The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-primitive-roots-state-any-key-formu-79a8

stable pending witness
2026-05-29 05:43:15 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain primitive roots. State any key formula.
65 chars

Generation

A primitive root modulo n is an integer g whose powers generate all integers coprime to n. Specifically, g is a primitive root mod n if the set {g¹, g², g³, ..., g^φ(n)} produces all φ(n) numbers that are relatively prime to n (mod n), where φ is Euler's totient function. **Key formula:** g is a primitive root mod n if and only if **g^φ(n) ≡ 1 (mod n)** and **g^d ≢ 1 (mod n)** for all proper divisors d of φ(n). Primitive roots exist only when n = 1, 2, 4, p^k, or 2p^k where p is an odd prime.
24 tokens in · 185 tokens out · 4946 ms · $0.0028

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
4954.2 ms
Base LLM
4945.8 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0028

Trail

05:43:15 received prompt_chars=65 verifiers=['scripture_anchors', 'theology_doctrine']
05:43:15 red_gate decision=pass reason=no disqualifying input patterns detected
05:43:20 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=24 tokens_out=185 latency_ms=4945.8 cost_usd=0.0028
05:43:20 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:43:20 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:43:20 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:43:20 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:43:20 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:43:20 signed hash_algo=sha256 hash=sha256:fae80c2b71cfbdb56adf3ddb46db312a3848df6b7cfa57b77d3a7f04f1bc5fcc

Content hash

sha256:fae80c2b71cfbdb56adf3ddb46db312a3848df6b7cfa57b77d3a7f04f1bc5fcc

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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