The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-mean-value-theorem-state-any-ke-022f

stable pending witness
2026-05-29 06:53:05 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the mean value theorem. State any key formula.
72 chars

Generation

The Mean Value Theorem states that for a function that is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), there exists at least one point c in (a,b) where the instantaneous rate of change equals the average rate of change over the interval. The key formula is: **f'(c) = [f(b) - f(a)]/(b - a)** Geometrically, this means there is at least one point on the curve where the tangent line is parallel to the secant line connecting the endpoints. This theorem is fundamental in calculus, connecting the derivative at a point to the overall behavior of the function over an interval.
26 tokens in · 154 tokens out · 2800 ms · $0.0024

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
2806.6 ms
Base LLM
2799.6 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0024

Trail

06:53:05 received prompt_chars=72 verifiers=['scripture_anchors', 'theology_doctrine']
06:53:05 red_gate decision=pass reason=no disqualifying input patterns detected
06:53:08 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=154 latency_ms=2799.6 cost_usd=0.0024
06:53:08 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:53:08 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:53:08 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:53:08 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:53:08 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:53:08 signed hash_algo=sha256 hash=sha256:fa8cc8075cbd2aec6e3fa892869288f9cf1ea8b948b9df51fb16177f76f31228

Content hash

sha256:fa8cc8075cbd2aec6e3fa892869288f9cf1ea8b948b9df51fb16177f76f31228

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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