The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-the-mean-value-theorem-answer-co-65e9

stable pending witness
2026-05-29 08:55:21 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of the mean value theorem? Answer concisely in 2-3 sentences.
86 chars

Generation

The Mean Value Theorem guarantees that for a smooth function over an interval, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over that interval. This fundamental result connects local behavior to global behavior and serves as the foundation for many important calculus theorems, including the relationship between derivatives and whether functions are increasing or decreasing.
29 tokens in · 81 tokens out · 2882 ms · $0.0013

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
2889.7 ms
Base LLM
2882.4 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0013

Trail

08:55:21 received prompt_chars=86 verifiers=['scripture_anchors', 'theology_doctrine']
08:55:21 red_gate decision=pass reason=no disqualifying input patterns detected
08:55:24 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=81 latency_ms=2882.4 cost_usd=0.0013
08:55:24 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:55:24 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:55:24 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:55:24 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:55:24 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:55:24 signed hash_algo=sha256 hash=sha256:69b38f809b55a31b22af9b6ff18b7b60fc578eaee265b80499946c867e0e871e

Content hash

sha256:69b38f809b55a31b22af9b6ff18b7b60fc578eaee265b80499946c867e0e871e

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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