The Mechanism · Gated-generation receipt

gen-state-the-euclidean-algorithm-precisely-and-explain-why-it-m-8bb7

stable pending witness
2026-05-29 02:47:16 UTC · anthropic/claude-sonnet-4-5

Prompt

State the Euclidean algorithm precisely and explain why it matters in 2-3 sentences.
84 chars

Generation

# The Euclidean Algorithm **Precise Statement:** To find gcd(a,b) where a ≥ b > 0, repeatedly replace the pair (a,b) with (b, a mod b) until reaching (d, 0); then gcd(a,b) = d. Equivalently: gcd(a,b) = gcd(b, a mod b), with base case gcd(a,0) = a. **Why It Matters:** The Euclidean algorithm provides an efficient method to compute greatest common divisors, which is fundamental to number theory, cryptography (like RSA), and simplifying fractions. Its extended version also allows us to find integer solutions to linear Diophantine equations and compute modular inverses, making it essential for modern cryptographic systems.
28 tokens in · 184 tokens out · 5288 ms · $0.0028

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
5294.5 ms
Base LLM
5287.6 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0028

Trail

02:47:16 received prompt_chars=84 verifiers=['scripture_anchors', 'theology_doctrine']
02:47:16 red_gate decision=pass reason=no disqualifying input patterns detected
02:47:21 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=184 latency_ms=5287.6 cost_usd=0.0028
02:47:21 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:47:21 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:47:21 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:47:21 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:47:21 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:47:21 signed hash_algo=sha256 hash=sha256:396360d4a4488ee4e6cfbf1f36b15d6e182ff4ef12c1946b434fee89639d912a

Content hash

sha256:396360d4a4488ee4e6cfbf1f36b15d6e182ff4ef12c1946b434fee89639d912a

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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