Assumptions, results and research status
Original manuscripts are supplied as unchanged source copies; website corrections and claim status are documented beside them.
Choose the source for the claim
The reference library contains twelve supplied manuscripts, totaling 466 PDF pages. They have different purposes and stages of development: focused conditional theorems, exact model calculations, umbrella programmes and early sketches. A source’s version is part of its meaning. The website therefore identifies the source next to the relevant result and keeps the original files unchanged.
For completed records, begin with the focused v21 paper. For exact singlet energies, begin with the comprehensive v20 paper. For vacuum-aware coercivity, begin with the v2 obstruction paper and then the v4 umbrella. For matter, anomalies and spectral unification, use the October 2026 conditional synthesis. The chronological development is explained on the section updates page.
Page references in the articles use the displayed PDF page or printed page as stated. Local copies have stable descriptive filenames. The source manifest records their original filenames, sizes and SHA-256 hashes, allowing an independent reader to verify that the copies have not changed.
Completed records and conditional gauge transport in SMM/DTT
The supplied v21 manuscript states predictive sufficiency relative to future tests, the comparator obstruction, selector completion, the joint-reference preservation theorem, and conditional determinant reduction. Appendix carrier reconstruction adds source and scalar-extension assumptions. It explicitly separates geometry from field dynamics.
Mohamed Haj Yousef · 30 pages · 0.45 MB. Supplied manuscript; publication status is not inferred from its filename.
Gauge Structure from Temporal Completion in SMM/DTT: Exact singlet-loop results and conditional operational reconstruction
Revised comprehensive manuscript, version 20, 7 October 2026. The main quantitative source for the exact four-position singlet Hamiltonian, secular energy equation, finite-model excitation gap, identity Hessian and controlled effective interactions. Its operational interpretation is a separate conditional reconstruction; physical matching remains open.
Mohamed Haj Yousef · 91 pages · 0.97 MB. Supplied manuscript; publication status is not inferred from its filename.
Gauge Structure from Temporal Completion in SMM/DTT: Conditional necessity, generated loop dynamics, dimensional reconstruction, and one-generation matter
Comprehensive reconstruction manuscript, version 18, revised 7 October 2026. Develops the marked recurrence, polar triplet, real doubled cut, scalar extension, retained product volume, loop generation and an assumed exterior matter construction. Read its earlier loop calculations alongside the exact v20 model and the more focused v21 record paper.
Mohamed Haj Yousef · 62 pages · 0.72 MB. Supplied manuscript; publication status is not inferred from its filename.
Vacuum Obstructions, Protected Band Geometry, and Quadratic-Form Feshbach Reduction in Auxiliary-Time Lattice Gauge Hamiltonians
Supplied v2 manuscript, dated 8 September 2026. A central source for exact fixed-regulator Hamiltonians, the two-sided vacuum process, twisted auxiliary spectra, the completed-vacuum obstruction and the punctured low-window criterion. Its repair prevents a vacuum-containing window from being assigned a strictly positive excitation bound.
Mohamed Haj Yousef · 27 pages · 0.56 MB. Supplied manuscript; publication status is not inferred from its filename.
An Exact-Hamiltonian Completion-Band Framework for Auxiliary-Time Lattice Gauge Theory
Fourth revised umbrella preprint. Organizes exact Hamiltonians, protected spectral geometry, phase compatibility, vacuum-aware reduction, and the conditions required for a constructive continuum programme. The website uses the untraced small-loop expansion for matrix curvature and the traced expansion for curvature-squared observables.
Mohamed Haj Yousef · 82 pages · 0.99 MB. Supplied manuscript; publication status is not inferred from its filename.
Temporal Genesis VIII. Gauge Holonomy from Inner-Time Orientation
The supplied file identifies itself as an expanded manuscript skeleton and contains theorem, figure and quantitative-table placeholders. It is included as a map of intended topics. Statements awaiting hypotheses or proofs are treated as programme targets, with completed arguments sought in the detailed companion manuscripts.
Mohamed Haj Yousef · 15 pages · 0.34 MB. Supplied manuscript; publication status is not inferred from its filename.
Complex-Time Spectral Triples and the Temporal Origin of Spatial Dimension, Stationary Action, Lorentzian Geometry, and Physical Structure
Supplied v5 formulation of the noncommutative geometry programme. Spectral triples, projectors, heat traces and physical continuation require distinct hypotheses. The website uses genuine orthogonal spectral projectors for bundle geometry and distinguishes a principal sampling zone from the full energy alias period.
Mohamed Haj Yousef · 43 pages · 0.62 MB. Supplied manuscript; publication status is not inferred from its filename.
Yang–Mills Theory in the Single Monad Model and Duality of Time Framework: Projected Lattice Gauge Theory, Dangerous Clusters, and Full Yang–Mills Identification
Supplied v20 projected-lattice synthesis. Useful for dangerous-cluster estimates, regulator and scale dependencies, and the complete list of identification targets. Coercivity statements must be read with their vacuum, projection, form-domain and uniformity conditions; later vacuum-obstruction analysis is used to explain the necessary puncture.
Mohamed Haj Yousef · 56 pages · 0.75 MB. Supplied manuscript; publication status is not inferred from its filename.
Temporal Completion, Gauge Transport, and Spectral Geometry: Toward a Conditional Unification Framework for the Duality of Time Theory
Supplied v1 manuscript, dated 3 October 2026. A dependency map joining behavioral records, real phase selection, descended transport, an assumed finite spectral geometry, anomaly arithmetic, gauge normalization and gravity. It explicitly requires a source-level derivation of the internal algebra, chiral module, dynamics and weights.
Mohamed Haj Yousef · 33 pages · 0.50 MB. Supplied manuscript; publication status is not inferred from its filename.
Duality of Time and Emergent Gauge Structure from Inner-Time Spectral Geometry
Supplied v7 manuscript, dated 4 April 2026. Protected spectral clusters and proposed coupling functionals motivate the geometric route. The website corrects the local connection formula to A = u†du and the invariant connection to ∇s = P ds, and states the cutoff assumptions needed for heat-kernel asymptotics.
Mohamed Haj Yousef · 20 pages · 0.81 MB. Supplied manuscript; publication status is not inferred from its filename.
Gauge Fields from Temporal Connections: A Non-Commutative Approach
Three-page early sketch retained for the development history. A postulated spectral triple or inner-time derivative is an input requiring construction. In its Abelian example, a connection depending only on the auxiliary coordinate has zero commutator with the outer-time derivative; the website does not use that example as a nonzero field-strength calculation.
Mohamed Haj Yousef · 3 pages · 0.16 MB. Supplied manuscript; publication status is not inferred from its filename.
The Fundamental Temporal Origin of Gauge Symmetries: A Duality of Time Theory Approach
Four-page early conceptual sketch. Its strong origin language is qualified by the later manuscripts, which expose carrier, comparator, determinant, dynamics and matching assumptions. Historical inclusion does not promote the sketch into a completed Standard Model derivation.
Mohamed Haj Yousef · 4 pages · 0.16 MB. Supplied manuscript; publication status is not inferred from its filename.
Formula and hypothesis notes for readers
The website develops the supplied work with a consistent active-frame convention: Uyx maps Vx to Vy, products act from right to left, and D = d + A uses an anti-Hermitian connection. A source using inverse frames or a Hermitian gauge potential needs the corresponding explicit conversion.
| Issue | Treatment in the website |
|---|---|
| Projection and connection | P(dP)P = 0. For sections of a moving eigenspace, use ∇s = P ds and A = u†du in a local orthonormal frame. The curvature is P(dP ∧ dP)P. |
| Exact singlet Hessian | E(exp(iX)) − E(I) = ½κ̂ Tr X² + O(X⁴). This factor is kept when comparing the exact response with the leading loop coefficient. |
| Traced small loops | The untraced matrix holonomy begins with I − F × area. For traceless anti-Hermitian curvature, the real trace first detects curvature squared at order area²; a scalar trace cannot recover the full matrix F. |
| Smooth windows | A smoothed spectral window is generally a positive weight rather than an idempotent projector. Bundle and Kato-transport statements use genuine orthogonal spectral projections with the required gap. |
| Fixed finite-rank heat traces | With a fixed finite-rank cutoff, the small-s heat trace tends to that rank. A divergent Weyl law needs the stated infinite-dimensional, growing-cutoff or per-volume regime. |
| Sampling period | πℏ/ε is a principal-zone edge. Energies differing by 2πℏ/ε have the same sampled phase; the full alias period is twice the edge. |
| Vacuum and low windows | A low window containing the projected vacuum cannot support a strictly positive excitation estimate on that vector. The vacuum correction belongs in every block, and the punctured window requires its own coercivity hypotheses. |
These notes do not silently alter the PDFs. They let readers see which formula, convention or assumption governs a website calculation. The research-status ledger gives the larger claim boundaries, and worked exercises provide short independent checks.
Yang–Mills and the Mass Gap
The official overview lists the Millennium problem as unsolved when checked on 18 October 2026. This section distinguishes that four-dimensional quantum-field-theory target from an exact finite model or a positive gap at one fixed regulator.
Quantum Yang–Mills Theory
Arthur Jaffe and Edward Witten formulate the existence and mass-gap target for a nontrivial quantum Yang–Mills theory on four-dimensional space. Use the full statement when assessing continuum existence, quantum-field axioms and physical spectrum claims.
Lectures on Gauge Theory
An independent pedagogical reference for the conventional gauge-theory framework. The supplied SMONAD manuscripts address a proposed origin and conditional reconstruction; standard gauge geometry and quantum-field benchmarks have an established mathematical literature.
Gauge Theory: Anomalies
Background for perturbative and global consistency conditions. The local explorer states its all-left-handed convention, integer charge normalization and fixed matter representations explicitly before doing the arithmetic.
Gauge Theory on a Lattice
Background for link variables, plaquettes, Wilson observables and lattice regularization. The auxiliary-cycle examples here add specified degrees of freedom and do not establish the required continuum identification by their finite spectra alone.
Books, papers and the wider website
The gauge section joins the research programme, geometry, symmetry, quantum foundations and matter. The hydrogen section makes the interaction-to-observable problem concrete through bound states and spectra.
Browse the paper catalogue for publication records and the books for the broader physical and philosophical background. The website’s author page and contact page provide current contact and account links. Specific questions for experts are listed on the gauge collaboration page.