The Mechanism · Gated-generation receipt

gen-state-the-pigeonhole-principle-precisely-and-explain-why-it-01fa

stable pending witness
2026-05-29 04:12:54 UTC · anthropic/claude-sonnet-4-5

Prompt

State the pigeonhole principle precisely and explain why it matters in 2-3 sentences.
85 chars

Generation

## The Pigeonhole Principle **Precise Statement:** If n items are placed into m containers, and n > m, then at least one container must contain more than one item. **Why It Matters:** The pigeonhole principle provides a simple yet powerful tool for proving existence results—it guarantees that certain patterns or collisions *must* occur without requiring you to construct them explicitly. Despite its apparent simplicity, it underlies countless proofs in combinatorics, computer science, and number theory, from showing that two people in large cities share the same number of hairs on their head to proving fundamental results about data compression limits and algorithmic complexity.
27 tokens in · 144 tokens out · 4432 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
4439.4 ms
Base LLM
4432.3 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0022

Trail

04:12:54 received prompt_chars=85 verifiers=['scripture_anchors', 'theology_doctrine']
04:12:54 red_gate decision=pass reason=no disqualifying input patterns detected
04:12:58 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=144 latency_ms=4432.3 cost_usd=0.0022
04:12:58 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:12:58 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:12:58 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:12:58 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:12:58 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:12:58 signed hash_algo=sha256 hash=sha256:86b4295efe477c196b2345a686f3e96a991292ea019178d46084d7d3ed3952ed

Content hash

sha256:86b4295efe477c196b2345a686f3e96a991292ea019178d46084d7d3ed3952ed

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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