The Mechanism · Gated-generation receipt

gen-explain-graph-coloring-counts-as-you-would-to-a-capable-stud-d059

stable pending witness
2026-05-29 08:29:29 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain graph coloring counts as you would to a capable student, in 3-4 sentences.
82 chars

Generation

Graph coloring counts tell you how many ways you can color the vertices of a graph using at most k colors, such that no two adjacent vertices share the same color. This count is given by the **chromatic polynomial** P(k), which is a function of the number of colors k available. For example, for a simple edge connecting two vertices, P(k) = k(k-1) because you have k choices for the first vertex and k-1 choices for the second (since it must differ from the first). The chromatic polynomial encodes important information about the graph's structure—its smallest positive integer root is the chromatic number, which is the minimum number of colors needed to properly color the graph.
29 tokens in · 152 tokens out · 4602 ms · $0.0024

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
4608.5 ms
Base LLM
4601.6 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0024

Trail

08:29:29 received prompt_chars=82 verifiers=['scripture_anchors', 'theology_doctrine']
08:29:29 red_gate decision=pass reason=no disqualifying input patterns detected
08:29:34 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=152 latency_ms=4601.6 cost_usd=0.0024
08:29:34 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:29:34 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:29:34 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:29:34 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:29:34 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:29:34 signed hash_algo=sha256 hash=sha256:1dd88e6e651b45194f54ca402ce0caffff6212096de10c328997fac38493336f

Content hash

sha256:1dd88e6e651b45194f54ca402ce0caffff6212096de10c328997fac38493336f

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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