bibkey: darpo2009vectorproduct authors: Erik Darpö year: 2009 title: Vector product algebras doi: 10.1112/blms/bdp066 url: https://arxiv.org/abs/0810.5464v1 claim: Theorem 1 classifies vector product algebras with their compatible nondegenerate symmetric bilinear form; this is distinct from equivariance under the full special orthogonal group. strata_touched: [] license: citation-only triage: anchor
Vector products and the full rotation hypothesis
Darpö, Vector product algebras, Bulletin of the London Mathematical Society 41(5), 898–902. The arXiv record identifies the journal reference and DOI; the primary preprint is https://arxiv.org/pdf/0810.5464v1.
The preprint’s introduction defines an antisymmetric product with invariant inner-product pairing and the product-norm identity. Theorem 1 allows dimensions 0, 1, 3 and 7 and classifies isomorphism by the bilinear form. It does not assert that the seven-dimensional product is equivariant under every element of SO(7).
The FIB boundary volume uses this as a classical intermediate input in §§5 and 8. Its stricter full-SO condition and its fixed-input leakage optimization are separate computations. The octonion two-generator result is already sourced by Library/VertexAlgebra/vanekeren2020atomicmonstercompletion.md and owner §32.2; Baez’s Artin locator is node2, whereas the cross-product and G2 locator is node14.
Octonion coordinates for permission-dependent responses
John C. Baez, The Octonions, Bulletin of the American Mathematical Society 39 (2002), 145–205, DOI 10.1090/S0273-0979-01-00934-X, arXiv:math/0105155. The author’s §2.2, node5 supplies the classical Cayley–Dickson construction; node2 supplies alternativity and Artin’s two-generator fact; §4.1, node14 supplies the octonion automorphism and cross-product background. The volume matches its Fano table by conjugating the second quaternion coordinate relative to Baez’s displayed pair convention; multiplication parentheses remain fixed.
The FIB boundary volume §§38–39 uses quaternion multiplication and the common unit-quaternion action as intermediate algebra, then proves its own finite tuple response classification for single-occurrence and reusable inputs and for one fixed shared exterior anchor. Neither those classical tools nor a generic Gram/orbit argument is presented as a new standalone theorem. The classification retains the actual joint source domain and the conditions for guards and metadata; it does not identify the full octonion carrier with native FIB reachability.
Causal acquisition and initial-state identification
Mihály Petreczky, Laurent Bako and Jan H. van Schuppen, Realization theory of discrete-time linear switched systems, arXiv:1103.1343v2, Definition 6 and Remark 6. Definition 6 tests equality for all input/switching words; Remark 6 distinguishes experiments with different switching sequences from collecting data along one switching sequence. These are the relevant quantifiers for the FIB boundary volume §40. The paper’s finite-mode realization and identification results do not supply the volume’s exterior controls, same-initial-state return, copies, or numerical acquisition bounds.
Petra van den Bos and Frits Vaandrager, State Identification for Labeled Transition Systems with Inputs and Outputs, arXiv:1907.11034v2, Definitions 12, 17 and 20 and Figure 3. The definitions specify adaptive tests and separation by observed traces from candidate starting states. Figure 3 gives a system with no full adaptive distinguishing graph: every first choice merges a pair that still needs distinguishing. This is initial-state identification, rather than merely learning the later current state. The volume’s octonion blind line is proved from its own chronological source/update contract; no finite automaton theorem or length bound is transferred to the continuous carrier.
Yuan Wang and Eduardo D. Sontag, Orders of Input/Output Differential Equations and State-Space Dimensions, SIAM Journal on Control and Optimization 33(4), 1102–1126 (1995), DOI 10.1137/S0363012993246828, author-hosted primary reprint, §5.1 and Theorem 5.3. The discrete-time universal-input theorem assumes analytic, reversible and observable dynamics; reversible here means each fixed-control state map is one-to-one, not an executable reset or inverse permission. On the full octonion carrier every nonzero destructive map satisfies , so it fails that injectivity hypothesis. This remains true for the separately supplied mixed map ; its two-update recovery uses retained observations, not invertibility of that map. The theorem does not authorize combining counterfactual input words into a single destructive history.