The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-permutations-and-combinations-state-e5f6

stable pending witness
2026-05-29 05:37:11 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain permutations and combinations. State any key formula.
79 chars

Generation

# Permutations and Combinations **Permutations** count the number of ways to arrange objects where **order matters**. The formula for permutations of r objects from n total objects is P(n,r) = n!/(n-r)! **Combinations** count the number of ways to select objects where **order doesn't matter**. The formula for combinations of r objects from n total objects is C(n,r) = n!/[r!(n-r)!] The key difference: choosing a president, vice president, and treasurer from 10 people uses permutations (order matters), while choosing any 3 committee members from 10 people uses combinations (order doesn't matter).
27 tokens in · 160 tokens out · 3204 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
3212.0 ms
Base LLM
3204.2 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0025

Trail

05:37:11 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
05:37:11 red_gate decision=pass reason=no disqualifying input patterns detected
05:37:14 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=160 latency_ms=3204.2 cost_usd=0.0025
05:37:14 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:37:14 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:37:14 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:37:14 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:37:14 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:37:14 signed hash_algo=sha256 hash=sha256:f478fa4fd997738237dca9736714158c40f990370c0aad7e9f2584fffe4199d5

Content hash

sha256:f478fa4fd997738237dca9736714158c40f990370c0aad7e9f2584fffe4199d5

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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