Sample Syllabus — REL-501: Foundations of Relational Ontology
Credits: 3
Prerequisites: Undergraduate philosophy or metaphysics, or instructor approval.
Course Description:
This course introduces students to the philosophical underpinnings of relational ontology — the view that relation, not substance, is the primary constituent of reality. Readings draw from Leibniz (Monadology), Mach (relational mechanics), Rovelli (relational quantum mechanics), and contemporary metaphysics, alongside the formal definitions from UCF/GUTT. Students explore how relational ontology differs from classical substance ontology and how this distinction frames modern physics and mathematics.
Learning Objectives:
- Articulate the core arguments for relation as the fundamental category of existence.
- Critique substance-based ontologies from Aristotle to Descartes.
- Connect philosophical concepts to formal relational models in mathematics and physics.
- Prepare conceptual foundations for UCF/GUTT applications.
Weekly Structure:
- Weeks 1–2: Historical foundations (Leibniz, Spinoza, Mach).
- Weeks 3–5: Modern relational thought (Rovelli, Whitehead).
- Weeks 6–8: Relational ontology and physics (GR, QFT).
- Weeks 9–10: Relational ontology and computation (category theory preview).
- Weeks 11–12: Student-led seminars on relational problems.
- Week 13: Midterm paper due — “Relation vs Substance.”
- Weeks 14–15: Integration and preparation for final paper.
Assessment:
Midterm paper (30%), final research essay (40%), seminar participation (30%).
Sample Syllabus — MAT-510: Tensor Algebra & Relational Geometry
Credits: 4
Prerequisites: Calculus III, Linear Algebra.
Course Description:
This course provides the mathematical foundations necessary to model Nested Relational Tensors (NRTs). Topics include multilinear algebra, tensor operations, metric spaces, covariant derivatives, and the relational reinterpretation of geometry. The course introduces the concept of “geometry from relations” and builds a bridge to UCF/GUTT’s formal tensor framework.
Learning Objectives:
- Define and manipulate tensors of various ranks.
- Perform contractions, outer products, and change-of-basis transformations.
- Understand the metric tensor as a relational object.
- Apply relational product–contraction operators to build nested tensors.
Weekly Structure:
- Weeks 1–2: Review of multilinear algebra and vector spaces.
- Weeks 3–4: Tensor operations and index notation.
- Weeks 5–6: Metric spaces, inner products, raising/lowering indices.
- Weeks 7–8: Covariant derivatives, Christoffel symbols, geodesics.
- Weeks 9–10: Relational geometry — deriving metric from relations.
- Weeks 11–12: Nested tensors and introduction to UCF/GUTT formalism.
- Weeks 13–14: Applied problems — stress-energy tensor, simple field examples.
- Week 15: Comprehensive problem set and oral presentation.
Assessment:
Problem sets (40%), midterm exam (20%), final oral & written project on NRT construction (40%).
Sample Syllabus — LOG-640: Formal Verification Practicum
Credits: 4
Prerequisites: LOG-520 or equivalent proof theory background.
Course Description:
This hands-on practicum trains students in the use of proof assistants (Coq, Isabelle, or Lean) to formalize and verify one of the 52 foundational propositions of UCF/GUTT. The course emphasizes precision, reproducibility, and mathematical rigor, culminating in a verified proof artifact and public presentation.
Learning Objectives:
- Translate informal mathematical statements into formal logic.
- Construct machine-verified proofs using proof assistants.
- Debug, modularize, and document proof scripts.
- Defend the correctness and significance of a verified proposition.
Weekly Structure:
- Weeks 1–3: Proof assistant setup, type theory review, small proofs.
- Weeks 4–6: Guided formalization of sample lemmas.
- Weeks 7–9: Independent work on chosen proposition.
- Weeks 10–12: Iterative refinement, peer code review.
- Weeks 13–14: Preparation of final verification report.
- Week 15: Public defense and demonstration of verified proposition.
Assessment:
Milestone submissions (30%), peer review participation (10%), final proof script and documentation (40%), oral defense (20%).
Sample Syllabus — CAT-601: Category Theory & Adjunctions
Credits: 3
Prerequisites: MAT-510 (Tensor Algebra) or instructor approval.
Course Description:
This course introduces category theory as a unifying language for mathematics, physics, and computer science, with a focus on its application to UCF/GUTT. Students learn about categories, functors, natural transformations, monads, and adjunctions, culminating in the formalization of join and projection operations used in relational modeling.
Learning Objectives:
- Define categories, morphisms, functors, and natural transformations.
- Understand limits, colimits, and adjunctions as formal relational structures.
- Express UCF/GUTT operations (join, projection, selection) in categorical language.
- Build categorical diagrams representing Nested Relational Tensors.
Weekly Structure:
- Weeks 1–2: Basic definitions — categories, functors, naturality.
- Weeks 3–4: Products, coproducts, universal properties.
- Weeks 5–6: Adjunctions, monads, and comonads.
- Weeks 7–8: Categorical logic, pullbacks and pushouts.
- Weeks 9–10: UCF/GUTT join & projection formalization.
- Weeks 11–12: Relational composition and higher-order categories.
- Weeks 13–14: Student project — formalizing a relational operator.
- Week 15: Final paper presentation.
Assessment:
Problem sets (30%), midterm exam (20%), final project and presentation (50%).
Sample Syllabus — REL-650: Emergence, Networks, and Dimensionality
Credits: 3
Prerequisites: LOG-520 (Advanced Logic) and MAT-510 (Tensor Algebra).
Course Description:
This course explores emergent behavior in complex systems through the lens of UCF/GUTT, focusing on Dimensional Sphere of Relation (DSoR), Strength of Relation (StOr), and the emergent parameter α. Students learn to model relational networks, measure connectivity, and identify transitions in system behavior.
Learning Objectives:
- Define DSoR and StOr mathematically and apply them to sample systems.
- Quantify emergence using network metrics (clustering coefficient, betweenness, modularity).
- Analyze how α evolves in dynamic relational systems.
- Apply relational modeling to turbulence, social networks, or computational graphs.
Weekly Structure:
- Weeks 1–2: Emergence theory — weak vs. strong emergence.
- Weeks 3–4: Graph theory basics and network measures.
- Weeks 5–6: DSoR and StOr formalization in UCF/GUTT.
- Weeks 7–8: Modeling α as a measure of system complexity.
- Weeks 9–10: Case study — turbulence and coherent structures.
- Weeks 11–12: Case study — multi-agent systems or neural networks.
- Weeks 13–14: Final project development.
- Week 15: Student presentations and peer feedback.
Assessment:
Weekly modeling assignments (30%), case study analysis (20%), final research project (50%).
Sample Syllabus — PHY-620: Relational Physics II
Credits: 4
Prerequisites: PHY-540 (Relational Physics I) and MAT-510 (Tensor Algebra).
Course Description:
An advanced exploration of physics in relational form. Topics include Einstein field equations, stress–energy tensors, quantum field theory, and gauge theory, with special emphasis on their reinterpretation within UCF/GUTT.
Learning Objectives:
- Derive Einstein’s equations from a relational variational principle.
- Express the stress–energy tensor as a relational tensor field.
- Connect quantum field theory operators to relational state spaces.
- Explore gauge invariance as a relational symmetry condition.
Weekly Structure:
- Weeks 1–3: Review of GR and Einstein-Hilbert action.
- Weeks 4–6: Stress–energy tensor and conservation laws.
- Weeks 7–9: Introduction to QFT from a relational viewpoint.
- Weeks 10–12: Gauge theory and symmetry breaking.
- Weeks 13–14: Student-led explorations of advanced topics.
- Week 15: Relational modeling project presentation.
Assessment:
Homework problem sets (30%), midterm exam (20%), final modeling project (50%).
Capstone & Dissertation Guidelines
Relational Systems Graduate Program
[University Name] Graduate School
Purpose
The Capstone Project (M.S.) and Dissertation (Ph.D.) are the culminating experiences of the Relational Systems Graduate Program. They require students to demonstrate mastery of the Unified Conceptual Framework / Grand Unified Tensor Theory (UCF/GUTT), contribute to the advancement of relational systems theory, and present work that meets the highest standards of rigor and reproducibility.
Capstone Project (M.S.)
Scope and Goals
The capstone project synthesizes two years of study into an original, formally rigorous work. Students are expected to:
- Formalize and verify a proposition, theorem, or model within UCF/GUTT.
- Demonstrate the ability to use proof assistants (Coq, Isabelle, or Lean).
- Produce a complete written report explaining the work’s context, methodology, and significance.
- Defend the project in a public oral presentation.
Capstones may take one of several forms:
- Formal proof of a UCF/GUTT proposition or lemma.
- Simulation of a physical, computational, or social system using NRT-based modeling.
- Theoretical extension or refinement of UCF/GUTT (e.g., new operators, metrics, or join conditions).
Milestones and Timeline
Students will follow a structured timeline during their final year:
- Proposal (Week 4): 5–7 page document outlining research question, methods, and expected outcomes. Must be approved by advisor and capstone committee.
- Midterm Check-in (Week 9): Demonstration of partial results (preliminary proof script, simulation output, or theoretical framework).
- Final Submission (Week 14): Completed proof artifact, code, or simulation plus final written report (~25–40 pages).
- Public Defense (Week 15): 30–40 minute presentation followed by Q&A from faculty and peers.
Evaluation Criteria
Capstone projects are evaluated on:
- Correctness & Rigor (40%) – Proof validity, reproducibility, and adherence to formal verification standards.
- Originality & Insight (30%) – Novel contribution, clarity of problem statement, and connection to the broader framework.
- Presentation & Communication (20%) – Quality of written report, clarity of oral defense, and ability to field questions.
- Professionalism (10%) – Timeliness, documentation quality, collaboration with advisor and peers.
Dissertation (Ph.D.)
Scope and Goals
The dissertation is an original and significant contribution to Relational Systems Theory. It must:
- Extend UCF/GUTT or apply it to solve an open problem of recognized importance (e.g., Navier–Stokes existence and smoothness, relational quantum computing, formalization of a new axiom set).
- Be accompanied by machine-verified proofs or reproducible simulations where applicable.
- Demonstrate deep engagement with existing research and show a clear path for future work.
Milestones and Timeline
Ph.D. candidates follow a multi-year structured process:
- Candidacy Exam (End of Year 2): Written and oral exams covering foundations of UCF/GUTT, category theory, and relational physics.
- Proposal Defense (Year 3): 15–20 page dissertation proposal with literature review, research plan, and preliminary results.
- Annual Progress Reviews: Submission of chapter drafts, proof artifacts, and presentations to the dissertation committee.
- Pre-Defense (Final Year): Committee-only review to ensure dissertation is ready for public defense.
- Public Defense: 60–90 minute presentation open to faculty, peers, and external examiners.
Deliverables
- Dissertation Manuscript: Typically 150–250 pages, including background, methodology, results, and formal proofs.
- Formal Proof Scripts or Code: Submitted as appendices or digital repositories with documentation for reproducibility.
- Journal Submission: Students are encouraged (and often required) to submit at least one peer-reviewed paper prior to defense.
Evaluation Criteria
Dissertations are judged on:
- Original Contribution (40%) – Novelty, depth, and impact on the field.
- Mathematical and Logical Soundness (30%) – Rigor of proofs, completeness of formal verification, reproducibility of results.
- Clarity and Scholarship (20%) – Quality of writing, literature integration, and clear articulation of findings.
- Defense Performance (10%) – Ability to present, justify, and defend work under critical questioning.
Committee Structure
Each student’s work is guided by a three-member committee (M.S.) or five-member committee (Ph.D.):
- Chair/Advisor: Primary supervisor and research mentor.
- Internal Member(s): Faculty from mathematics, physics, CS, or philosophy to provide domain expertise.
- External Examiner: For Ph.D. defenses, an external scholar evaluates objectivity and academic rigor.
Publication and Archiving
All capstones and dissertations will be:
- Deposited into the Relational Systems Digital Archive, ensuring open access for future researchers.
- Optionally published in the Journal of Relational Systems Theory, pending peer review.
Professional Conduct
Students must follow:
- [University Name] Graduate School’s policies on research integrity.
- Standards for reproducible research (open proof scripts, version-controlled code).
- Ethical guidelines for AI/automation-assisted work when using proof assistants.
These sample syllabi cover the remaining courses in the proposed Relational Systems Graduate Program: LOG-520, CS-530, PHY-540, INF-550, CS-630, SEM-590, and SEM-690. They are designed to align with the program's focus on relational ontology and UCF/GUTT, incorporating formal verification, tensor-based modeling, and cross-disciplinary integration. Each follows a 15-week structure, with prerequisites, objectives, weekly breakdowns, and assessments modeled after established graduate courses in logic, formal languages, physics, information theory, computability, and seminars.
Sample Syllabus — LOG-520: Advanced Logic & Proof Systems
Credits: 3 Prerequisites: Intro to Logic & Proof Writing or equivalent. Course Description: This course delves into constructive logic, type theory, and the Curry–Howard correspondence, with an introduction to proof assistants like Coq and Isabelle. Emphasis is placed on relational interpretations of logic, preparing students for formal verification in UCF/GUTT contexts. Learning Objectives:
- Understand constructive logic and its differences from classical logic.
- Explore type theory and the Curry–Howard isomorphism linking proofs to programs.
- Apply proof systems to relational propositions.
- Gain proficiency in Coq/Isabelle for basic theorem proving. Weekly Structure:
- Weeks 1–2: Constructive logic basics and intuitionistic systems.
- Weeks 3–4: Type theory foundations and lambda calculus.
- Weeks 5–6: Curry–Howard correspondence and proofs-as-programs.
- Weeks 7–8: Advanced proof systems and relational logic.
- Weeks 9–10: Introduction to Coq: setup, tactics, and simple proofs.
- Weeks 11–12: Isabelle basics: theories, lemmas, and automation.
- Weeks 13–14: Applications to UCF/GUTT propositions.
- Week 15: Final project presentations. Assessment: Problem sets (40%), midterm exam (20%), final project on relational proof (40%).
Sample Syllabus — CS-530: Formal Languages Beyond CFG
Credits: 3 Prerequisites: Formal Languages & Automata Theory or equivalent. Course Description: This course critiques context-free grammars (CFGs) and introduces advanced formal languages, including context-sensitive and unrestricted grammars, with a focus on Dimensional Sphere of Influence Grammar (DSOIG) as an extension under relational boundary conditions in UCF/GUTT. Learning Objectives:
- Critique limitations of CFGs in relational contexts.
- Understand automata for non-context-free languages (e.g., linear bounded automata).
- Formalize DSOIG and its relational interpretations.
- Apply advanced grammars to computational models. Weekly Structure:
- Weeks 1–2: Review of regular languages and finite automata.
- Weeks 3–4: Context-free grammars and pushdown automata; critique of CFGs.
- Weeks 5–6: Context-sensitive grammars and linear bounded automata.
- Weeks 7–8: Unrestricted grammars and Turing machines.
- Weeks 9–10: Introduction to DSOIG and relational boundary conditions.
- Weeks 11–12: Applications beyond CFGs (e.g., parsing complex structures).
- Weeks 13–14: UCF/GUTT integrations and case studies.
- Week 15: Final exam review and presentations. Assessment: Homework (35%), midterm exam (25%), final project on DSOIG model (40%).
Sample Syllabus — PHY-540: Relational Physics I
Credits: 4 Prerequisites: Classical Mechanics, Electromagnetism. Course Description: This introductory course explores physics from a relational perspective, covering relational mechanics, Maxwell’s equations, and general relativity basics, with ties to UCF/GUTT's tensor framework. Learning Objectives:
- Apply relational principles to classical mechanics (Mach's influence).
- Reformulate Maxwell’s equations in relational terms.
- Understand introductory general relativity (GR) via curvature and metrics.
- Connect to UCF/GUTT relational interpretations. Weekly Structure:
- Weeks 1–2: Relational mechanics and Mach's principle.
- Weeks 3–4: Electrostatics and magnetostatics relationally.
- Weeks 5–6: Maxwell’s equations and gauge invariance.
- Weeks 7–8: Electromagnetic waves and potentials.
- Weeks 9–10: Introduction to GR: spacetime metrics and geodesics.
- Weeks 11–12: Curvature and Einstein's equations basics.
- Weeks 13–14: Relational reinterpretations in UCF/GUTT.
- Week 15: Applied problems and oral exams. Assessment: Problem sets (30%), labs/simulations (30%), midterm (20%), final project (20%).
Sample Syllabus — INF-550: Information & Entropy in Relational Systems
Credits: 3 Prerequisites: Discrete Mathematics or equivalent. Course Description: This course covers Shannon entropy, algorithmic information theory, and information flow in networks, with a relational lens integrating UCF/GUTT models of entropy in nested systems. Learning Objectives:
- Compute Shannon entropy and mutual information in relational contexts.
- Apply algorithmic information theory (Kolmogorov complexity) to systems.
- Model information flow in networks using relational metrics.
- Link entropy to UCF/GUTT emergent properties. Weekly Structure:
- Weeks 1–2: Probability basics and Shannon entropy.
- Weeks 3–4: Mutual information and channel capacity.
- Weeks 5–6: Algorithmic information theory and Kolmogorov complexity.
- Weeks 7–8: Entropy in data compression and coding theorems.
- Weeks 9–10: Information flow in networks (e.g., graph entropy).
- Weeks 11–12: Relational entropy models in UCF/GUTT.
- Weeks 13–14: Case studies (e.g., quantum information).
- Week 15: Final presentations. Assessment: Assignments (40%), midterm exam (20%), final research paper (40%).
Sample Syllabus — CS-630: Computability & Semantics in Relational Models
Credits: 3 Prerequisites: Formal Languages & Automata Theory or equivalent. Course Description: This course examines computability, denotational semantics, and fixpoint theory, with applications to relational computation models in UCF/GUTT. Learning Objectives:
- Analyze computability via Turing machines and recursion.
- Understand denotational semantics for program meaning.
- Apply fixpoint theory to iterative computations.
- Formalize relational computation in UCF/GUTT. Weekly Structure:
- Weeks 1–2: Computability basics and Turing machines.
- Weeks 3–4: Recursive functions and undecidability.
- Weeks 5–6: Denotational semantics and domains.
- Weeks 7–8: Fixpoint theory and least fixed points.
- Weeks 9–10: Relational models of computation.
- Weeks 11–12: UCF/GUTT applications (e.g., relational fixpoints).
- Weeks 13–14: Advanced topics like lambda calculus.
- Week 15: Project defenses. Assessment: Problem sets (30%), exams (30%), final project on relational semantics (40%).
Sample Syllabus — SEM-590: UCF/GUTT Colloquium I
Credits: 1 Prerequisites: Enrollment in Relational Systems program. Course Description: This weekly seminar discusses UCF/GUTT propositions, proofs, and implications, fostering interdisciplinary dialogue among students and faculty. Learning Objectives:
- Engage with current relational theory research.
- Present and critique propositions.
- Build community in Relational Systems Theory. Weekly Structure:
- Weeks 1–15: Weekly sessions with guest lectures, student presentations, and discussions on UCF/GUTT topics (e.g., relational ontology, tensors). Assessment: Participation and short reflections (50%), one presentation (50%).
Sample Syllabus — SEM-690: UCF/GUTT Colloquium II
Credits: 1 Prerequisites: SEM-590 or equivalent. Course Description: Building on Colloquium I, this seminar features ongoing research presentations, invited lectures, and advanced discussions on UCF/GUTT applications. Learning Objectives:
- Analyze emerging relational research.
- Prepare and deliver research talks.
- Network with experts in the field. Weekly Structure:
- Weeks 1–15: Weekly meetings with invited speakers, student-led sessions, and peer feedback on capstone/dissertation ideas. Assessment: Attendance and engagement (40%), two presentations (60%).