The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-chain-rule-of-entropy-state-any-c449

stable pending witness
2026-05-29 02:30:16 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the chain rule of entropy. State any key formula.
75 chars

Generation

The chain rule of entropy states that the entropy of a collection of random variables can be decomposed into a sum of conditional entropies. Specifically, for random variables X₁, X₂, ..., Xₙ: **H(X₁, X₂, ..., Xₙ) = H(X₁) + H(X₂|X₁) + H(X₃|X₁,X₂) + ... + H(Xₙ|X₁,...,Xₙ₋₁)** This formula shows that joint entropy equals the entropy of the first variable plus the additional uncertainty contributed by each subsequent variable given all previous ones. It's analogous to the chain rule in calculus and is fundamental for analyzing information in sequential or dependent systems.
27 tokens in · 183 tokens out · 5690 ms · $0.0028

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
5698.6 ms
Base LLM
5689.8 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0028

Trail

02:30:16 received prompt_chars=75 verifiers=['scripture_anchors', 'theology_doctrine']
02:30:16 red_gate decision=pass reason=no disqualifying input patterns detected
02:30:22 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=183 latency_ms=5689.8 cost_usd=0.0028
02:30:22 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:30:22 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:30:22 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:30:22 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:30:22 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:30:22 signed hash_algo=sha256 hash=sha256:8d3752970bab97544fc6a68e417a364944a8f38deac27a6fa7a7a8db70f58906

Content hash

sha256:8d3752970bab97544fc6a68e417a364944a8f38deac27a6fa7a7a8db70f58906

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.

Bring another teaching · Run a discernment