the same size. The nth Catalan number tells you, for example, how many ways you can properly match n pairs of parentheses, or how many different binary trees you can build with n+1 leaves. They appear surprisingly often in counting problems involving recursive structures or paths that must stay on one side of a diagonal, and can be calculated using the formula Cₙ = (1/(n+1)) × (2n choose n). Their ubiquity across seemingly unrelated problems makes them one of the most important sequences in combinatorics.
29 tokens in · 155 tokens out · 6074 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
6081.5 ms
Base LLM
6074.0 ms
Verifiers
0.4 ms
Gates
0.1 ms
Cost
$0.0024

Trail

04:49:59 received prompt_chars=76 verifiers=['scripture_anchors', 'theology_doctrine']
04:49:59 red_gate decision=pass reason=no disqualifying input patterns detected
04:50:05 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=155 latency_ms=6074.0 cost_usd=0.0024
04:50:05 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:50:05 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:50:05 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:50:05 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:50:05 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:50:05 signed hash_algo=sha256 hash=sha256:b17a2bc0f618465e4d3f5e40c640c3344eed814025c74280d55229088150520e

Content hash

sha256:b17a2bc0f618465e4d3f5e40c640c3344eed814025c74280d55229088150520e

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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