The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-derangements-state-any-key-formula-4922

stable pending witness
2026-05-29 08:08:31 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain derangements. State any key formula.
62 chars

Generation

A derangement is a permutation of a set where no element appears in its original position. For example, if you have n letters and n addressed envelopes, a derangement represents a way to put all letters in the wrong envelopes. The number of derangements of n objects, denoted D(n) or !n, is given by the formula: **D(n) = n! · Σ(k=0 to n) (-1)^k / k!** Alternatively, **D(n) = n! / e** (rounded to the nearest integer), since the sum approximates 1/e.
25 tokens in · 145 tokens out · 3115 ms · $0.0022

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
3126.9 ms
Base LLM
3115.4 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0022

Trail

08:08:31 received prompt_chars=62 verifiers=['scripture_anchors', 'theology_doctrine']
08:08:31 red_gate decision=pass reason=no disqualifying input patterns detected
08:08:34 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=145 latency_ms=3115.4 cost_usd=0.0022
08:08:34 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:08:34 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:08:34 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:08:34 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:08:34 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:08:34 signed hash_algo=sha256 hash=sha256:f3f78de9e652b4d666d227dd28311154406d5e73ca2f54ee3c5f81940cfb89a9

Content hash

sha256:f3f78de9e652b4d666d227dd28311154406d5e73ca2f54ee3c5f81940cfb89a9

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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