The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-inclusion-exclusion-principle-s-873a

stable pending witness
2026-05-29 07:18:21 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the inclusion-exclusion principle. State any key formula.
83 chars

Generation

The inclusion-exclusion principle is a counting technique used to find the size of the union of overlapping sets by avoiding double-counting of shared elements. It works by first adding all individual set sizes, then subtracting the overcounted pairwise intersections, adding back the triple intersections (which were subtracted too many times), and so on. For two sets A and B, the formula is: **|A ∪ B| = |A| + |B| - |A ∩ B|** For three sets: **|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|**
28 tokens in · 188 tokens out · 7741 ms · $0.0029

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
7748.5 ms
Base LLM
7741.3 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0029

Trail

07:18:21 received prompt_chars=83 verifiers=['scripture_anchors', 'theology_doctrine']
07:18:21 red_gate decision=pass reason=no disqualifying input patterns detected
07:18:28 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=188 latency_ms=7741.3 cost_usd=0.0029
07:18:28 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:18:28 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:18:28 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:18:28 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:18:28 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:18:28 signed hash_algo=sha256 hash=sha256:6bc487130e3c4ce5a17b25c4d3cf4bcdb89f00a16a3e68ff2e852811c57b2dea

Content hash

sha256:6bc487130e3c4ce5a17b25c4d3cf4bcdb89f00a16a3e68ff2e852811c57b2dea

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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