The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-divisor-function-state-any-key-a77a

stable pending witness
2026-05-29 08:07:35 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the divisor function. State any key formula.
70 chars

Generation

The divisor function σ_k(n) is an arithmetic function that sums the k-th powers of all positive divisors of a natural number n. The most common cases are σ_0(n), which counts the number of divisors, and σ_1(n) (often written simply as σ(n)), which sums all divisors. **Key formula:** For n = p₁^a₁ · p₂^a₂ · ... · pᵣ^aᵣ (prime factorization), the divisor function is multiplicative: σ_k(n) = ∏ᵢ₌₁ʳ (pᵢ^(k(aᵢ+1)) - 1)/(pᵢ^k - 1)
26 tokens in · 194 tokens out · 6988 ms · $0.003

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
6998.8 ms
Base LLM
6987.8 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.003

Trail

08:07:35 received prompt_chars=70 verifiers=['scripture_anchors', 'theology_doctrine']
08:07:35 red_gate decision=pass reason=no disqualifying input patterns detected
08:07:42 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=194 latency_ms=6987.8 cost_usd=0.003
08:07:42 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:07:42 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:07:42 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:07:42 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:07:42 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:07:42 signed hash_algo=sha256 hash=sha256:f24c12550614b221c5a298f55052b40a932b8c90605de2827996fc425e2563bf

Content hash

sha256:f24c12550614b221c5a298f55052b40a932b8c90605de2827996fc425e2563bf

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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