The Mechanism · Gated-generation receipt

gen-define-pascal-s-triangle-and-give-one-short-worked-example-35a9

stable pending witness
2026-05-29 08:01:37 UTC · anthropic/claude-sonnet-4-5

Prompt

Define Pascal's triangle and give one short worked example.
59 chars

Generation

# Pascal's Triangle **Definition:** Pascal's triangle is a triangular array of numbers where: - The top row is 1 - Each row starts and ends with 1 - Each interior number is the sum of the two numbers directly above it The triangle looks like this: ``` 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 ``` ## Worked Example **Find the 5th row of Pascal's triangle.** Starting from row 0 (the single 1 at top): - Row 0: 1 - Row 1: 1, 1 - Row 2: 1, (1+1)=2, 1 - Row 3: 1, (1+2)=3, (2+1)=3, 1 - Row 4: 1, (1+3)=4, (3+3)=6, (3+1)=4, 1 - **Row 5: 1, (1+4)=5, (4+6)=10, (6+4)=10, (4+1)=5, 1** **Answer:** The 5th row is **1, 5, 10, 10, 5, 1** These numbers are also the binomial coefficients (ⁿCᵣ) and appear in the expansion of (a+b)⁵.
18 tokens in · 358 tokens out · 5701 ms · $0.0054

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
5709.5 ms
Base LLM
5701.0 ms
Verifiers
0.4 ms
Gates
0.0 ms
Cost
$0.0054

Trail

08:01:37 received prompt_chars=59 verifiers=['scripture_anchors', 'theology_doctrine']
08:01:37 red_gate decision=pass reason=no disqualifying input patterns detected
08:01:43 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=18 tokens_out=358 latency_ms=5701.0 cost_usd=0.0054
08:01:43 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:01:43 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:01:43 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:01:43 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:01:43 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:01:43 signed hash_algo=sha256 hash=sha256:9701f202bc9eeb26949e663ecef11c3ff48130afcd16eb42b7c81b2c59c6be41

Content hash

sha256:9701f202bc9eeb26949e663ecef11c3ff48130afcd16eb42b7c81b2c59c6be41

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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