The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-law-of-large-numbers-state-any-f03b

stable pending witness
2026-05-29 08:41:38 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the law of large numbers. State any key formula.
74 chars

Generation

The law of large numbers states that as the number of independent trials or observations increases, the sample average converges to the expected value (population mean). In other words, with more data, the empirical average becomes a more reliable estimate of the true mean. **Key formula:** As n → ∞, the sample mean X̄ₙ = (X₁ + X₂ + ... + Xₙ)/n converges to the expected value μ = E[X]. This law justifies why we can use sample statistics to estimate population parameters and explains why larger samples generally provide more accurate estimates.
27 tokens in · 137 tokens out · 3790 ms · $0.0021

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
3796.5 ms
Base LLM
3789.7 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0021

Trail

08:41:38 received prompt_chars=74 verifiers=['scripture_anchors', 'theology_doctrine']
08:41:38 red_gate decision=pass reason=no disqualifying input patterns detected
08:41:41 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=137 latency_ms=3789.7 cost_usd=0.0021
08:41:41 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:41:41 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:41:41 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:41:41 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:41:41 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:41:41 signed hash_algo=sha256 hash=sha256:319deec7f31d1425fb9dd8e7e7878dfe80dbd80d78c751573dd3068d51b82f22

Content hash

sha256:319deec7f31d1425fb9dd8e7e7878dfe80dbd80d78c751573dd3068d51b82f22

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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