The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-bezout-s-identity-state-any-key-for-a4eb

stable pending witness
2026-05-29 03:44:59 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Bezout's identity. State any key formula.
67 chars

Generation

# Bézout's Identity Bézout's identity states that for any two integers *a* and *b*, there exist integers *x* and *y* such that **ax + by = gcd(a,b)**, where gcd(a,b) is the greatest common divisor of *a* and *b*. This identity guarantees that the GCD can always be expressed as a linear combination of the original numbers. The coefficients *x* and *y* can be found using the Extended Euclidean Algorithm. An important consequence is that *a* and *b* are coprime (gcd(a,b) = 1) if and only if there exist integers *x* and *y* such that ax + by = 1.
27 tokens in · 171 tokens out · 3339 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
3346.3 ms
Base LLM
3339.0 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0026

Trail

03:44:59 received prompt_chars=67 verifiers=['scripture_anchors', 'theology_doctrine']
03:44:59 red_gate decision=pass reason=no disqualifying input patterns detected
03:45:02 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=171 latency_ms=3339.0 cost_usd=0.0026
03:45:02 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:45:02 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:45:02 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:45:02 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:45:02 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:45:02 signed hash_algo=sha256 hash=sha256:af4d228517d1ba87ba45334e4a165cdee51fbfcc1733e94d224625cb2553d0f9

Content hash

sha256:af4d228517d1ba87ba45334e4a165cdee51fbfcc1733e94d224625cb2553d0f9

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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