The Mechanism · Gated-generation receipt

gen-state-the-chinese-remainder-theorem-precisely-and-explain-wh-65dc

stable pending witness
2026-05-29 08:52:10 UTC · anthropic/claude-sonnet-4-5

Prompt

State the Chinese remainder theorem precisely and explain why it matters in 2-3 sentences.
90 chars

Generation

## Chinese Remainder Theorem **Precise Statement:** Let n₁, n₂, ..., nₖ be pairwise coprime positive integers, and let a₁, a₂, ..., aₖ be any integers. Then the system of congruences: - x ≡ a₁ (mod n₁) - x ≡ a₂ (mod n₂) - ... - x ≡ aₖ (mod nₖ) has a unique solution modulo N = n₁ · n₂ · ... · nₖ. **Why It Matters:** The CRT establishes a fundamental isomorphism between ℤ/Nℤ and the product ℤ/n₁ℤ × ... × ℤ/nₖℤ, allowing us to break down difficult modular arithmetic problems into simpler, independent subproblems. This has critical applications in cryptography (RSA), fast computer arithmetic, and solving Diophantine equations.
26 tokens in · 249 tokens out · 5319 ms · $0.0038

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
5325.9 ms
Base LLM
5319.2 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0038

Trail

08:52:10 received prompt_chars=90 verifiers=['scripture_anchors', 'theology_doctrine']
08:52:10 red_gate decision=pass reason=no disqualifying input patterns detected
08:52:15 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=249 latency_ms=5319.2 cost_usd=0.0038
08:52:15 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:52:15 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:52:15 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:52:15 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:52:15 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:52:15 signed hash_algo=sha256 hash=sha256:a38a76ce4d8cdc4144883c0f0640b78d170adcce48f784ff0385e379d2060fad

Content hash

sha256:a38a76ce4d8cdc4144883c0f0640b78d170adcce48f784ff0385e379d2060fad

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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