The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-fermat-s-little-theorem-state-any-k-80d9

stable pending witness
2026-05-29 06:10:09 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Fermat's little theorem. State any key formula.
73 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) is congruent to 1 modulo p. **Key formula:** a^(p-1) ≡ 1 (mod p), or equivalently, a^p ≡ a (mod p) for any integer a. This theorem is fundamental in number theory and has important applications in cryptography, particularly in the RSA algorithm and primality testing. It provides a necessary (but not sufficient) condition for primality.
28 tokens in · 135 tokens out · 2732 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
2738.8 ms
Base LLM
2731.7 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0021

Trail

06:10:09 received prompt_chars=73 verifiers=['scripture_anchors', 'theology_doctrine']
06:10:09 red_gate decision=pass reason=no disqualifying input patterns detected
06:10:12 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=135 latency_ms=2731.7 cost_usd=0.0021
06:10:12 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:10:12 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:10:12 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:10:12 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:10:12 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:10:12 signed hash_algo=sha256 hash=sha256:8d72fc8879ecfe854aa24ed19a81e235962ccd77851e09d2cc13873c8a786673

Content hash

sha256:8d72fc8879ecfe854aa24ed19a81e235962ccd77851e09d2cc13873c8a786673

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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