Active Research Goal Log

Historical completed-goal log. The theorem-level goal recorded here closed on 16 October 2026. It is retained as provenance, not as the active programme controller. The current programme is summarized in The admissibility frontier.

Opened: 16 October 2026 Goal: determine precisely whether and under what hypotheses

and maintain the research record as Markdown throughout.

Determination status: completion audit passed on 16 October 2026; the theorem-level goal is complete, while the physical selector and dynamical programme remain open follow-on research.

Current determination

The unconditional implication is false. Jordan triple systems have many Tits–Kantor–Koecher Lie algebras.

A precise conditional result is true:

has Tits–Kantor–Koecher/conformal Lie algebra

At the connected group level this yields the familiar low-dimensional identification

up to the convention used for connected spin groups and covers.

The exact mechanism is the three-grading

where both \(J^\pm\) are copies of \(\operatorname{Sym}_2(\mathbb R)\). The Jordan triple product is recovered by a double Lie bracket.

Critical correction

The Jordan carrier \(J\) is three-dimensional. The defining symplectic carrier on which \(Sp(4,\mathbb R)\) acts is four-dimensional. The construction relates them; it does not identify them.

Documents being maintained

• Relational carrier and Jordan-triple composition
• Determination: Jordan triple to Sp(4,R)
• Programme integration and provenance
• Jordan selection axioms and the 0 descent
• Executable algebra certificate
• Stress–energy to the Jordan carrier: reduction gate
• Global group isomorphism 0
• Goal completion audit
• Grand synthesis

Research status

1. Exact algebraic bridge: determined.
2. Conditional uniqueness: determined; a homogeneous self-dual irreducible cone of rank/capacity two and real dimension three yields the three-dimensional spin factor \(\cong\operatorname{Sym}_2(\mathbb R)\). Physical derivation of those hypotheses remains open.
3. Derivation from stress–energy: open.
4. Physical action, positivity and observable: open.
5. Historical provenance of the original \(AXB\) intuition: unresolved.
6. Exact conformal descent: determined mathematically; \(H_2(\mathbb R)=H_2(\mathbb C)^\kappa\) induces \(\mathfrak{sp}(4,\mathbb R)\subset\mathfrak{su}(2,2)\), or \(\mathfrak{so}(2,3)\subset\mathfrak{so}(2,4)\). The physical origin of \(\kappa\) is open.
7. Executable verification: passed; exact integer checks cover the 10 symplectic generators, 27 triple products, 27 outer-symmetry cases and 243 Jordan-identity cases.
8. Minimal stress reduction: formulated; a full \(T_{\mu\nu}\) must first be contracted with a timelike flow \(u\) and projected through a codimension-one real structure \(n\) before it becomes an element of \(H_2(\mathbb R)\). Covariant selection and dynamical triple closure remain open.
9. Global group topology: determined; the primitive exterior-square representation gives \(Sp(4,\mathbb R)/\{\pm I\}\cong SO_0(2,3)\), and the connected double cover is \(Spin_0(2,3)\). A second exact-integer certificate verifies the five-dimensional metric representation and its full ten-dimensional induced Lie algebra.

Next mathematical gate

Derive—or falsify—the physical selector that reduces the natural four-dimensional carrier \(H_2(\mathbb C)\) to the real fixed carrier \(H_2(\mathbb R)\), while preserving the complex internal algebra needed for Standard-Model quantum numbers. In parallel, determine whether homogeneity, self-duality, irreducibility, capacity two and dimension three follow from the proposed relational or stress–energy dynamics rather than being assumed.

Progress record

16 October 2026 — exact bridge

• Rejected the unrestricted implication from an arbitrary Jordan triple.
• Identified the exact carrier \(J=\operatorname{Sym}_2(\mathbb R)\).
• Reconstructed its triple as a double bracket in the \(3+4+3\) grading of \(\mathfrak{sp}(4,\mathbb R)\).
• Distinguished the three-dimensional Jordan carrier from the four-dimensional defining symplectic representation.

16 October 2026 — selection and descent

• Replaced “assume a Jordan algebra” by the conditional Koecher–Vinberg route from a homogeneous self-dual cone.
• Added irreducibility, capacity two and dimension three as the visible selectors that force \(H_2(\mathbb R)\).
• Identified the exact fixed-point descent \(H_2(\mathbb C)^\kappa=H_2(\mathbb R)\), corresponding to \(Spin_0(2,3)\subset Spin_0(2,4)\).
• Isolated the physical involution as the next decisive unknown.
• Reran the executable certificate successfully with exact integer arithmetic.
• Derived the minimal tensorially valid map \(\Pi_{n,u}(T)=\operatorname{Pauli}_{\mathbb R}(h\cdot T\cdot u)\) and identified its covariance, positivity and image-closure tests.
• Closed the global-topology gap using the primitive-bivector representation, kernel \(\{\pm I\}\), connected image and maximal-compact identification.
• Completed the requirement-by-requirement audit, reran both exact-integer certificates and checked the full Markdown document set for unresolved relative links.