Spin(2,3) gravity-route research checkpoint

Date: 2026-10-22

This checkpoint promotes the most informative results from the independent Spin(2,3) toy-model search. It is a research-direction gate, not a claim that the programme has derived gravity or replaced string theory.

Corrected representation inventory

The active ten-dimensional carrier is two copies of the vector 5, not one adjoint 10. With the completed scalar sectors, the finite module is

94 * 1 + 10 * 5 + 10 + 14 = 168 dimensions.

Under the Lorentz subgroup this contains 105 scalar dimensions, twelve vector 4 copies, one bivector 6 and one symmetric traceless rank-two 9. Earlier scratch calculations that labelled the active ten-dimensional carrier as an adjoint, or reported the resulting Lorentz multiplicities differently, are superseded by 0.

The original active-active scalar seeds occupy only the Lorentz-vector pieces of the 10 and 14 sectors. AdS coset translations do generate the missing finite descendants:

• vector 4 to bivector 6 in the adjoint 10;
• vector 4 to scalar 1 plus symmetric traceless tensor 9 in the 14.

This is an exact representation-theoretic completion, but not yet a dynamics. Finite covariantly constant zero-forms contain no generic local propagating degrees of freedom.

What the toy search rules out

The promoted calculations give five useful negative results.

1. The finite adjoint 10 is not a spin-2 Weyl module. It branches as 6 + 4, whereas the four-dimensional Weyl tensor is 5 + 5, and a scalar-valued adjoint also has the wrong form degree for a gravitational connection.
2. A connection made only as g^-1 d g contains Lorentz-connection and vierbein components, but its curvature and torsion vanish by the Maurer-Cartan identity. It is locally pure gauge.
3. Two generic vector-5 expectation values reduce the stabilizer to dimension three. Retaining SO(1,3) requires the two copies to be parallel in multiplicity space.
4. The simplest SO(2,3)-invariant two-vector Gram quartic cannot both be bounded and select the required aligned timelike vacuum. One sign is unstable at the aligned point; the other has a null runaway.
5. A fixed positive Cartan metric can stabilize the compact problem, but its common stabilizer with a timelike compensator is only SO(3). Making that metric auxiliary does not remove the issue: finite extrema select a frame, while null directions can run to the noncompact boundary.

Candidate 48 is therefore not currently a gravity phase. Its vector-seed Hessian is only residual-U(1) covariant; its five-dimensional moduli tangent does not coincide with the SO(4) orbit tangent; and the omitted isotropic response repair moves the vacuum. The frozen evidence records these numerical obstructions and their tolerances, without promoting the large dependency chain of exploratory scratch scripts as stable API.

Surviving constructive route

The narrow viable route is a standard gauge-gravity extension with programme data feeding it:

1. introduce an independent adjoint-valued one-form A;
2. retain one normalized vector combination as a constrained compensator;
3. use its four broken SO(2,3)/SO(1,3) directions as pure gauge;
4. impose or derive a multiplicity-space constraint that removes or aligns the orthogonal vector copy;
5. test a MacDowell-Mansouri/Stelle-West action and its linearized Einstein equations;
6. add the infinite Weyl zero-form tower only when local spin-2 propagation is required.

For one constrained vector, the constraint tangent has dimension four, the gauge orbit has rank four, and their principal cosines are all one. Thus the single compensator contributes no local physical scalar modes. The unresolved input is the selection of one combination from the corrected two-vector multiplicity.

This changes the research priority. Further scans of unconstrained scalar potentials have low value until the independent connection, compensator constraint algebra, and multiplicity selection are written explicitly.

Published evidence

The 19 structured records in evidence/data/ fall into four groups:

• representation correction and descendant maps;
• Candidate-48 Hessian, moduli and compensator obstructions;
• finite unfolded-module and spin-2 Weyl-tower diagnostics;
• connection, alignment, Cartan-metric and constrained-compensator toys.

The filenames all begin with spin23- and end in -v1.json. Each record states its claim status, numerical or algebraic results, interpretation, falsifiers and next branches.

Reproduction

Nine standalone Python certificates were promoted under verification/. They require NumPy and print a single JSON result to standard output. From the repository root, for example:

& .venv-alpaca/Scripts/python.exe verification/analyze/analyze_spin23_constrained_compensator_quotient.py
& .venv-alpaca/Scripts/python.exe verification/analyze/analyze_spin23_spin2_weyl_tower_gap.py
& .venv-alpaca/Scripts/python.exe verification/verify/verify_spin23_composite_cartan_connection.py

All nine certificates passed on 2026-10-22 with the repository's .venv-alpaca interpreter. The root .scratch_spin23_* files remain exploratory working material and are intentionally not part of this published checkpoint.

Next decisive test

Build the independent SO(2,3) connection plus one constrained compensator, derive the gauge-fixed quadratic action, and verify that its physical spectrum is exactly the massless graviton with no extra ghost or propagating compensator. A failure there closes the present gravity route; a success would justify coupling the programme's relational response sectors to it.