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Spike #51 R3-δ — Spin(8) triality on round-S⁷ (substrate-identity constructive test)

Date: 2026-05-17. Status: COMPLETED — verdict B partition-coexistence ~80% (hardened in Spike #84 R4 to ~90% canonical).

Mission

Per Spike #51 R2-α §7 housekeeping note: investigate Spin(8) triality structure on round-S⁷ as constructive next-move after R2-α verdict ROUND ruled out substrate-identity-via-squashed-S⁷. R3-δ tests whether triality + G₂-holonomy mechanism produces 3-generation SM-shaped chirality on framework's round-S⁷ substrate.

Findings

Spin(8) triality structure: - Out(Spin(8)) = S₃ — three outer automorphisms cycle the three 8-dim irreps (8_v, 8_s, 8_c) - G₂ = triality-invariant subalgebra of 𝔰𝔬(8); dim 14 - Three Spin(7) embeddings (Spin(7)_v, Spin(7)_s, Spin(7)_c) cycled by triality S₃ - Quotient Spin(7)/G₂ ≅ ℝ⁷ supplies 7D_g per [[project_space_gauge_time_framework]]

Verdict B partition-coexistence: - Framework's round-S⁷ (Spin(8) isometry; λ²=1) vs M-theory's squashed-S⁷ (Sp(2)×Sp(1) isometry; λ²=⅕ Awada-Duff-Pope 1983) coexist as different geometric instantiations of same Class I cyclic-cascade primitive - G₂ orientation-symmetric core invariant; three Spin(7) embeddings simultaneously present at substrate level - Spike #51 R3-δ probability mass: A unique-identity ~10% / B partition-coexistence ~80% / C different-substrates ~10%

Pillar F closure (per [[user_stance_dimensional_mode_conversion_at_2d_boundary]]): - 7D_g cascade extension UNGATED from substrate-identity verdict - Spike #58.F write-up records the 7D_g cascade composition as Class I ∘ Class C on Spin(7)/G₂ fiber

Composes with

  • [[user_stance_g2_triality_invariant_gauge_structure]] — committed 2026-05-17; canonical landing for R3-δ result
  • [[user_stance_substrate_identity_partition_coexistence_canonical]] — Spike #84 R4 closes Spike #51 at canonical ~90%
  • [[user_stance_dark_sector_in_7d_g_gauge_space]] — dark sector in non-selected Spin(7)/G₂ orientations per triality cycling
  • [[user_stance_chirality_is_local_sign_flip_through_metric_fiber]] — Class C cascade-orientation IS triality-selected Spin(7) embedding
  • [[project_space_gauge_time_framework]] — three Spin(7)/G₂ fibers supply 7D_g

Citations (open-access verified)

  • Baez 2002 arXiv:math/0105155 §3 (Octonions; Spin(8) triality)
  • Awada-Duff-Pope 1983 PRL 50:294 (cite-by-ref; squashed-S⁷)
  • Nilsson 2024 arXiv:2412.04208 (squashed-S⁷ metric; PDF-verified)
  • Acharya-Witten 2001 arXiv:hep-th/0109152 (singular-G₂ chirality)
  • Bryant-Salamon 1989 (cite-by-ref; G₂ holonomy)

Status

COMPLETED. R3-δ verdict ~80% hardened to ~90% canonical at Spike #84 R4 (PARTITION-COEXISTENCE-CANONICAL). Substrate-identity question CLOSED at R4. G₂ triality + three Spin(7)/G₂ fibers = canonical 7D_g substrate framing per project canon.

Per [[user_stance_framework_domain_algebra_not_length_or_magnitude]] (committed 2026-05-17): the triality structure operates at substrate-geometry-independent algebra level. Three substrate realizations (round-S⁷ / squashed-S⁷ / Joyce-G₂) are different geometric instantiations of same Class I cyclic-cascade primitive.