The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-wilson-s-theorem-state-any-key-form-c4ca

stable pending witness
2026-05-29 05:56:06 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Wilson's theorem. State any key formula.
66 chars

Generation

Wilson's theorem states that a positive integer \(n > 1\) is prime if and only if \((n-1)! \equiv -1 \pmod{n}\). In other words, for a prime number \(p\), the factorial of \(p-1\) leaves a remainder of \(p-1\) (or equivalently, \(-1\)) when divided by \(p\). For example, if \(p = 5\), then \(4! = 24 \equiv -1 \equiv 4 \pmod{5}\). This provides a theoretical primality test, though it's computationally impractical for large numbers due to the factorial calculation.
25 tokens in · 161 tokens out · 3202 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
3210.3 ms
Base LLM
3202.3 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0025

Trail

05:56:06 received prompt_chars=66 verifiers=['scripture_anchors', 'theology_doctrine']
05:56:06 red_gate decision=pass reason=no disqualifying input patterns detected
05:56:09 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=161 latency_ms=3202.3 cost_usd=0.0025
05:56:09 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:56:09 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:56:09 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:56:09 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:56:09 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:56:09 signed hash_algo=sha256 hash=sha256:6a141e9b63fca1500e6565be6942990db8d9e3d2915dd76b6a7800cb80e0d0fa

Content hash

sha256:6a141e9b63fca1500e6565be6942990db8d9e3d2915dd76b6a7800cb80e0d0fa

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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