The Mechanism · Gated-generation receipt

gen-state-the-divisor-function-precisely-and-explain-why-it-matt-24a7

stable pending witness
2026-05-29 04:30:40 UTC · anthropic/claude-sonnet-4-5

Prompt

State the divisor function precisely and explain why it matters in 2-3 sentences.
81 chars

Generation

## The Divisor Function **Precise definition:** The divisor function σₖ(n) is the sum of the kth powers of all positive divisors of n. Most commonly, σ₀(n) = d(n) counts the number of divisors, and σ₁(n) = σ(n) gives the sum of divisors. **Why it matters:** The divisor function is fundamental to number theory because it encodes the multiplicative structure of integers and appears in problems ranging from perfect numbers (where σ(n) = 2n) to the Riemann zeta function. It also has practical applications in cryptography, combinatorics, and analyzing the average-case complexity of algorithms.
26 tokens in · 162 tokens out · 4869 ms · $0.0025

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
4880.8 ms
Base LLM
4869.0 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0025

Trail

04:30:40 received prompt_chars=81 verifiers=['scripture_anchors', 'theology_doctrine']
04:30:40 red_gate decision=pass reason=no disqualifying input patterns detected
04:30:45 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=162 latency_ms=4869.0 cost_usd=0.0025
04:30:45 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:30:45 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:30:45 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:30:45 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:30:45 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:30:45 signed hash_algo=sha256 hash=sha256:9f5ed55360370732a0727e1e5ccc48c670472692b78250e71e2b45350409f4e5

Content hash

sha256:9f5ed55360370732a0727e1e5ccc48c670472692b78250e71e2b45350409f4e5

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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