The Mechanism · Gated-generation receipt

gen-define-the-binomial-theorem-and-give-one-short-worked-exampl-a9e8

stable pending witness
2026-05-29 06:05:12 UTC · anthropic/claude-sonnet-4-5

Prompt

Define the binomial theorem and give one short worked example.
62 chars

Generation

# Binomial Theorem ## Definition The **binomial theorem** states that for any positive integer *n*: **(a + b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ** (where k goes from 0 to n) Or written out: **(a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + ... + C(n,n)bⁿ** where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient. ## Worked Example Expand **(x + 2)³** Using the binomial theorem with a = x, b = 2, n = 3: **(x + 2)³ = C(3,0)x³(2)⁰ + C(3,1)x²(2)¹ + C(3,2)x¹(2)² + C(3,3)x⁰(2)³** Calculate the coefficients: - C(3,0) = 1 - C(3,1) = 3 - C(3,2) = 3 - C(3,3) = 1 Therefore: **(x + 2)³ = 1·x³·1 + 3·x²·2 + 3·x·4 + 1·1·8** **= x³ + 6x² + 12x + 8**
19 tokens in · 413 tokens out · 9588 ms · $0.0063

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
9600.5 ms
Base LLM
9587.9 ms
Verifiers
0.4 ms
Gates
0.1 ms
Cost
$0.0063

Trail

06:05:12 received prompt_chars=62 verifiers=['scripture_anchors', 'theology_doctrine']
06:05:12 red_gate decision=pass reason=no disqualifying input patterns detected
06:05:22 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=19 tokens_out=413 latency_ms=9587.9 cost_usd=0.0063
06:05:22 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:05:22 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:05:22 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:05:22 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:05:22 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:05:22 signed hash_algo=sha256 hash=sha256:5bf93240518ebc782834e30f73bbef66c167697b5927486ec38c07ac11e1d8e8

Content hash

sha256:5bf93240518ebc782834e30f73bbef66c167697b5927486ec38c07ac11e1d8e8

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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