The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-wilson-s-theorem-answer-concisel-6a4b

stable pending witness
2026-05-29 06:12:30 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of Wilson's theorem? Answer concisely in 2-3 sentences.
80 chars

Generation

Wilson's theorem states that a natural number n > 1 is prime if and only if (n-1)! ≡ -1 (mod n). While it provides a theoretical characterization of prime numbers, it's computationally impractical for primality testing due to the factorial calculation. The theorem is primarily significant for its elegant connection between prime numbers and modular arithmetic, with applications in number theory proofs.
28 tokens in · 93 tokens out · 3299 ms · $0.0015

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
3305.2 ms
Base LLM
3298.6 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0015

Trail

06:12:30 received prompt_chars=80 verifiers=['scripture_anchors', 'theology_doctrine']
06:12:30 red_gate decision=pass reason=no disqualifying input patterns detected
06:12:33 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=93 latency_ms=3298.6 cost_usd=0.0015
06:12:33 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:12:33 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:12:33 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:12:33 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:12:33 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:12:33 signed hash_algo=sha256 hash=sha256:1e5158663313ebf44350b01e04bf943adc63c09c6ef448453255dacd2f307da5

Content hash

sha256:1e5158663313ebf44350b01e04bf943adc63c09c6ef448453255dacd2f307da5

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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