The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-graph-coloring-counts-state-any-key-f215

stable pending witness
2026-05-29 08:14:50 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain graph coloring counts. State any key formula.
71 chars

Generation

# Graph Coloring Counts Graph coloring counts determine the number of ways to properly color a graph's vertices using at most k colors, where adjacent vertices must have different colors. This count is given by the **chromatic polynomial P(G, k)**, which can be computed using the deletion-contraction recurrence: **P(G, k) = P(G - e, k) - P(G / e, k)** where G - e removes edge e and G / e contracts edge e. For basic graphs, P(empty graph with n vertices, k) = k^n, and P(complete graph K_n, k) = k(k-1)(k-2)ยทยทยท(k-n+1).
26 tokens in · 162 tokens out · 4137 ms · $0.0025

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
4147.8 ms
Base LLM
4136.9 ms
Verifiers
0.1 ms
Gates
0.1 ms
Cost
$0.0025

Trail

08:14:50 received prompt_chars=71 verifiers=['scripture_anchors', 'theology_doctrine']
08:14:50 red_gate decision=pass reason=no disqualifying input patterns detected
08:14:55 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=162 latency_ms=4136.9 cost_usd=0.0025
08:14:55 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:14:55 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:14:55 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:14:55 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:14:55 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:14:55 signed hash_algo=sha256 hash=sha256:24fc409c6ca01467b5896474603e0a1fedb1fa155770aed91dac0da9641c4799

Content hash

sha256:24fc409c6ca01467b5896474603e0a1fedb1fa155770aed91dac0da9641c4799

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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