The Mechanism · Gated-generation receipt

gen-state-the-inclusion-exclusion-principle-precisely-and-explai-0ec6

stable pending witness
2026-05-29 04:24:12 UTC · anthropic/claude-sonnet-4-5

Prompt

State the inclusion-exclusion principle precisely and explain why it matters in 2-3 sentences.
94 chars

Generation

# Inclusion-Exclusion Principle **Precise Statement:** For finite sets A₁, A₂, ..., Aₙ, |A₁ ∪ A₂ ∪ ... ∪ Aₙ| = ∑|Aᵢ| - ∑|Aᵢ ∩ Aⱼ| + ∑|Aᵢ ∩ Aⱼ ∩ Aₖ| - ... + (-1)ⁿ⁺¹|A₁ ∩ A₂ ∩ ... ∩ Aₙ| where the sums are over all possible combinations of the subscripts. **Why It Matters:** This principle corrects for overcounting when computing the size of unions—simply adding set sizes counts overlapping elements multiple times. It's fundamental to combinatorics, probability theory, and counting problems where direct enumeration is impractical, from computing poker hand probabilities to the analysis of algorithms.
28 tokens in · 257 tokens out · 5341 ms · $0.0039

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
5348.5 ms
Base LLM
5341.4 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0039

Trail

04:24:12 received prompt_chars=94 verifiers=['scripture_anchors', 'theology_doctrine']
04:24:12 red_gate decision=pass reason=no disqualifying input patterns detected
04:24:18 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=257 latency_ms=5341.4 cost_usd=0.0039
04:24:18 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:24:18 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:24:18 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:24:18 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:24:18 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:24:18 signed hash_algo=sha256 hash=sha256:823989cab02273edea621ad7542b45fed813353ee2979ff97cc9f881d0d45a2c

Content hash

sha256:823989cab02273edea621ad7542b45fed813353ee2979ff97cc9f881d0d45a2c

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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