The Mechanism · Gated-generation receipt

gen-state-the-chain-rule-of-entropy-precisely-and-explain-why-it-f6c1

stable pending witness
2026-05-29 08:00:05 UTC · anthropic/claude-sonnet-4-5

Prompt

State the chain rule of entropy precisely and explain why it matters in 2-3 sentences.
86 chars

Generation

# Chain Rule of Entropy **Precise Statement:** For random variables X₁, X₂, ..., Xₙ, H(X₁, X₂, ..., Xₙ) = Σᵢ₌₁ⁿ H(Xᵢ | X₁, ..., Xᵢ₋₁) Equivalently, the joint entropy equals the sum of conditional entropies when conditioning on all previous variables. **Why it matters:** The chain rule decomposes complex joint distributions into manageable conditional pieces, making it fundamental for analyzing communication systems, data compression, and machine learning models. It reveals how uncertainty accumulates across sequences and provides the theoretical foundation for optimal sequential coding schemes like arithmetic coding.
27 tokens in · 177 tokens out · 5882 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
5888.7 ms
Base LLM
5881.8 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0027

Trail

08:00:05 received prompt_chars=86 verifiers=['scripture_anchors', 'theology_doctrine']
08:00:05 red_gate decision=pass reason=no disqualifying input patterns detected
08:00:11 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=177 latency_ms=5881.8 cost_usd=0.0027
08:00:11 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:00:11 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:00:11 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:00:11 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:00:11 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:00:11 signed hash_algo=sha256 hash=sha256:5da957784bdb5c26ce9ebde9127ea3abe8c7d7b5dca16ded5a9d5c12896f37b6

Content hash

sha256:5da957784bdb5c26ce9ebde9127ea3abe8c7d7b5dca16ded5a9d5c12896f37b6

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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