The Mechanism · Gated-generation receipt

gen-explain-wilson-s-theorem-as-you-would-to-a-capable-student-i-d1a9

stable pending witness
2026-05-29 06:36:04 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain Wilson's theorem as you would to a capable student, in 3-4 sentences.
77 chars

Generation

Wilson's theorem states that a number p is prime if and only if (p-1)! ≡ -1 (mod p). In other words, when you multiply all the positive integers less than p and divide by p, you get a remainder of p-1 exactly when p is prime. For example, with p = 5: we have 4! = 24 ≡ -1 ≡ 4 (mod 5), confirming that 5 is prime. The key insight is that in the product (p-1)!, most numbers pair up with their modular inverses and cancel out, leaving only -1 when p is prime.
28 tokens in · 154 tokens out · 4634 ms · $0.0024

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
4639.5 ms
Base LLM
4633.6 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0024

Trail

06:36:04 received prompt_chars=77 verifiers=['scripture_anchors', 'theology_doctrine']
06:36:04 red_gate decision=pass reason=no disqualifying input patterns detected
06:36:09 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=154 latency_ms=4633.6 cost_usd=0.0024
06:36:09 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:36:09 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:36:09 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:36:09 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:36:09 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:36:09 signed hash_algo=sha256 hash=sha256:0d91f888ca8b4ce6048232802ee49384ed5c417b53e32b96e43c9c1827688a34

Content hash

sha256:0d91f888ca8b4ce6048232802ee49384ed5c417b53e32b96e43c9c1827688a34

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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