The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-pascal-s-triangle-state-any-key-for-4635

stable pending witness
2026-05-29 06:02:28 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Pascal's triangle. State any key formula.
67 chars

Generation

Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it, with 1's along the edges. The entries represent binomial coefficients, which appear in the expansion of (a + b)ⁿ and count combinations. The key formula for the entry in row n and position k is: **C(n,k) = n!/(k!(n-k)!)**, also written as ⁿCₖ or (n choose k). The triangle has numerous applications in probability, combinatorics, and algebra.
25 tokens in · 128 tokens out · 3811 ms · $0.002

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
3818.3 ms
Base LLM
3810.7 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.002

Trail

06:02:28 received prompt_chars=67 verifiers=['scripture_anchors', 'theology_doctrine']
06:02:28 red_gate decision=pass reason=no disqualifying input patterns detected
06:02:32 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=128 latency_ms=3810.7 cost_usd=0.002
06:02:32 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:02:32 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:02:32 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:02:32 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:02:32 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:02:32 signed hash_algo=sha256 hash=sha256:0d2eec506a1f5d179038e162a28109e292e19f3f00db0dac2688835902bad86e

Content hash

sha256:0d2eec506a1f5d179038e162a28109e292e19f3f00db0dac2688835902bad86e

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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