W33-Theory/docs/steiner-scheme-order1440-4873-4874.html at master · wilcompute/W33-Theory · GitHub

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  <title>Passes 4873–4874 — two order-1440 groups and the Steiner scheme</title>
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  <p><a href="index.html">← Back to the W33 atlas</a></p>
  <div class="eyebrow">Passes 4873–4874 · exact finite certificates</div>
  <h1>One number hid two groups. Four relations expose the Steiner cover.</h1>
  <p class="lede">The marked-double-six residue and the exceptional outer automorphism of
    S<sub>6</sub> both produce order 1440, but not the same group. Separately, the 120 Steiner
    triangles carry a complete imprimitive four-class association scheme whose quotient is
    Q(4,3), the line-intersection graph of W(3,3), not its point graph.</p>

  <div class="grid">
    <div class="card"><div class="metric">1440 ≠ identification</div><p>different centers, involution counts, and element orders</p></div>
    <div class="card"><div class="metric">120 = 40 × 3</div><p>forty intrinsic three-element Steiner fibers</p></div>
    <div class="card"><div class="metric">1, 2, 27, 36, 54</div><p>exact scheme valencies</p></div>
    <div class="card"><div class="metric">15 ≠ 11 over F₃</div><p>Q43 line carrier separated from W33 points</p></div>
  </div>

  <h2>Pass 4873: the two order-1440 extensions are nonisomorphic</h2>
  <table>
    <thead><tr><th>Carrier</th><th>Group</th><th>Center</th><th>Involutions</th><th>Order-8 elements</th></tr></thead>
    <tbody>
      <tr><td>marked double-six residue</td><td>S<sub>6</sub> × C<sub>2</sub></td><td>2</td><td>151</td><td>0</td></tr>
      <tr><td>duad–syntheme outer action</td><td>Aut(S<sub>6</sub>) = S<sub>6</sub>:Out(S<sub>6</sub>)</td><td>1</td><td>111</td><td>360</td></tr>
    </tbody>
  </table>
  <p>The first C<sub>2</sub> is the central operation that fixes duads and complements
    triads. The second is the exceptional outer-automorphism direction, certified by sending
    a transposition to a triple transposition. Their common order is a count coincidence, not
    a group identification.</p>

  <h2>Pass 4874: the complete Steiner association scheme</h2>
  <p>The four nonidentity relations on the 120 Steiner triangles have the exact meanings:</p>
  <div class="eq">R₁: 120 pairs, valency 2  = forty K₃ fibers
R₂: 1620 pairs, valency 27 = a perfect matching across each nonedge fiber pair
R₃: 2160 pairs, valency 36 = all nine pairs across each Q(4,3) edge
R₄: 3240 pairs, valency 54 = the complementary six pairs across each nonedge</div>
  <p>The exact first eigenmatrix is</p>
  <div class="eq">[ 1,  2, 27,  36, 54]
[ 1,  2, -3,   6, -6]
[ 1,  2,  3, -12,  6]
[ 1, -1,  9,   0, -9]
[ 1, -1, -3,   0,  3]</div>
  <p>with primitive multiplicities <code>1,24,15,20,60</code>. The first three sectors
    are the 40-dimensional fiber-constant Q(4,3) line-side Bose–Mesner module: dividing the lifted
    adjacency eigenvalues <code>36,6,−12</code> by the fiber size three recovers
    <code>12,2,−4</code>. The transverse 80-space splits as <code>20+60</code> and is
    annihilated by the Q(4,3) adjacency lift. The forty maximal K₄ pencils recover
    the dual W33 point carrier; rank<sub>F3</sub>(A+I) is 15 on the line side and 11 on the point side.</p>

  <h2>Reproduce</h2>
  <pre class="eq"><code>python3 analysis/w33_pass4873_two_order1440_extensions.py
python3 analysis/w33_pass4874_steiner_w33_association_scheme.py</code></pre>
  <ul>
    <li><a href="https://github.com/wilcompute/W33-Theory/blob/master/data/PART_W33_PASS4873_TWO_ORDER1440_EXTENSIONS.json">Pass 4873 certificate</a></li>
    <li><a href="https://github.com/wilcompute/W33-Theory/blob/master/data/PART_W33_PASS4874_STEINER_W33_ASSOCIATION_SCHEME.json">Pass 4874 certificate</a></li>
    <li><a href="pass4870-steiner-w33-quadratic.html">Preceding Pass 4870 three-cover and quadratic bridge</a></li>
  </ul>

  <div class="boundary"><strong>Evidence boundary.</strong> The association relations,
    including the 3+6 refinement over every Q(4,3) nonedge (a pair of disjoint W33 lines), are canonical. Naming the three
    elements inside each fiber is additional gauge or coordinate data. The group theorem
    distinguishes two finite actions but selects neither as a physical symmetry. The complete
    <code>[360,36,20]₂</code> weight enumerator is closed, but its covering radius remains open
    at the certified bound <code>124 ≤ ρ(K) ≤ 179</code>.</div>

  <footer>Finite group and association-scheme certificates only; no selected physical
    coupling, continuum normalization, hardware implementation, or preferred fiber gauge is inferred.</footer>
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