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<div class="container">
<div class="header">
Reexpress MCP Server Output <span class="mcp-server-version">(v1.2.0)</span>
</div>
<div class="section">
<div class="section-title">Verification Results</div>
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">Successfully Verified (Prediction)</div>
<div class="field-value">
<div class="field-value"><span class="tag tag-positive">True</span></div>
</div>
</div>
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">Confidence</div>
<div class="field-value"><span class="tag tag-positive">>= 90%</span></div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 1 Summary <span class="model-name">(gpt-5-2025-08-07)</span></div>
<div>Question: What is the derivative of ln(x)? AI response: The derivative is 1/x, defined for x > 0.</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 1 Explanation <span class="model-name">(gpt-5-2025-08-07)</span></div>
<div>The response correctly gives d/dx ln(x) = 1/x with the appropriate domain x > 0. This is an objective, standard calculus result, not a matter of opinion and not time-sensitive. Multiple derivations (via the limit definition or inverse function rule for exp) agree. Verification is based on stable internal mathematical knowledge, not on user-provided context.</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 2 Explanation <span class="model-name">(gemini-2.5-pro)</span></div>
<div>The AI's response correctly identifies the derivative of the natural logarithm function, ln(x), as 1/x. This is a fundamental and well-established rule in differential calculus.</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 3 Agreement <span class="model-name">(granite-3.3-8b-instruct)</span></div>
<div>Do the model explanations agree that the response is correct?</div>
<div><span class="tag tag-positive">Yes</span></div>
</div>
</div>
<div class="separator"></div>
<div class="section">
<div class="section-title">Additional Information</div>
<div class="field-grid">
<div class="field-box">
<div class="field-label">File Access</div>
<div class="field-value">The verification model did not have access to any external files.</div>
</div>
<div class="field-box">
<div class="field-label">Date</div>
<div class="field-value">2025-08-25 19:17:46</div>
</div>
</div>
</div>
<div class="section">
<div class="section-title">Uncertainty (instance-level) Details</div>
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">p(y | x)_lower</div>
<div class="field-value">[0.00637540368055234, 0.9936213860453533]</div>
</div>
<div class="field-grid">
<div class="field-box">
<div class="field-label">Valid Index-Conditional Estimate</div>
<div class="field-value">
<span class="tag tag-positive">True</span>
</div>
</div>
<div class="field-box">
<div class="field-label">Out-of-Distribution</div>
<div class="field-value">
<span class="tag tag-positive">False</span>
</div>
</div>
<div class="field-box">
<div class="field-label">Rescaled q_lower, <span style="font-family: 'Times New Roman', serif;">
(<span class="math-qtilde">q</span><sub style="font-size: 0.7em;">lower</sub>)
</span></div>
<div class="field-value">2.2818913309070847</div>
</div>
<div class="field-box">
<div class="field-label">Iterated offset_lower (for class 1),
<span class="math-operator-m">
m
<span class="math-superscript-hat-y">ŷ</span>
<span class="math-subscript-floor">
⌊<span class="qtilde-small">q</span>⌋
</span>
</span>
</div>
<div class="field-value">3.210274094376977e-06</div>
</div>
<div class="field-box">
<div class="field-label">Effective Sample Size (by class)</div>
<div class="field-value">[17172.0, 20688.0]</div>
</div>
</div>
<div class="field-grid">
<div class="field-box">
<div class="field-label">
Similarity to Training (q)
</div>
<div class="field-value">40</div>
</div>
<div class="field-box">
<div class="field-label">
Distance to Training (d) Quantile
</div>
<div class="field-value">0.3010709285736084</div>
</div>
<div class="field-box">
<div class="field-label">
Magnitude
</div>
<div class="field-value">[-4.72243595123291, 3.3480544090270996]</div>
</div>
</div>
</div>
<div class="section">
<div class="section-title">SDM Estimator (Model-level) Details</div>
<div class="field-grid">
<div class="field-box">
<div class="field-label">
α'
</div>
<div class="field-value">0.9</div>
</div>
<div class="field-box">
<div class="field-label">
Min valid rescaled q
<span style="font-family: 'Times New Roman', serif;">
(<span class="math-qtilde">q</span><sup style="font-size: 0.7em;">γ</sup><sub style="font-size: 0.7em;">min</sub>)
</span>
</div>
<div class="field-value">1.0003684141164448</div>
</div>
<div class="field-box">
<div class="field-label">
Class-wise output thresholds (ψ)
</div>
<div class="field-value">[0.9551128149032593, 0.9338333010673523]</div>
</div>
<div class="field-box">
<div class="field-label">
Support/training size
</div>
<div class="field-value">74684</div>
</div>
</div>
</div>
<div class="section">
<div class="section-title">Prompt</div>
<div class="prompt-box">What is the derivative of ln(x)?</div>
</div>
<div class="section">
<div class="section-title">AI Response</div>
<div class="document-box">Reference [1, internal knowledge]: The derivative of the natural logarithm function ln(x) with respect to x is 1/x, which is a fundamental result in calculus; Answer: The derivative of ln(x) is 1/x, defined for all x > 0.</div>
</div>
<div class="separator"></div>
<div class="nearest-match-box">
<div class="section" style="margin-left: 40px;">
<div class="section-title">Nearest Match in Training</div>
<div class="field-grid">
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">Successfully Verified (Prediction)</div>
<div class="field-value">
<div class="field-value"><span class="tag tag-positive">True</span></div>
</div>
</div>
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">True Label</div>
<div class="field-value"><span class="tag tag-positive">Verified</span></div>
</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 1 Summary <span class="model-name">(gpt-5-2025-08-07)</span></div>
<div>Question: Find the imaginary part of (1 - i)(2 + 4i). AI Response: Multiplied to get 6 + 2i and concluded the imaginary part is 2.</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 1 Explanation <span class="model-name">(gpt-5-2025-08-07)</span></div>
<div>The AI correctly applied complex multiplication: (1 - i)(2 + 4i) = 2 + 4i - 2i - 4i^2 = 6 + 2i, so the imaginary part is 2. The response directly answers the question. This is an objective math problem with no dependence on recent information. Verification is based on the user-provided values and standard complex arithmetic (internal knowledge).</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 2 Explanation <span class="model-name">(gemini-2.5-pro)</span></div>
<div>The AI correctly calculates the product of the two complex numbers and accurately identifies the imaginary part of the resulting complex number. The step-by-step calculation is flawless.</div>
</div>
<div class="explanation-box-positive">
<div class="explanation-title-positive">Model 3 Agreement <span class="model-name">(granite-3.3-8b-instruct)</span></div>
<div>Do the model explanations agree that the response is correct?</div>
<div><span class="tag tag-positive">Yes</span></div>
</div>
<div class="section">
<div class="section-title">Prompt</div>
<div class="prompt-box">If the complex numbers $z_1 = 1 - i$ and $z_2 = 2 + 4i$, where $i$ is the imaginary unit, then the imaginary part of the complex number $z_1z_2$ is.</div>
</div>
<div class="section">
<div class="section-title">AI Response</div>
<div class="document-box">To find the imaginary part of the complex number \( z_1 z_2 \) where \( z_1 = 1 - i \) and \( z_2 = 2 + 4i \), we use the formula for multiplying complex numbers.
First, we use the distributive property (FOIL method) to expand the product:
\[
(1 - i)(2 + 4i)
\]
Expanding each term:
\[
= 1 \cdot 2 + 1 \cdot 4i - i \cdot 2 - i \cdot 4i
\]
Calculating each term:
\[
= 2 + 4i - 2i - 4i^2
\]
Combine like terms and simplify, noting that \( i^2 = -1 \):
\[
= 2 + (4i - 2i) - 4(-1)
\]
\[
= 2 + 2i + 4
\]
Combine the real parts:
\[
= (2 + 4) + 2i
\]
\[
= 6 + 2i
\]
The imaginary part of the complex number \( z_1 z_2 \) is the coefficient of \( i \), which is 2.
Thus, the imaginary part of the complex number \( z_1 z_2 \) is \(\boxed{2}\).</div>
</div>
<div class="field-grid">
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">Document ID</div>
<div class="field-value">pos_253709_da37d774-190d-48db-a53c-90b464046869</div>
</div>
<div class="field-box" style="margin-bottom: 20px;">
<div class="field-label">Document Source</div>
<div class="field-value">OpenVerification1: openthoughts</div>
</div>
</div>
</div>
</div>
<div class="separator"></div>
<div class="section">
<div class="section-title">Legend</div>
<div class="legend-content">
<p>An ensemble of models 1, 2, and 3 (including the hidden states of model 3) is taken as the input to the SDM estimator that determines the verification classification.</p>
<div class="legend-items">
<div class="legend-item">
<span class="legend-label">Class 0:</span>
<span class="legend-value">NOT Verified</span>
</div>
<div class="legend-item">
<span class="legend-label">Class 1:</span>
<span class="legend-value">Verified</span>
</div>
</div>
</div>
</div>
</div>
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