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PASS1325_1329_TRIALITY_INTEGRAL_GAUGE_RELEASE.md

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Passes 1325–1329 Exact Release

This release executes the five frontiers opened after Passes 1320–1324:

  1. globalization across the three conjugate 432 carriers;
  2. integral/Smith reduction of the six transports and all 26 Hecke units;
  3. species-20 normalizer and gauge-invariant classification;
  4. length-7/8 primitive-cycle transport testing;
  5. an independent reconstruction and formal GAP certificate.

Headline results

  • Three-carrier commutant: H_26 ⊗ M3(C), dimension 234.
  • Triality-fixed commutant: H_26 ⊕ H_26, dimension 52.
  • Unsymmetrized common-support linking algebra: 3 M4(C) ⊕ M10(C), dimension 148.
  • Triality-equivariant common-support linking algebra: 3(M2(C)⊕C) ⊕ M4(C) ⊕ M3(C), dimension 40.
  • Six-channel Smith diagonal: 1,1,1,12,12,24.
  • Primitive 26-unit Hecke Smith diagonal: 1,1,1,1,1,2,2,2,2,2,2,2,4,12,12,12,12,24,24,24,48,144,288,864,4320,34560.
  • Exact bad primes of the Hecke unit lattice: 2,3,5.
  • Species-20 orthogonal idempotent normalizer: W(B3) of order 48; orientation-preserving subgroup order 24.
  • Three-carrier independent gauge group: S3 wr S3, order 1296.
  • Length-7/8 species-20 transport blocks: −I3 and +I3.
  • No invariant primitive-cycle operator can distinguish the three species-20 copies.

Validation

python analysis/w33_pass1325_1329_triality_integral_gauge.py
python analysis/w33_pass1329_independent_checker.py
pytest -q tests/test_w33_pass1325_1329.py
gap -q analysis/w33_pass1329_triality_integral_check.g

The local Python release completed with 8/8 focused tests passing. The GAP certificate is included and wired into GitHub Actions; GAP was not installed in the local execution container, so no local GAP result is claimed.