The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-pigeonhole-principle-state-any-5658

stable pending witness
2026-05-29 03:11:52 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the pigeonhole principle. State any key formula.
74 chars

Generation

The pigeonhole principle states that if you distribute more than n items into n containers, at least one container must contain more than one item. More formally, if m items are placed into n containers where m > n, then at least one container holds at least ⌈m/n⌉ items (ceiling function). **Key formula:** If m objects are distributed among n boxes, at least one box contains at least ⌈m/n⌉ objects. This seemingly simple principle is powerful in proving existence results across mathematics, particularly in combinatorics and number theory.
27 tokens in · 129 tokens out · 6509 ms · $0.002

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
6519.0 ms
Base LLM
6509.5 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.002

Trail

03:11:52 received prompt_chars=74 verifiers=['scripture_anchors', 'theology_doctrine']
03:11:52 red_gate decision=pass reason=no disqualifying input patterns detected
03:11:58 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=129 latency_ms=6509.5 cost_usd=0.002
03:11:58 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:11:58 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:11:58 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:11:58 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:11:58 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:11:58 signed hash_algo=sha256 hash=sha256:a8e254412f6ca6004421d52b0494c3b22e1eef7dcca116da354ce56cdcb3626f

Content hash

sha256:a8e254412f6ca6004421d52b0494c3b22e1eef7dcca116da354ce56cdcb3626f

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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