2026 · Vol.10 · pp.996–1012
Completion principles, real commutants, and positive-frequency geometry for orthogonal Hilbert representations
Completion, compatible complex structures, and the conditions behind positive-frequency geometry.
CONNECT
Follow publication notes and new routes through the research collection, with direct links to the underlying material.
Use these notes to find the underlying publications and the connected explanatory material. Dates identify the source’s publication year or the website edition; they do not indicate that every older draft has been superseded.
17 OCTOBER 2026 · COMPLETION EDITION
All 220 article summaries now read as finished introductions. Thirteen publication records and seven book records supply source and edition details. Eight viewing guides, nine contextual readings and the recovered animation studio extend the library. Four mathematical wording or formula corrections are recorded in the release notes, alongside shared editorial and notation standards.
16 OCTOBER 2026 · EXPANSION 06
Six new research chapters with original diagrams, two interactive workbenches, twelve Arabic lessons with full worked answers, a printable course handbook, and a searchable 78-figure catalogue. The English and Arabic Coulomb derivations also correct the factorial-integral exponent to n!/λⁿ⁺¹.
16 OCTOBER 2026 · EXPANSION 05
A 32-entry claim map, eight validation guides, eight critical discussions, a twelve-lesson course with 24 worked exercises, and eight complete Arabic derivations extend the connected collection.
16 OCTOBER 2026 · EXPANSION 04
Eight Symmetry guides, eight Time Chest guides, eight worked derivations, and six interactive demonstrations extend the reading routes. The download centre adds eight complete foundation PDFs, a collection of 72 SVG figures, and verified citations for three published papers.
16 OCTOBER 2026 · EXPANSION 03
Fifty-two new guides extend the site: eight each in relativity, gauge interactions, cosmology and open problems; six each in philosophy and history; and eight complete Arabic foundation guides.
Twelve new diagrams support the explanations. Arabic pages have right-to-left navigation, reciprocal language links and their own search-language filter. The complete collection now includes 104 individual guides in addition to the existing 220 numbered articles.
16 OCTOBER 2026 · RESEARCH EXPANSION 02
Four expanded research sections now lead to 32 individual topic pages: spatial emergence, quantum foundations, matter and atoms, and physical constants.
The papers section adds thirteen detailed guides, and the books section adds seven thematic reading guides. Fourteen diagrams illustrate the calculations and conceptual connections. The new pages are included in site search and linked from their section indexes.
Begin with spatial emergence →16 OCTOBER 2026 · FOUNDATIONS EXPANSION 01
The foundations now include detailed explanations and simplified derivations across Start Here, SMM, DTT, the postulate, research, reading paths, the glossary, and FAQ.
Worked examples cover nested readout ranks, future-test equivalence, phase selection, memory after projection, clock response, the structural count 137, and conditional atomic binding. Six new diagrams accompany the explanations; eight ordered reading paths connect them to the article collection.
The reference guides now provide 60 connected definitions and 26 answers. The explanations identify the assumptions behind each calculation and the steps that connect a mathematical construction to a physical observation.
Read the expanded introduction →WEBSITE EDITION 2026.10
The collection brings together 220 expanded articles, 40 original figures, six reading routes and a linked constants explorer. Each article retains its own address, source references and statement of scope.
Explore the collection →Completion, compatible complex structures, and the conditions behind positive-frequency geometry.
How real representations and their commutants constrain the selection of quantum phase.
A dual-time geometric approach to temporal asymmetry and reduced dynamics.
Newsletter subscription page.
The glossary explains the vocabulary; the reading paths connect the ideas; the source register points to the original arguments.