The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-derangements-answer-concisely-in-6cb9

stable pending witness
2026-05-29 08:42:15 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of derangements? Answer concisely in 2-3 sentences.
76 chars

Generation

Derangements count permutations where no element appears in its original position, which has practical applications in problems like the "hat-check problem" (where guests randomly receive back hats, none getting their own). They appear in probability theory, combinatorics, and the inclusion-exclusion principle, with the number of derangements of n items given by D(n) = n!∑(-1)^k/k! for k=0 to n. The probability that a random permutation is a derangement approaches 1/e ≈ 36.8% as n increases.
28 tokens in · 133 tokens out · 6138 ms · $0.0021

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
6145.9 ms
Base LLM
6138.4 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0021

Trail

08:42:15 received prompt_chars=76 verifiers=['scripture_anchors', 'theology_doctrine']
08:42:15 red_gate decision=pass reason=no disqualifying input patterns detected
08:42:21 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=133 latency_ms=6138.4 cost_usd=0.0021
08:42:21 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:42:21 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:42:21 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:42:21 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:42:21 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:42:21 signed hash_algo=sha256 hash=sha256:f71d8c16b8c49927b6dafc785f77da61bf8a3272a4edbc43cb9c82d238e7634a

Content hash

sha256:f71d8c16b8c49927b6dafc785f77da61bf8a3272a4edbc43cb9c82d238e7634a

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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