The Mechanism · Gated-generation receipt

gen-state-bezout-s-identity-precisely-and-explain-why-it-matters-fada

stable pending witness
2026-05-29 03:15:50 UTC · anthropic/claude-sonnet-4-5

Prompt

State Bezout's identity precisely and explain why it matters in 2-3 sentences.
78 chars

Generation

## Bézout's Identity **Precise Statement:** For integers *a* and *b* with greatest common divisor *d* = gcd(*a*, *b*), there exist integers *x* and *y* such that *ax* + *by* = *d*. **Why It Matters:** Bézout's identity is fundamental because it shows that the GCD can be expressed as a linear combination of the original numbers, which is not obvious from the definition of GCD. This result is essential for proving key theorems in number theory (like the existence of multiplicative inverses in modular arithmetic) and provides the theoretical foundation for the Extended Euclidean Algorithm, which is used extensively in cryptography and computer algebra.
27 tokens in · 162 tokens out · 4363 ms · $0.0025

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
4371.5 ms
Base LLM
4363.5 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0025

Trail

03:15:50 received prompt_chars=78 verifiers=['scripture_anchors', 'theology_doctrine']
03:15:50 red_gate decision=pass reason=no disqualifying input patterns detected
03:15:54 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=162 latency_ms=4363.5 cost_usd=0.0025
03:15:54 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:15:54 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:15:54 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:15:54 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:15:54 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:15:54 signed hash_algo=sha256 hash=sha256:be5170200897bda0e8d69b20b4cebf3ec265cfd70ee1db0d49a62e554ec7e796

Content hash

sha256:be5170200897bda0e8d69b20b4cebf3ec265cfd70ee1db0d49a62e554ec7e796

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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