The Mechanism · Gated-generation receipt

gen-state-the-binomial-theorem-precisely-and-explain-why-it-matt-6f34

stable pending witness
2026-05-29 05:03:20 UTC · anthropic/claude-sonnet-4-5

Prompt

State the binomial theorem precisely and explain why it matters in 2-3 sentences.
81 chars

Generation

## Binomial Theorem For any real numbers *a* and *b* and non-negative integer *n*: **(a + b)ⁿ = Σ(k=0 to n) C(n,k) aⁿ⁻ᵏ bᵏ** where C(n,k) = n!/(k!(n-k)!) are the binomial coefficients. ## Why It Matters The binomial theorem provides an explicit formula for expanding powers of sums, eliminating the need for tedious repeated multiplication. It's fundamental across mathematics—appearing in probability theory (binomial distributions), calculus (Taylor series), combinatorics (counting problems), and algebra—making it one of the most widely applied formulas in both pure and applied mathematics.
26 tokens in · 185 tokens out · 4202 ms · $0.0029

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
4209.4 ms
Base LLM
4202.1 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0029

Trail

05:03:20 received prompt_chars=81 verifiers=['scripture_anchors', 'theology_doctrine']
05:03:20 red_gate decision=pass reason=no disqualifying input patterns detected
05:03:24 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=185 latency_ms=4202.1 cost_usd=0.0029
05:03:24 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:03:24 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:03:24 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:03:24 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:03:24 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:03:24 signed hash_algo=sha256 hash=sha256:bcc6171fb0cff3b567117b1a530763927bcb3f0456f62ffe8b82e7f7cbf83d53

Content hash

sha256:bcc6171fb0cff3b567117b1a530763927bcb3f0456f62ffe8b82e7f7cbf83d53

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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