The Mechanism · Gated-generation receipt

gen-state-graph-coloring-counts-precisely-and-explain-why-it-mat-2df9

stable pending witness
2026-05-29 07:35:07 UTC · anthropic/claude-sonnet-4-5

Prompt

State graph coloring counts precisely and explain why it matters in 2-3 sentences.
82 chars

Generation

# Graph Coloring Counts The **chromatic polynomial** P(G,k) counts the number of ways to properly color a graph G using at most k colors, where adjacent vertices must have different colors. For example: - Complete graph Kₙ: P(Kₙ,k) = k(k-1)(k-2)···(k-n+1) - Path graph Pₙ: P(Pₙ,k) = k(k-1)ⁿ⁻¹ - Cycle Cₙ: P(Cₙ,k) = (k-1)ⁿ + (-1)ⁿ(k-1) ## Why It Matters Graph coloring counts are fundamental because they connect combinatorics to real-world scheduling and allocation problems (like frequency assignment or exam scheduling) while also revealing deep structural properties of graphs. The chromatic number—the minimum k where P(G,k) > 0—determines the computational complexity class of many optimization problems, making coloring counts essential to both theoretical computer science and practical algorithm design.
26 tokens in · 261 tokens out · 7143 ms · $0.004

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
7150.2 ms
Base LLM
7143.0 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.004

Trail

07:35:07 received prompt_chars=82 verifiers=['scripture_anchors', 'theology_doctrine']
07:35:07 red_gate decision=pass reason=no disqualifying input patterns detected
07:35:14 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=261 latency_ms=7143.0 cost_usd=0.004
07:35:14 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:35:14 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:35:14 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:35:14 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:35:14 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:35:14 signed hash_algo=sha256 hash=sha256:8459d1bae999a22e64cb8b1275a6c8222b3138eef9cc78610706a387522bb30c

Content hash

sha256:8459d1bae999a22e64cb8b1275a6c8222b3138eef9cc78610706a387522bb30c

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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