The Mechanism · Gated-generation receipt

gen-explain-fermat-s-little-theorem-as-you-would-to-a-capable-st-0f90

stable pending witness
2026-05-29 08:32:53 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain Fermat's little theorem as you would to a capable student, in 3-4 sentences.
84 chars

Generation

Fermat's Little Theorem states that if p is a prime number and a is any integer not divisible by p, then a^(p-1) ≡ 1 (mod p). In simpler terms, when you raise a number to the power of (prime minus one) and divide by that prime, you always get a remainder of 1. For example, take a=2 and p=7: 2^6 = 64, and 64 ÷ 7 leaves remainder 1. This theorem is foundational in number theory and has practical applications in cryptography, particularly in RSA encryption.
31 tokens in · 142 tokens out · 4206 ms · $0.0022

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
4217.7 ms
Base LLM
4206.0 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0022

Trail

08:32:53 received prompt_chars=84 verifiers=['scripture_anchors', 'theology_doctrine']
08:32:53 red_gate decision=pass reason=no disqualifying input patterns detected
08:32:57 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=31 tokens_out=142 latency_ms=4206.0 cost_usd=0.0022
08:32:57 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:32:57 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:32:57 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:32:57 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:32:57 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:32:57 signed hash_algo=sha256 hash=sha256:fddde5e5d1cffa4d1005f4f61d756c4fd8a24a0308b9fc579f1eee2bda97fe83

Content hash

sha256:fddde5e5d1cffa4d1005f4f61d756c4fd8a24a0308b9fc579f1eee2bda97fe83

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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