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bundle2tutorialeng.html

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Modal Tutorial Bundle 2 – The Trinity in Different Approaches (Modules 9–11)</title>
    <base href="https://pompeii-project.org">
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<div class="container" id="app">

    <!-- ===== HEADER ===== -->
    <h1>📐 The Trinity in Different Approaches <span class="version-badge">Bundle 2 – Modules 9–11</span></h1>
    <div class="subhead">
        An interactive tutorial – Step by step<br>
        <a href="modal.html?lang=en">← Back to the entry page</a> |
        <a href="bundle2eng.pdf">📄 PDF of the bundle</a>
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    <div style="margin-top: 2.5rem; padding-top: 1rem; border-top: 1px solid #e0d6c8; font-size: 0.75rem; color: #8b8b8b; text-align: center;">
        <p>© Paul Koop – <a href="modal.html?lang=en" style="color: #6b4f8a;">the-last-freedom.org/Projekt_Pompeji</a></p>
        <p style="margin-top:0.2rem;">Bundle 2 guides through Modules 9, 10, and 11 – the Trinity in different approaches.</p>
    </div>

</div>

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        //  INTRODUCTION (Steps 0–2)
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        {
            id: 0,
            title: "What is Bundle 2?",
            part: "Introduction",
            explanation: `
                <p><strong>Bundle 2</strong> guides through the three modules that show the Trinity in <strong>different approaches</strong>:</p>
                <ul>
                    <li><strong>Module 9:</strong> Bimodal Non-Compatibility in DeLillo and Van Dormael.</li>
                    <li><strong>Module 10:</strong> Unitarity as Theorem.</li>
                    <li><strong>Module 11:</strong> Timelessness and Presence.</li>
                </ul>
                <p>These three modules answer the question: <strong>How does the Trinity appear in different approaches?</strong></p>
                <p>They build on Bundle 1 but presuppose only the foundations from <code>modal5.md</code>.</p>
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                <p><strong>What is the goal of Bundle 2?</strong></p>
                <div class="qoptions">
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                    <label><input type="radio" name="q0" value="1"> To show the Trinity in different approaches.</label>
                    <label><input type="radio" name="q0" value="2"> To prove the existence of God.</label>
                    <label><input type="radio" name="q0" value="3"> To expose the Trinity as an illusion.</label>
                    <label><input type="radio" name="q0" value="4"> To replace physics.</label>
                    <label><input type="radio" name="q0" value="5"> To ground theology.</label>
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                <p>In Bundle 1 it was shown:</p>
                <ol>
                    <li><strong>Module 6:</strong> The minimal subset for the target statement is {ASP3, ASP4, ASP5}.</li>
                    <li><strong>Module 7:</strong> The Trinity is axiom-invariant.</li>
                    <li><strong>Module 8:</strong> The limit definition can be precisely grasped.</li>
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                <p>Bundle 2 goes one step further: It shows how the Trinity appears in different <strong>perspectives</strong>.</p>
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                <p><strong>What has Bundle 1 shown?</strong></p>
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                    <label><input type="radio" name="q1" value="2"> That S5 is contradictory.</label>
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                <p>Bundle 2 shows three approaches to the Trinity:</p>
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                    <li><strong>Bimodal approach</strong> (Module 9): The Trinity appears in two perspectives — DeLillo and Van Dormael — that are not compatible but simultaneously true.</li>
                    <li><strong>Unitarity approach</strong> (Module 10): The Trinity guarantees that nothing is lost.</li>
                    <li><strong>Temporal approach</strong> (Module 11): The Trinity is timeless and present.</li>
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                <p>All three approaches lead to the same structure — but in different ways.</p>
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                <p><strong>Which three approaches does Bundle 2 show?</strong></p>
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                    <label><input type="radio" name="q2" value="1"> Bimodal, unitary, temporal</label>
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                    <label><input type="radio" name="q2" value="3"> Formal, informal, practical</label>
                    <label><input type="radio" name="q2" value="4"> Old, middle, new</label>
                    <label><input type="radio" name="q2" value="5"> Theoretical, practical, aesthetic</label>
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        // ============================================================
        //  PART A: BIMODAL NON-COMPATIBILITY (Module 9) – Steps 3–12
        // ============================================================
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            title: "PART A – Module 9: Bimodal Non-Compatibility",
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                <div style="background:#f8d7da; padding:0.5rem 1rem; border-radius:6px; margin-bottom:1rem;">
                    <span class="part-tag pA">PART A</span>
                    <strong>Module 9</strong>
                </div>
                <p><strong>Question:</strong> How can two perspectives — DeLillo and Van Dormael — be simultaneously true although they are not compatible?</p>
                <p><strong>Answer:</strong> Through a <strong>bimodal modal logic</strong> with two accessibility relations R_D and R_V on the same set of worlds.</p>
                <p><strong>Consequence:</strong> The Omega Point is a fixed point, not a state.</p>
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                <p><strong>What is the main question of Module 9?</strong></p>
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                    <label><input type="radio" name="q3" value="1"> How two incompatible perspectives can be simultaneously true.</label>
                    <label><input type="radio" name="q3" value="2"> Whether S5 is contradictory.</label>
                    <label><input type="radio" name="q3" value="3"> Whether physics is relevant.</label>
                    <label><input type="radio" name="q3" value="4"> Whether God exists.</label>
                    <label><input type="radio" name="q3" value="5"> Whether the Trinity is an illusion.</label>
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        {
            id: 4,
            title: "The Basic Idea of Bimodal Modal Logic",
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                <p>Classical modal logic uses <strong>one</strong> accessibility relation R. <strong>Bimodal modal logic</strong> uses <strong>two</strong> relations R_D and R_V on <strong>the same</strong> set of worlds.</p>
                <div class="formula-box partA">ℳ = (W, R_D, R_V, I)</div>
                <p><strong>Intuition:</strong> Both relations describe <strong>the same reality</strong> — but from different perspectives.</p>
                <p>The two necessity operators:</p>
                <ul>
                    <li><span class="symbol">□_D p</span>: "Necessary in the DeLillo perspective."</li>
                    <li><span class="symbol">□_V p</span>: "Necessary in the Van Dormael perspective."</li>
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                    <label><input type="radio" name="q4" value="3"> It is the same as S5.</label>
                    <label><input type="radio" name="q4" value="4"> It is contradictory.</label>
                    <label><input type="radio" name="q4" value="5"> It uses infinitely many relations.</label>
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        {
            id: 5,
            title: "The DeLillo Perspective (□_D)",
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                <p>The <strong>DeLillo perspective</strong> corresponds to the <strong>Copenhagen interpretation</strong>:</p>
                <ul>
                    <li><strong>Collapse:</strong> The wave function collapses upon measurement.</li>
                    <li><strong>One reality:</strong> Out of many possibilities comes <strong>one</strong> reality.</li>
                    <li><strong>Constitutive observer:</strong> Without observer no measurement, without measurement no collapse.</li>
                    <li><strong>Thinning:</strong> What does not collapse is <strong>not real</strong>.</li>
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                <p><strong>Axiom:</strong></p>
                <div class="formula-box partA">D4 := □_D(Observer → □_D Observer)</div>
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                <p><strong>Which physical interpretation does the DeLillo perspective correspond to?</strong></p>
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                    <label><input type="radio" name="q5" value="1"> Copenhagen interpretation</label>
                    <label><input type="radio" name="q5" value="2"> Bohmian mechanics</label>
                    <label><input type="radio" name="q5" value="3"> Relational quantum mechanics</label>
                    <label><input type="radio" name="q5" value="4"> QBism</label>
                    <label><input type="radio" name="q5" value="5"> Classical mechanics</label>
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            id: 6,
            title: "The Van Dormael Perspective (□_V)",
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                <p>The <strong>Van Dormael perspective</strong> corresponds to the <strong>Many-Worlds Interpretation (MWI)</strong>:</p>
                <ul>
                    <li><strong>No collapse:</strong> The wave function does <strong>not</strong> collapse.</li>
                    <li><strong>All realities:</strong> Every possibility is realized in a world.</li>
                    <li><strong>Non-constitutive observer:</strong> The observer is himself part of the branching.</li>
                    <li><strong>Fullness:</strong> What does not collapse is <strong>real</strong>.</li>
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                <p><strong>Axiom:</strong></p>
                <div class="formula-box partA">V4 := □_V(Observer → ◇_V ¬Observer)</div>
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                <p><strong>Which physical interpretation does the Van Dormael perspective correspond to?</strong></p>
                <div class="qoptions">
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                    <label><input type="radio" name="q6" value="1"> Many-Worlds Interpretation</label>
                    <label><input type="radio" name="q6" value="2"> Bohmian mechanics</label>
                    <label><input type="radio" name="q6" value="3"> Relational quantum mechanics</label>
                    <label><input type="radio" name="q6" value="4"> QBism</label>
                    <label><input type="radio" name="q6" value="5"> Classical mechanics</label>
                </div>
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        },
        {
            id: 7,
            title: "The Omega Point as Fixed Point",
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            explanation: `
                <p>The <strong>Omega Point</strong> is defined as the set of worlds in which <strong>both perspectives hold simultaneously</strong>:</p>
                <div class="formula-box partA important">Ω := { w ∈ W | □_D p @ w ∧ □_V ¬p @ w }</div>
                <p>for a suitable proposition p (e.g., p := "There is exactly one reality").</p>
                <p><strong>Meaning:</strong> The Omega Point is <strong>not a state</strong> that one could reach. It is a <strong>fixed point</strong> — a set of worlds in which both perspectives hold <strong>simultaneously</strong>.</p>
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                <p><strong>What is the Omega Point?</strong></p>
                <div class="qoptions">
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                    <label><input type="radio" name="q7" value="1"> A fixed point — a set of worlds in which both perspectives hold simultaneously.</label>
                    <label><input type="radio" name="q7" value="2"> A contradiction.</label>
                    <label><input type="radio" name="q7" value="3"> An illusion.</label>
                    <label><input type="radio" name="q7" value="4"> An axiom.</label>
                    <label><input type="radio" name="q7" value="5"> A time.</label>
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        },
        {
            id: 8,
            title: "The Consistency of the Two Perspectives",
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            explanation: `
                <p><strong>Main result:</strong> There are consistent bimodal S5 models in which Ω ≠ ∅.</p>
                <p><strong>Construction:</strong></p>
                <ul>
                    <li>W = {w₁}</li>
                    <li>R_D = {(w₁, w₁)} (S5)</li>
                    <li>R_V = {(w₁, w₁)} (S5)</li>
                    <li>p @ w₁: true in R_D-perspective, false in R_V-perspective</li>
                </ul>
                <p><strong>Verification:</strong> □_D p @ w₁ holds and □_V ¬p @ w₁ holds. Hence w₁ ∈ Ω.</p>
                <p><strong>Consequence:</strong> Non-compatibility is <strong>not an inconsistency</strong>, but a structural property of reality.</p>
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                <p><strong>What does the consistency proof show?</strong></p>
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                    <label><input type="radio" name="q8" value="1"> That there are consistent models in which both perspectives hold simultaneously.</label>
                    <label><input type="radio" name="q8" value="2"> That one of the perspectives is false.</label>
                    <label><input type="radio" name="q8" value="3"> That there are no perspectives.</label>
                    <label><input type="radio" name="q8" value="4"> That physics is irrelevant.</label>
                    <label><input type="radio" name="q8" value="5"> That S5 is contradictory.</label>
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        {
            id: 9,
            title: "The Consequence: The Omega Point Is a Relation",
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            explanation: `
                <p>The Omega Point is <strong>not a state</strong>, but a <strong>relation</strong> between two perspectives:</p>
                <ul>
                    <li>It is not the end of time.</li>
                    <li>It is not a goal that one could reach.</li>
                    <li>It is the <strong>structure</strong> of reality as a relation between two perspectives.</li>
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                <p><strong>Consequence:</strong> Whoever holds one of the two perspectives for the whole <strong>loses the Omega Point</strong>. It becomes an ideology.</p>
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                <p><strong>What is the Omega Point in Module 9?</strong></p>
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                    <label><input type="radio" name="q9" value="1"> A relation between two perspectives.</label>
                    <label><input type="radio" name="q9" value="2"> A goal.</label>
                    <label><input type="radio" name="q9" value="3"> A contradiction.</label>
                    <label><input type="radio" name="q9" value="4"> An illusion.</label>
                    <label><input type="radio" name="q9" value="5"> A time.</label>
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        {
            id: 10,
            title: "Bundle 2 – Part A: Summary",
            part: "Part A",
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                <div style="background:#f8d7da; padding:0.8rem 1.2rem; border-radius:8px;">
                    <p style="font-weight:bold; color:#8a3a3a;">Result of Module 9:</p>
                    <div class="formula-box partA important">DeLillo and Van Dormael are not compatible, but simultaneously true.</div>
                    <p>The Omega Point is a fixed point — a relation, not a state.</p>
                    <p>Both perspectives are paths to the same limit.</p>
                </div>
            `,
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                <p><strong>What is the conclusion of Part A?</strong></p>
                <div class="qoptions">
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                    <label><input type="radio" name="q10" value="1"> The two perspectives are not compatible but simultaneously true.</label>
                    <label><input type="radio" name="q10" value="2"> One of the perspectives is false.</label>
                    <label><input type="radio" name="q10" value="3"> There are no perspectives.</label>
                    <label><input type="radio" name="q10" value="4"> Physics is irrelevant.</label>
                    <label><input type="radio" name="q10" value="5"> S5 is contradictory.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 11,
            title: "Comparison of the Two Approaches",
            part: "Part A",
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                <p>The two perspectives in comparison:</p>
                <table style="width:100%; border-collapse:collapse; margin:1rem 0; font-size:0.9rem;">
                    <tr style="background:#e0d6c8;">
                        <th style="border:1px solid #d4cab8; padding:0.4rem; text-align:left;">Aspect</th>
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                        <th style="border:1px solid #d4cab8; padding:0.4rem; text-align:left;">Van Dormael (MWI)</th>
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                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Collapse</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Yes</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">No</td>
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                    <tr>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Reality</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">One</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Many</td>
                    </tr>
                    <tr>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Observer</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Constitutive</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Non-constitutive</td>
                    </tr>
                    <tr>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Image</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Stones</td>
                        <td style="border:1px solid #d4cab8; padding:0.4rem;">Branchings</td>
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                <p><strong>How do the two perspectives differ regarding the observer?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q11" value="0"> DeLillo: non-constitutive; Van Dormael: constitutive</label>
                    <label><input type="radio" name="q11" value="1"> DeLillo: constitutive; Van Dormael: non-constitutive</label>
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        },
        {
            id: 12,
            title: "Transition to Part B",
            part: "Part A",
            explanation: `
                <p>Module 9 has shown: The Trinity appears in two perspectives that are not compatible but simultaneously true. The question remained open:</p>
                <p><strong>How can the resurrection be derived from weaker assumptions?</strong></p>
                <p>This question is treated in <strong>Module 10</strong>.</p>
            `,
            question_html: `
                <p><strong>Which question does Module 10 treat?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q12" value="0"> Whether the Trinity is true.</label>
                    <label><input type="radio" name="q12" value="1"> How the resurrection can be derived from weaker assumptions.</label>
                    <label><input type="radio" name="q12" value="2"> Whether S5 is contradictory.</label>
                    <label><input type="radio" name="q12" value="3"> Whether physics is relevant.</label>
                    <label><input type="radio" name="q12" value="4"> Whether God exists.</label>
                    <label><input type="radio" name="q12" value="5"> Whether the Trinity is an illusion.</label>
                </div>
            `,
            answer: 1
        },

        // ============================================================
        //  PART B: UNITARITY AS THEOREM (Module 10) – Steps 13–21
        // ============================================================
        {
            id: 13,
            title: "PART B – Module 10: Unitarity as Theorem",
            part: "Part B",
            explanation: `
                <div style="background:#d4e4f7; padding:0.5rem 1rem; border-radius:6px; margin-bottom:1rem;">
                    <span class="part-tag pB">PART B</span>
                    <strong>Module 10</strong>
                </div>
                <p><strong>Question:</strong> How can the resurrection be derived from weaker assumptions?</p>
                <p><strong>Answer:</strong> Through the three principles U1–U3 and the operator 𝓘 ("information is conserved").</p>
                <p><strong>Consequence:</strong> Unitarity is the structure of Totality T itself.</p>
            `,
            question_html: `
                <p><strong>What is the main question of Module 10?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q13" value="0"> Whether the Trinity is true.</label>
                    <label><input type="radio" name="q13" value="1"> How the resurrection can be derived from weaker assumptions.</label>
                    <label><input type="radio" name="q13" value="2"> Whether S5 is contradictory.</label>
                    <label><input type="radio" name="q13" value="3"> Whether physics is relevant.</label>
                    <label><input type="radio" name="q13" value="4"> Whether God exists.</label>
                    <label><input type="radio" name="q13" value="5"> Whether the Trinity is an illusion.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 14,
            title: "From Strong Axioms to Weaker Principles",
            part: "Part B",
            explanation: `
                <p>In <code>modal5.md</code>, the resurrection was derived from the strong axioms A16–A20:</p>
                <div class="formula-box partB">A16 := ∀c(Consciousness(c) → integrated(c, S))</div>
                <div class="formula-box partB">A18 := □∀I(Existence(I) → Conservation(I))</div>
                <p>These axioms are <strong>strong</strong>. They presuppose that all consciousnesses are integrated into S.</p>
                <p>Module 10 shows: <strong>Unitarity can be derived from weaker assumptions.</strong></p>
            `,
            question_html: `
                <p><strong>Why are axioms A16–A20 strong?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q14" value="0"> Because they are too short.</label>
                    <label><input type="radio" name="q14" value="1"> Because they presuppose that all consciousnesses are integrated into S.</label>
                    <label><input type="radio" name="q14" value="2"> Because they contradict A1.</label>
                    <label><input type="radio" name="q14" value="3"> Because they are not axioms.</label>
                    <label><input type="radio" name="q14" value="4"> Because they are too long.</label>
                    <label><input type="radio" name="q14" value="5"> Because they are true.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 15,
            title: "The Operator 𝓘 (Information)",
            part: "Part B",
            explanation: `
                <p>The operator <strong>𝓘</strong> formalizes the concept of <strong>information conservation</strong>:</p>
                <div class="formula-box partB">𝓘p @ w ⟺ the information about p is conserved in w</div>
                <p><strong>Axioms:</strong></p>
                <ul>
                    <li>I1: 𝓘p → ◇p</li>
                    <li>I2: 𝓘(p ∧ q) ↔ (𝓘p ∧ 𝓘q)</li>
                    <li>I3: 𝓘¬p ↔ ¬𝓘p</li>
                    <li>I4: 𝓘p → □𝓘p</li>
                </ul>
                <p><strong>Important:</strong> 𝓘 is <strong>not</strong> a normal operator. It does <strong>not</strong> satisfy the Kripke axiom K.</p>
            `,
            question_html: `
                <p><strong>What does the operator 𝓘 formalize?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q15" value="0"> Necessity</label>
                    <label><input type="radio" name="q15" value="1"> Information conservation</label>
                    <label><input type="radio" name="q15" value="2"> Possibility</label>
                    <label><input type="radio" name="q15" value="3"> Contradiction</label>
                    <label><input type="radio" name="q15" value="4"> Time</label>
                    <label><input type="radio" name="q15" value="5"> Consciousness</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 16,
            title: "Principle U1: Realization Implies Information Conservation",
            part: "Part B",
            explanation: `
                <p><strong>U1:</strong></p>
                <div class="formula-box partB important">U1 := □∀p(Realized(p) → 𝓘p)</div>
                <p><strong>Reading:</strong> If p is realized, then the information about p is conserved.</p>
                <p><strong>Meaning:</strong> What was once real cannot fully disappear. It remains conserved as information — even if the original form is lost.</p>
            `,
            question_html: `
                <p><strong>What does U1 state?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q16" value="0"> Realization implies destruction.</label>
                    <label><input type="radio" name="q16" value="1"> Realization implies information conservation.</label>
                    <label><input type="radio" name="q16" value="2"> Realization is impossible.</label>
                    <label><input type="radio" name="q16" value="3"> Realization is the same as necessity.</label>
                    <label><input type="radio" name="q16" value="4"> Realization is irrelevant.</label>
                    <label><input type="radio" name="q16" value="5"> Realization is an illusion.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 17,
            title: "Principle U2: Death Is Local Access Loss",
            part: "Part B",
            explanation: `
                <p><strong>U2:</strong></p>
                <div class="formula-box partB important">U2 := □∀c(Death(c) → ¬◇_local Access(c))</div>
                <p><strong>Reading:</strong> The death of a consciousness c is the loss of <strong>local</strong> access to c.</p>
                <p><strong>Meaning:</strong> Death is <strong>not</strong> extinction. It is an access loss. From the perspective of c, it appears as an end. From the perspective of S, it is a transition.</p>
            `,
            question_html: `
                <p><strong>What does U2 state?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q17" value="0"> Death is the end of everything.</label>
                    <label><input type="radio" name="q17" value="1"> Death is a local access loss.</label>
                    <label><input type="radio" name="q17" value="2"> Death is an illusion.</label>
                    <label><input type="radio" name="q17" value="3"> Death is a new beginning.</label>
                    <label><input type="radio" name="q17" value="4"> Death is the same as life.</label>
                    <label><input type="radio" name="q17" value="5"> Death is impossible.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 18,
            title: "Principle U3: Global Integration Is Possible",
            part: "Part B",
            explanation: `
                <p><strong>U3:</strong></p>
                <div class="formula-box partB important">U3 := □∀c(Death(c) → ◇_global Integration(c, S))</div>
                <p><strong>Reading:</strong> The death of a consciousness c makes its <strong>global</strong> integration into S possible.</p>
                <p><strong>Meaning:</strong> Death is <strong>not the end</strong>. It is the condition of the possibility of integration. What is lost locally is integrated globally.</p>
            `,
            question_html: `
                <p><strong>What does U3 state?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q18" value="0"> Death makes integration impossible.</label>
                    <label><input type="radio" name="q18" value="1"> Death makes global integration possible.</label>
                    <label><input type="radio" name="q18" value="2"> Death is irrelevant.</label>
                    <label><input type="radio" name="q18" value="3"> Death is an illusion.</label>
                    <label><input type="radio" name="q18" value="4"> Death is the same as integration.</label>
                    <label><input type="radio" name="q18" value="5"> Death is impossible.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 19,
            title: "Proof: Unitarity Implies Resurrection",
            part: "Part B",
            explanation: `
                <p><strong>Theorem U-A:</strong> In S5+U1–U3:</p>
                <div class="formula-box partB important">□∀c(Consciousness(c) → ◇Resurrection(c))</div>
                <p><strong>Proof sketch:</strong></p>
                <ol>
                    <li>Let c be a consciousness. Suppose c dies.</li>
                    <li>From U2: Local access to c is lost.</li>
                    <li>From U1: The information about c is conserved.</li>
                    <li>From U3: The global integration of c into S is possible.</li>
                    <li>Definition: Resurrection(c) := Integration(c, S).</li>
                    <li>Therefore: ◇Resurrection(c).</li>
                    <li>With Necessitation: □∀c(Consciousness(c) → ◇Resurrection(c)).</li>
                </ol>
            `,
            question_html: `
                <p><strong>What follows from U1–U3?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q19" value="0"> Resurrection is impossible.</label>
                    <label><input type="radio" name="q19" value="1"> Resurrection is possible for every consciousness.</label>
                    <label><input type="radio" name="q19" value="2"> Resurrection is an illusion.</label>
                    <label><input type="radio" name="q19" value="3"> Resurrection is identical to death.</label>
                    <label><input type="radio" name="q19" value="4"> Resurrection is identical to nothing.</label>
                    <label><input type="radio" name="q19" value="5"> Resurrection is irrelevant.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 20,
            title: "Unitarity as the Structure of T",
            part: "Part B",
            explanation: `
                <p><strong>Central insight:</strong> Unitarity is <strong>not an additional assumption</strong>. It is the <strong>structure</strong> of Totality T itself:</p>
                <div class="formula-box partB important">∀γ ⊆ T : γ ∈ T</div>
                <p><strong>Reading:</strong> For every path γ that is a subset of T: γ is an element of T.</p>
                <p><strong>Consequence:</strong> Totality T contains <strong>all</strong> paths — including those that do not reach the Omega Point. There is <strong>no escape</strong> from T.</p>
            `,
            question_html: `
                <p><strong>What is unitarity in Module 10?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q20" value="0"> An additional assumption.</label>
                    <label><input type="radio" name="q20" value="1"> The structure of Totality T itself.</label>
                    <label><input type="radio" name="q20" value="2"> An illusion.</label>
                    <label><input type="radio" name="q20" value="3"> A contradiction.</label>
                    <label><input type="radio" name="q20" value="4"> A time.</label>
                    <label><input type="radio" name="q20" value="5"> An axiom.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 21,
            title: "Bundle 2 – Part B: Summary",
            part: "Part B",
            explanation: `
                <div style="background:#d4e4f7; padding:0.8rem 1.2rem; border-radius:8px;">
                    <p style="font-weight:bold; color:#3a5a8a;">Result of Module 10:</p>
                    <div class="formula-box partB important">Unitarity is not an additional assumption — it is the structure of T.</div>
                    <p>From U1–U3, resurrection follows as a theorem.</p>
                    <p>The "intervention" of the Omega Point is structural, not temporal.</p>
                </div>
            `,
            question_html: `
                <p><strong>What is the conclusion of Part B?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q21" value="0"> Resurrection is impossible.</label>
                    <label><input type="radio" name="q21" value="1"> Unitarity is the structure of T; resurrection follows as a theorem.</label>
                    <label><input type="radio" name="q21" value="2"> Resurrection is an illusion.</label>
                    <label><input type="radio" name="q21" value="3"> Resurrection is identical to death.</label>
                    <label><input type="radio" name="q21" value="4"> Resurrection is identical to nothing.</label>
                    <label><input type="radio" name="q21" value="5"> Resurrection is irrelevant.</label>
                </div>
            `,
            answer: 1
        },

        // ============================================================
        //  PART C: TEMPORAL MODAL LOGIC (Module 11) – Steps 22–29
        // ============================================================
        {
            id: 22,
            title: "PART C – Module 11: Temporal Modal Logic",
            part: "Part C",
            explanation: `
                <div style="background:#e8e0f0; padding:0.5rem 1rem; border-radius:6px; margin-bottom:1rem;">
                    <span class="part-tag pC">PART C</span>
                    <strong>Module 11</strong>
                </div>
                <p><strong>Question:</strong> How can the time structure of the Trinity be precisely grasped?</p>
                <p><strong>Answer:</strong> Through the temporal operators P, F, N, Z and the axioms T1–T4.</p>
                <p><strong>Consequence:</strong> The Trinity is timeless and present.</p>
            `,
            question_html: `
                <p><strong>What is the main question of Module 11?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q22" value="0"> Whether the Trinity is true.</label>
                    <label><input type="radio" name="q22" value="1"> How the time structure of the Trinity can be precisely grasped.</label>
                    <label><input type="radio" name="q22" value="2"> Whether S5 is contradictory.</label>
                    <label><input type="radio" name="q22" value="3"> Whether physics is relevant.</label>
                    <label><input type="radio" name="q22" value="4"> Whether God exists.</label>
                    <label><input type="radio" name="q22" value="5"> Whether the Trinity is an illusion.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 23,
            title: "The Informal Time Structure",
            part: "Part C",
            explanation: `
                <p>In <code>Unitaritaet.md</code> and <code>DieZeitlosigkeitDerSteine2.md</code>, the distinction plays a central role:</p>
                <ul>
                    <li><strong>Past</strong> as horizon of knowledge.</li>
                    <li><strong>Future</strong> as horizon of knowledge.</li>
                    <li><strong>Present</strong> as locus of reality.</li>
                    <li><strong>Timelessness</strong> as limit.</li>
                </ul>
                <p>This distinction is <strong>informal</strong>. Module 11 provides a temporal modal-logical version of it.</p>
            `,
            question_html: `
                <p><strong>What is the present in the time structure?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q23" value="0"> A horizon of knowledge.</label>
                    <label><input type="radio" name="q23" value="1"> The locus of reality.</label>
                    <label><input type="radio" name="q23" value="2"> A limit.</label>
                    <label><input type="radio" name="q23" value="3"> An illusion.</label>
                    <label><input type="radio" name="q23" value="4"> A contradiction.</label>
                    <label><input type="radio" name="q23" value="5"> An axiom.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 24,
            title: "The Four Time Modes: P, F, N, Z",
            part: "Part C",
            explanation: `
                <p>We extend S5 by <strong>four</strong> temporal operators:</p>
                <ul>
                    <li><strong>P p</strong> (past): "It was the case that p."</li>
                    <li><strong>F p</strong> (future): "It will be the case that p."</li>
                    <li><strong>N p</strong> (present): "It is now the case that p."</li>
                    <li><strong>Z p</strong> (timelessness): "It is timelessly the case that p."</li>
                </ul>
                <p><strong>Semantics:</strong></p>
                <div class="formula-box partC">
P p @ w ⟺ ∃v (v &lt; w ∧ p @ v)<br>
F p @ w ⟺ ∃v (w &lt; v ∧ p @ v)<br>
N p @ w ⟺ p @ w<br>
Z p @ w ⟺ ∀v (p @ v)
                </div>
            `,
            question_html: `
                <p><strong>How many temporal operators does Module 11 use?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q24" value="0"> Two</label>
                    <label><input type="radio" name="q24" value="1"> Three</label>
                    <label><input type="radio" name="q24" value="2"> Four</label>
                    <label><input type="radio" name="q24" value="3"> Five</label>
                    <label><input type="radio" name="q24" value="4"> Six</label>
                    <label><input type="radio" name="q24" value="5"> Seven</label>
                </div>
            `,
            answer: 2
        },
        {
            id: 25,
            title: "The Axioms T1–T4",
            part: "Part C",
            explanation: `
                <p>The axioms of the temporal extension:</p>
                <ul>
                    <li><strong>T1:</strong> □(N p → p) — Present is real.</li>
                    <li><strong>T2:</strong> □(P p → ¬N p) — Past is not present.</li>
                    <li><strong>T3:</strong> □(Z p → □p) — Timelessness implies necessity.</li>
                    <li><strong>T4:</strong> □(◇p → (P p ∨ N p ∨ F p ∨ Z p)) — Completeness of time modes.</li>
                </ul>
                <p><strong>T3</strong> is particularly important: Timelessness is <strong>stronger</strong> than necessity.</p>
            `,
            question_html: `
                <p><strong>What does T3 state?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q25" value="0"> Present is real.</label>
                    <label><input type="radio" name="q25" value="1"> Timelessness implies necessity.</label>
                    <label><input type="radio" name="q25" value="2"> Past is not present.</label>
                    <label><input type="radio" name="q25" value="3"> Every possibility is temporally located.</label>
                    <label><input type="radio" name="q25" value="4"> Future is impossible.</label>
                    <label><input type="radio" name="q25" value="5"> Time is an illusion.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 26,
            title: "Reconstruction of the Trinity as Z(U ∧ T ∧ S)",
            part: "Part C",
            explanation: `
                <p>The Trinity is reconstructed as a <strong>timeless</strong> structure:</p>
                <div class="formula-box partC important">Tr := Z(U ∧ T ∧ S)</div>
                <p><strong>Reading:</strong> The Trinity is <strong>timelessly</strong> the case. It is not in time, but in the <strong>structure</strong> of time.</p>
                <p><strong>Derivation:</strong></p>
                <ol>
                    <li>From modal5.md: □(U ∧ T ∧ S).</li>
                    <li>From the timelessness of the Trinity: Z(U ∧ T ∧ S).</li>
                    <li>From T3: □(U ∧ T ∧ S).</li>
                </ol>
            `,
            question_html: `
                <p><strong>How is the Trinity reconstructed in Module 11?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q26" value="0"> Tr := U ∧ T ∧ S</label>
                    <label><input type="radio" name="q26" value="1"> Tr := Z(U ∧ T ∧ S)</label>
                    <label><input type="radio" name="q26" value="2"> Tr := □(U ∧ T ∧ S)</label>
                    <label><input type="radio" name="q26" value="3"> Tr := ◇(U ∧ T ∧ S)</label>
                    <label><input type="radio" name="q26" value="4"> Tr := P(U ∧ T ∧ S)</label>
                    <label><input type="radio" name="q26" value="5"> Tr := F(U ∧ T ∧ S)</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 27,
            title: "Timelessness and Presence Are Compatible",
            part: "Part C",
            explanation: `
                <p><strong>Theorem Z-A:</strong> In S5+T1–T4:</p>
                <div class="formula-box partC important">□(Z(U ∧ T ∧ S) ∧ N(U ∧ T ∧ S))</div>
                <p><strong>Reading:</strong> The Trinity is <strong>both timeless and present</strong>.</p>
                <p><strong>Meaning:</strong> The two modes are <strong>compatible</strong>. The Trinity is not bound to one mode.</p>
            `,
            question_html: `
                <p><strong>What follows from T1–T4 for the Trinity?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q27" value="0"> The Trinity is impossible.</label>
                    <label><input type="radio" name="q27" value="1"> The Trinity is timeless and present.</label>
                    <label><input type="radio" name="q27" value="2"> The Trinity is an illusion.</label>
                    <label><input type="radio" name="q27" value="3"> The Trinity is only in the past.</label>
                    <label><input type="radio" name="q27" value="4"> The Trinity is only in the future.</label>
                    <label><input type="radio" name="q27" value="5"> The Trinity is contradictory.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 28,
            title: "The Timelessness of Structural Integration",
            part: "Part C",
            explanation: `
                <p>In Module 10 it was shown: Unitarity is the structure of T. The integration of paths that do not reach the Omega Point is <strong>structural</strong>, not temporal.</p>
                <p>Module 11 makes this precise:</p>
                <div class="formula-box partC important">Z(∀γ ⊆ T : γ ∈ T)</div>
                <p><strong>Reading:</strong> It is <strong>timelessly</strong> the case that for every path γ that is a subset of T: γ is an element of T.</p>
                <p><strong>Consequence:</strong> Integration has <strong>always already</strong> happened. From the perspective of a path, it appears as future.</p>
            `,
            question_html: `
                <p><strong>What is structural integration in Module 11?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q28" value="0"> A temporal event.</label>
                    <label><input type="radio" name="q28" value="1"> A timeless structure — always already happened.</label>
                    <label><input type="radio" name="q28" value="2"> An illusion.</label>
                    <label><input type="radio" name="q28" value="3"> A contradiction.</label>
                    <label><input type="radio" name="q28" value="4"> A time.</label>
                    <label><input type="radio" name="q28" value="5"> An axiom.</label>
                </div>
            `,
            answer: 1
        },
        {
            id: 29,
            title: "Bundle 2 – Conclusion: The Trinity in Different Approaches",
            part: "Part C",
            explanation: `
                <p><strong>Summary of Bundle 2:</strong></p>
                <ol>
                    <li><strong>Module 9:</strong> The Trinity appears in two perspectives (DeLillo and Van Dormael) that are not compatible but simultaneously true.</li>
                    <li><strong>Module 10:</strong> Unitarity is the structure of T; resurrection follows as a theorem from U1–U3.</li>
                    <li><strong>Module 11:</strong> The Trinity is timeless and present — Z(U ∧ T ∧ S) ∧ N(U ∧ T ∧ S).</li>
                </ol>
                <div class="formula-box important" style="font-size:1.1rem;">
                    The Trinity appears in different approaches — <strong>bimodal</strong>, <strong>unitary</strong>, <strong>temporal</strong> — and yet leads to the same structure.
                </div>
                <p style="margin-top:1rem; text-align:center; font-style:italic; color:#555;">
                    What remains is the insight that the Trinity is <strong>timeless</strong> and manifests itself <strong>presently</strong>.
                </p>
            `,
            question_html: `
                <p><strong>What is the conclusion of Bundle 2?</strong></p>
                <div class="qoptions">
                    <label><input type="radio" name="q29" value="0"> The Trinity is impossible.</label>
                    <label><input type="radio" name="q29" value="1"> The Trinity appears in different approaches and yet leads to the same structure.</label>
                    <label><input type="radio" name="q29" value="2"> S5 is contradictory.</label>
                    <label><input type="radio" name="q29" value="3"> Physics is irrelevant.</label>
                    <label><input type="radio" name="q29" value="4"> God does not exist.</label>
                    <label><input type="radio" name="q29" value="5"> The Trinity is an illusion.</label>
                </div>
            `,
            answer: 1
        }
    ];

    // ================================================================
    //  LOGIC
    // ================================================================

    let currentStep = 0;
    const totalSteps = steps.length;
    const stepContainer = document.getElementById('stepContainer');
    const prevBtn = document.getElementById('prevBtn');
    const nextBtn = document.getElementById('nextBtn');
    const stepIndicator = document.getElementById('stepIndicator');
    const progressFill = document.getElementById('progressFill');
    const progressText = document.getElementById('progressText');

    let stepCompleted = new Array(totalSteps).fill(false);

    function renderStep(index) {
        const step = steps[index];
        if (!step) return;

        const partClass = step.part === 'Part A' ? 'partA' :
                         step.part === 'Part B' ? 'partB' :
                         step.part === 'Part C' ? 'partC' : '';

        let html = `
            <div class="step-block ${partClass}">
                <div class="step-number ${partClass}">Step ${index} of ${totalSteps-1} – ${step.part}</div>
                <div class="step-title">${step.title}</div>
                <div class="step-explanation">${step.explanation}</div>
                <div class="question-box" id="questionBox${index}">
                    ${step.question_html}
                    <button class="qsubmit" data-step="${index}">Check answer</button>
                    <div class="qfeedback" id="feedback${index}"></div>
                </div>
            </div>
        `;

        stepContainer.innerHTML = html;

        const submitBtn = document.querySelector(`.qsubmit[data-step="${index}"]`);
        if (submitBtn) {
            submitBtn.addEventListener('click', function(e) {
                checkAnswer(index);
            });
        }

        const questionBox = document.getElementById(`questionBox${index}`);
        if (questionBox) {
            const radioInputs = questionBox.querySelectorAll('input[type="radio"]');
            radioInputs.forEach(input => {
                input.addEventListener('keydown', function(e) {
                    if (e.key === 'Enter') {
                        e.preventDefault();
                        checkAnswer(index);
                    }
                });
            });
        }

        updateNav(index);
        updateProgress(index);

        if (stepCompleted[index]) {
            const feedback = document.getElementById(`feedback${index}`);
            if (feedback) {
                feedback.className = 'qfeedback correct';
                feedback.textContent = '✅ Correct! You have already completed this step.';
                feedback.style.display = 'block';
            }
            const submitBtn2 = document.querySelector(`.qsubmit[data-step="${index}"]`);
            if (submitBtn2) submitBtn2.disabled = true;
        }

        document.querySelector('.step-block').scrollIntoView({ behavior: 'smooth', block: 'start' });
    }

    function checkAnswer(index) {
        if (stepCompleted[index]) return;

        const step = steps[index];
        const questionBox = document.getElementById(`questionBox${index}`);
        const feedback = document.getElementById(`feedback${index}`);

        const selected = questionBox.querySelector('input[type="radio"]:checked');
        if (!selected) {
            feedback.className = 'qfeedback wrong';
            feedback.textContent = '❌ Please select an answer.';
            feedback.style.display = 'block';
            return;
        }

        const userAnswer = parseInt(selected.value);
        const isCorrect = (userAnswer === step.answer);

        if (isCorrect) {
            feedback.className = 'qfeedback correct';
            feedback.textContent = '✅ Correct! You may proceed to the next step.';
            feedback.style.display = 'block';
            stepCompleted[index] = true;

            const submitBtn = questionBox.querySelector('.qsubmit');
            if (submitBtn) submitBtn.disabled = true;

            const radios = questionBox.querySelectorAll('input[type="radio"]');
            radios.forEach(r => r.disabled = true);

            updateNav(index);
            updateProgress(index);

            if (stepCompleted.every(v => v === true)) {
                setTimeout(() => {
                    alert('🎉 Congratulations! You have successfully completed Bundle 2!');
                }, 400);
            }

        } else {
            feedback.className = 'qfeedback wrong';
            feedback.textContent = '❌ Incorrect. Please read the step again and try again.';
            feedback.style.display = 'block';
        }
    }

    function updateNav(index) {
        if (index > 0) {
            prevBtn.className = '';
            prevBtn.style.pointerEvents = 'auto';
            prevBtn.style.opacity = '1';
        } else {
            prevBtn.className = 'disabled';
            prevBtn.style.pointerEvents = 'none';
            prevBtn.style.opacity = '0.5';
        }

        if (index < totalSteps - 1 && stepCompleted[index]) {
            nextBtn.className = '';
            nextBtn.style.pointerEvents = 'auto';
            nextBtn.style.opacity = '1';
            nextBtn.textContent = 'Next →';
        } else if (index === totalSteps - 1 && stepCompleted[index]) {
            nextBtn.className = '';
            nextBtn.style.pointerEvents = 'auto';
            nextBtn.style.opacity = '1';
            nextBtn.textContent = '🎉 Done – back to the entry page';
            nextBtn.href = 'modal.html?lang=en';
        } else {
            nextBtn.className = 'disabled';
            nextBtn.style.pointerEvents = 'none';
            nextBtn.style.opacity = '0.5';
            if (index === totalSteps - 1) {
                nextBtn.textContent = '🎉 Final step – complete all steps';
            } else {
                nextBtn.textContent = 'Next → (please answer the question first)';
            }
        }

        stepIndicator.textContent = `${index} / ${totalSteps-1}`;
    }

    function updateProgress(index) {
        const completedCount = stepCompleted.filter(v => v === true).length;
        const percent = (completedCount / totalSteps) * 100;
        progressFill.style.width = `${percent}%`;
        progressText.textContent = `Step ${index} of ${totalSteps-1} – ${completedCount}/${totalSteps} completed`;
    }

    prevBtn.addEventListener('click', function(e) {
        e.preventDefault();
        if (currentStep > 0) {
            currentStep--;
            renderStep(currentStep);
        }
    });

    nextBtn.addEventListener('click', function(e) {
        e.preventDefault();
        if (currentStep < totalSteps - 1 && stepCompleted[currentStep]) {
            currentStep++;
            renderStep(currentStep);
        } else if (currentStep === totalSteps - 1 && stepCompleted[currentStep]) {
            window.location.href = 'modal.html?lang=en';
        }
    });

    renderStep(0);
</script>

</body>
</html>