The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-the-binomial-theorem-answer-conc-4762

stable pending witness
2026-05-29 07:54:13 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of the binomial theorem? Answer concisely in 2-3 sentences.
84 chars

Generation

The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ into a sum of terms involving binomial coefficients, powers of a, and powers of b. It is fundamental to algebra, probability theory, and calculus, enabling efficient calculations without tedious multiplication. The theorem also reveals important patterns in Pascal's triangle and connects to combinatorics through its coefficients, which count combinations.
29 tokens in · 92 tokens out · 3979 ms · $0.0015

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
3986.3 ms
Base LLM
3979.0 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0015

Trail

07:54:13 received prompt_chars=84 verifiers=['scripture_anchors', 'theology_doctrine']
07:54:13 red_gate decision=pass reason=no disqualifying input patterns detected
07:54:17 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=92 latency_ms=3979.0 cost_usd=0.0015
07:54:17 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:54:17 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:54:17 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:54:17 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:54:17 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:54:17 signed hash_algo=sha256 hash=sha256:b4748579e3ab8e16228cd47adbeb610afb4c190403023cf83aa0ee0089478935

Content hash

sha256:b4748579e3ab8e16228cd47adbeb610afb4c190403023cf83aa0ee0089478935

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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