The Mechanism · Gated-generation receipt

gen-explain-primitive-roots-as-you-would-to-a-capable-student-in-bbe1

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

Prompt

Explain primitive roots as you would to a capable student, in 3-4 sentences.
76 chars

Generation

A primitive root modulo n is a number g whose powers generate all numbers relatively prime to n. Think of it like this: if you keep multiplying g by itself (mod n), you'll eventually hit every number that shares no common factors with n before the pattern repeats. For example, 3 is a primitive root mod 7 because the powers 3¹, 3², 3³, 3⁴, 3⁵, 3⁶ give you 3, 2, 6, 4, 5, 1 (mod 7)—all six non-zero remainders. Primitive roots only exist for certain moduli (specifically n = 1, 2, 4, p^k, or 2p^k where p is an odd prime), and they're fundamental to number theory and cryptography.
27 tokens in · 191 tokens out · 4362 ms · $0.0029

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
4369.3 ms
Base LLM
4362.1 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0029

Trail

05:05:32 received prompt_chars=76 verifiers=['scripture_anchors', 'theology_doctrine']
05:05:32 red_gate decision=pass reason=no disqualifying input patterns detected
05:05:36 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=191 latency_ms=4362.1 cost_usd=0.0029
05:05:36 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:05:36 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:05:36 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:05:36 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:05:36 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:05:36 signed hash_algo=sha256 hash=sha256:47b869a7c199dfefcc0f733389a7962518be8dd0b4788382e85c703e94389654

Content hash

sha256:47b869a7c199dfefcc0f733389a7962518be8dd0b4788382e85c703e94389654

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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