The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-binomial-theorem-state-any-key-3160

stable pending witness
2026-05-29 07:05:57 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the binomial theorem. State any key formula.
70 chars

Generation

The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer. Instead of multiplying out the expression manually, the theorem states that: **(a + b)ⁿ = Σ(k=0 to n) C(n,k) · aⁿ⁻ᵏ · bᵏ** where C(n,k) = n!/(k!(n-k)!) represents the binomial coefficients (also written as "n choose k"). This formula shows that the expansion contains (n+1) terms with coefficients following Pascal's triangle, and the powers of a decrease while powers of b increase across the terms.
26 tokens in · 171 tokens out · 4551 ms · $0.0026

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
4558.9 ms
Base LLM
4551.4 ms
Verifiers
0.8 ms
Gates
0.0 ms
Cost
$0.0026

Trail

07:05:57 received prompt_chars=70 verifiers=['scripture_anchors', 'theology_doctrine']
07:05:57 red_gate decision=pass reason=no disqualifying input patterns detected
07:06:02 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=171 latency_ms=4551.4 cost_usd=0.0026
07:06:02 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:06:02 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:06:02 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:06:02 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:06:02 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:06:02 signed hash_algo=sha256 hash=sha256:000c5ac4fa0641e674606c982f738ddc1ba24e8b108673d70b9f1b276a85712e

Content hash

sha256:000c5ac4fa0641e674606c982f738ddc1ba24e8b108673d70b9f1b276a85712e

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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