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Passes 1355–1359 — Selector-Matching Association Scheme

Scope

Pass 1353 identified the 120-element selector orbit as

[ {(L,M): L \text{ a totally isotropic line of } W(3,3),\ M \text{ a perfect matching of the four points of }L}. ]

There are 40 isotropic lines and three perfect matchings per line. The present packet determines the complete pair-relation algebra on this set. It closes the manuscript's previously named transport-algebra gap while preserving the existing no-go boundary: the four selector orbitals do not manufacture a 12-regular 600-cell/H4 adjacency.

Pass 1355 — Exact four-class scheme

For selectors (x=(L,M)) and (y=(L',M')), define:

  1. (R_0): (x=y).
  2. (R_1): (L=L'), but (M\ne M').
  3. (R_2): (L\ne L') and (L\cap L'\ne\varnothing).
  4. (R_3): (L,L') are disjoint and the unique generalized-quadrangle transversal bijection (L\to L') carries (M) to (M').
  5. (R_4): (L,L') are disjoint and the transported matching differs from (M').

The five relation matrices partition (X\times X), are symmetric, and close under multiplication with constant intersection numbers. Hence they form a commutative symmetric association scheme of order 120 and class 4. Its valencies are

[ \boxed{1,2,36,27,54}. ]

Pass 1356 — Bose–Mesner eigenmatrices

With relation ordering ((R_0,R_1,R_2,R_3,R_4)),

[ P=\begin{pmatrix} 1&2&36&27&54\ 1&2&-12&3&6\ 1&2&6&-3&-6\ 1&-1&0&9&-9\ 1&-1&0&-3&3 \end{pmatrix}, ]

with primitive multiplicities

[ \boxed{1,15,24,20,60}. ]

The second eigenmatrix is

[ Q=\begin{pmatrix} 1&15&24&20&60\ 1&15&24&-10&-30\ 1&-5&4&0&0\ 1&5/3&-8/3&20/3&-20/3\ 1&5/3&-8/3&-10/3&10/3 \end{pmatrix},\qquad PQ=QP=120I_5. ]

The verifier checks the algebra-character equations entrywise against the exact intersection tensor; these matrices are not numerical fits.

Pass 1357 — Imprimitivity and fusion rigidity

The relation (R_0\cup R_1) gives 40 fibers of size three. The quotient is exactly the line-intersection graph of (W(3,3)), hence (\operatorname{SRG}(40,12,2,4)). The fiber-constant primitive ranks are (1+15+24=40); the fiber-zero ranks are (20+60=80).

All 15 set partitions of the four nonidentity relations were checked. Apart from the full scheme, the only fusions are

[ {R_3\cup R_4},\qquad {R_2\cup R_3\cup R_4},\qquad {R_1\cup R_2\cup R_3\cup R_4}. ]

Thus the aligned/misaligned split on disjoint lines is rigid: it can only be forgotten wholesale. The full scheme is neither P-polynomial nor Q-polynomial in any ordering.

Pass 1358 — S3 transport holonomy and automorphism kernel

The graph on the 40 isotropic lines in which two lines are adjacent when disjoint is connected and 27-regular, with 540 edges. Each edge transports the three perfect matchings by a permutation in (S_3). Cycle transports generate the whole group:

[ \boxed{\operatorname{Hol}=S_3}. ]

The centralizer of this holonomy inside (S_3) is trivial. Therefore a scheme automorphism inducing the identity on the 40-line quotient cannot perform any hidden independent fiber permutation. Combining this kernel theorem with the verified full automorphism group of the (W(3,3)) line graph yields

[ \boxed{\operatorname{Aut}(\mathcal X)\cong \operatorname{PGSp}(4,3)\cong W(E_6),\quad |\operatorname{Aut}(\mathcal X)|=51840.} ]

This is the precise line-dependent (S_3) connection requested by the earlier selector analysis.

Pass 1359 — Manuscript and literature boundary

The theorem insert is written for both w33_paper.tex and photonic_holonet.tex, with an idempotent integrator and focused regression tests. In the Holonet it is stated only as a finite routing/transport theorem. It is not evidence for cosmology, Standard-Model parameter claims, or a laboratory implementation.

A targeted literature search found two nearby but different constructions:

  • Colangelo–Monzillo–Siciliano study the association scheme on the 160 incident point-line flags of a finite generalized quadrangle and classify its fusions (Discrete Mathematics 347 (2024), 114054, DOI 10.1016/j.disc.2024.114054).
  • Srinivasan studies the classical perfect-matching association scheme on all perfect matchings of a complete graph (Algebraic Combinatorics 3 (2020), 559–591, DOI 10.5802/alco.104).

Neither source, in the targeted search, describes this 120-object bundle of three matchings over each of the 40 isotropic lines of (W(3,3)). This packet therefore claims a repository-new exact construction, not priority or literature novelty.

Reproducibility

python analysis/w33_pass1355_1359_selector_matching_scheme.py --check
pytest -q tests/test_w33_pass1355_1359_selector_matching_scheme.py
python tools/integrate_pass1355_1359.py --check

Frozen certificate SHA-256:

4efac1631cc6991861a927e04297c4a072b9a2d4e49953642b9113c7e22f87f0