The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-ramsey-theory-basics-state-any-key-4a8c

stable pending witness
2026-05-29 02:29:53 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Ramsey theory basics. State any key formula.
70 chars

Generation

Ramsey theory studies the emergence of order within large enough structures, asking how big a system must be before certain patterns inevitably appear. The fundamental principle is captured by Ramsey numbers R(m,n), which represent the minimum number of vertices needed in a complete graph so that any 2-coloring contains either a monochromatic clique of size m in the first color or size n in the second color. A key bound is **R(m,n) ≤ C(m+n-2, m-1)** (the binomial coefficient), though exact values are notoriously difficult to compute. The classic example is R(3,3) = 6, meaning among any 6 people, there must be either 3 mutual friends or 3 mutual strangers.
27 tokens in · 175 tokens out · 5261 ms · $0.0027

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
5269.3 ms
Base LLM
5261.2 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0027

Trail

02:29:53 received prompt_chars=70 verifiers=['scripture_anchors', 'theology_doctrine']
02:29:53 red_gate decision=pass reason=no disqualifying input patterns detected
02:29:58 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=175 latency_ms=5261.2 cost_usd=0.0027
02:29:58 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:29:58 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:29:58 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:29:58 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:29:58 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:29:58 signed hash_algo=sha256 hash=sha256:521178609954136adc8170700d165307f91e8048431a6f30df8c7c2f32bc65b1

Content hash

sha256:521178609954136adc8170700d165307f91e8048431a6f30df8c7c2f32bc65b1

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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