The Mechanism · Gated-generation receipt

gen-state-wilson-s-theorem-precisely-and-explain-why-it-matters-b2af

stable pending witness
2026-05-29 05:48:49 UTC · anthropic/claude-sonnet-4-5

Prompt

State Wilson's theorem precisely and explain why it matters in 2-3 sentences.
77 chars

Generation

# Wilson's Theorem **Statement:** For any integer *p* > 1, *p* is prime if and only if (*p* - 1)! ≡ -1 (mod *p*). **Why it matters:** Wilson's theorem provides a complete characterization of prime numbers through modular arithmetic, establishing a necessary and sufficient condition for primality. While computationally impractical for testing large primes (since computing factorials is expensive), it's theoretically significant in number theory and has applications in proving other results about primes, modular arithmetic, and polynomial congruences.
25 tokens in · 133 tokens out · 4346 ms · $0.0021

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
4353.5 ms
Base LLM
4346.3 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0021

Trail

05:48:49 received prompt_chars=77 verifiers=['scripture_anchors', 'theology_doctrine']
05:48:49 red_gate decision=pass reason=no disqualifying input patterns detected
05:48:53 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=133 latency_ms=4346.3 cost_usd=0.0021
05:48:53 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:48:53 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:48:53 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:48:53 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:48:53 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:48:53 signed hash_algo=sha256 hash=sha256:ae1923626d5eed199a77c1c42e89ed2e10b01f7106181d1e2bc0cecd1e97057b

Content hash

sha256:ae1923626d5eed199a77c1c42e89ed2e10b01f7106181d1e2bc0cecd1e97057b

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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