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Passes 1330--1334 exact release

This release turns the literal (26)-dimensional Hecke multiplication tensor into a prime-by-prime modular atlas, completes the nine-axis triality scheme, types two selected cyclic words without pretending to classify all cycles, and replaces the old species-(20) scaffold with an executed AtlasRep witness.

Pass 1330 — exact modular algebra

Let [ H_{26}=\operatorname{End}_{W(E_6)}\bigl(\mathbb Q[W(E_6)/S_5]\bigr) ] in the frozen orbital basis. Direct finite-field algebra gives:

(p) (\dim J,\dim J^2,\ldots,0) (H_{26,\mathbb F_p}/J) central-block dimensions
(2) (21,17,13,7,2,0) (M_2(\mathbb F_2)\oplus\mathbb F_2) (4,22)
(3) (22,16,10,4,0) (\mathbb F_3^4) (1,25)
(5) (6,2,0) (M_3(\mathbb F_5)\oplus M_2(\mathbb F_5)\oplus\mathbb F_5^7) (1,1,1,1,4,9,9)

The producer reconstructs the center, enumerates all central idempotents, and extracts the primitive blocks; these dimensions are not copied from a prior certificate. This is a radical/quotient/Loewy/center/block profile, not a claim to classify every finite-dimensional algebra up to isomorphism.

Passes 1331–1332 — exact scheme, selected words

The Pass-1327 (S_3^{\rm internal}\times S_3^{\rm triality}) grid has eigenmatrix [ \begin{pmatrix} 1&2&2&4\ 1&-1&2&-2\ 1&2&-1&-2\ 1&-1&-1&1 \end{pmatrix} ] and primitive-idempotent ranks (1,2,2,4). Coordinate swap fuses it to (H(2,3)), with ranks (1,4,4).

The selected cyclic words are [ (0,1,2,3,22,4,13),\qquad (0,1,2,3,22,4,7,14). ] Their ordered and dihedral stabilizer orders in (W(E_6)) are respectively (2) and (1), so their orbit sizes are (25920) and (51840). Both supports have five chords and are not induced cycles. These are two deterministic representatives, not a classification of every length-(7) or length-(8) cycle orbit. Copy nonselection remains the theorem of Pass 1328.

Pass 1333 — genuine AtlasRep execution

GAP 4.12.1 with AtlasRep, CTblLib, TomLib, and Repsn 3.1.2 constructs all three degree-(20) representations of (U_4(2).2). Each has two (20\times20) generators and image order (51840), hence is faithful. The table of marks contains:

  • an index-(432) action with degree-(20) multiplicities ([0,3,0]);
  • an index-(480) action with a single degree-(20) constituent.

This is the executed representation-theoretic result, not the earlier coordinate-only plan.

Pass 1334 — publication boundary

The theorem insert occurs exactly once in both w33_paper.tex and photonic_holonet.tex. Source integration and static TeX checks pass. A full PDF build remains an explicit CI boundary because this local environment has no TeX engine; the release does not report a local PDF compilation.

Reproduce

python3 analysis/w33_pass1330_1334_modular_triality_cycle_atlas.py
gap -q analysis/w33_pass1333_atlasrep_species20.g
python3 -m pytest -q tests/test_w33_pass1330_1334.py

Primary artifacts:

The combined status is PASS_WITH_DECLARED_TEX_BUILD_BOUNDARY.