The Mechanism · Gated-generation receipt

gen-define-the-pigeonhole-principle-and-give-one-short-worked-ex-2c16

stable pending witness
2026-05-29 02:50:01 UTC · anthropic/claude-sonnet-4-5

Prompt

Define the pigeonhole principle and give one short worked example.
66 chars

Generation

# The Pigeonhole Principle **Definition:** If you distribute *n* items into *m* containers, and *n* > *m*, then at least one container must contain more than one item. More generally: If *n* items are placed into *m* containers, at least one container must contain at least ⌈*n*/*m*⌉ items (ceiling function). ## Worked Example **Problem:** In any group of 13 people, at least 2 must share the same birth month. **Solution:** - Items (pigeons) = 13 people - Containers (pigeonholes) = 12 months Since 13 > 12, by the pigeonhole principle, at least one month must contain at least ⌈13/12⌉ = ⌈1.083...⌉ = 2 people. Therefore, at least 2 people must share a birth month. ✓
20 tokens in · 226 tokens out · 4987 ms · $0.0034

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
4994.5 ms
Base LLM
4987.2 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0034

Trail

02:50:01 received prompt_chars=66 verifiers=['scripture_anchors', 'theology_doctrine']
02:50:01 red_gate decision=pass reason=no disqualifying input patterns detected
02:50:06 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=20 tokens_out=226 latency_ms=4987.2 cost_usd=0.0034
02:50:06 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:50:06 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:50:06 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:50:06 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:50:06 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:50:06 signed hash_algo=sha256 hash=sha256:6b934d8ef799e79f285e70503261fe1ec0032753a336a3d40f8a97dd3eeca7b7

Content hash

sha256:6b934d8ef799e79f285e70503261fe1ec0032753a336a3d40f8a97dd3eeca7b7

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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