The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-bezout-s-identity-answer-concise-8271

stable pending witness
2026-05-29 05:20:17 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of Bezout's identity? Answer concisely in 2-3 sentences.
81 chars

Generation

Bézout's identity states that for any two integers a and b with greatest common divisor d, there exist integers x and y such that ax + by = d. This is significant because it proves that the GCD of two numbers can always be expressed as a linear combination of those numbers, which is fundamental to number theory and has practical applications in solving linear Diophantine equations and modular arithmetic. It also provides the theoretical basis for the extended Euclidean algorithm.
30 tokens in · 102 tokens out · 3335 ms · $0.0016

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
3343.9 ms
Base LLM
3334.7 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0016

Trail

05:20:17 received prompt_chars=81 verifiers=['scripture_anchors', 'theology_doctrine']
05:20:17 red_gate decision=pass reason=no disqualifying input patterns detected
05:20:21 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=30 tokens_out=102 latency_ms=3334.7 cost_usd=0.0016
05:20:21 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:20:21 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:20:21 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:20:21 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:20:21 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:20:21 signed hash_algo=sha256 hash=sha256:2ede62829fa8650664fb54ed77fc745101e6413a64061a72f704bc91efe25e37

Content hash

sha256:2ede62829fa8650664fb54ed77fc745101e6413a64061a72f704bc91efe25e37

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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