Gauge Structure from Admissible Non-Injective Projection

Core contributions

• Gauge structure from fibre invariants: the projection-fibre equivalence classes under $\Pi_q$ carry a canonical group action. The symmetry groups preserving the fibre structure under the admissibility filter are identified as the gauge group factors.
• U(1) as admissibility fixed point: the U(1) factor is identified as the simplest admissibility fixed point, corresponding to the abelian invariant of the projection fibres. This sector is unconditional.
• SU(2) as admissibility fixed point: the SU(2) factor emerges from the non-abelian structure of the horizontal admissible sector, identified as a fixed point of the admissibility flow on the Weil representation. This sector is unconditional.
• SU(3) and [H-color]: the SU(3) factor is derived from the colour triplet co-admissibility structure. Hypothesis [H-color] is now proved analytically pointwise on the standard graph in O31 v1.5 (Proposition 4.23) via the single-frequency BI fingerprint structure, and confirmed numerically for $q \in \{61, 151, 211, 307\}$ in O32. The SU(3) identification is therefore unconditional on the standard graph.
• Composite gauge group: the three admissibility fixed points are independent — no cross-admissibility mixes them — yielding the direct product $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ as a structural consequence.

Admissibility fixed points as gauge group factors

The identification of gauge group factors with admissibility fixed points provides a novel explanation for the structure of $G_\mathrm{SM}$. Rather than being postulated as fundamental input data, the gauge group emerges as the symmetry structure that is preserved under the admissibility filter applied to the relational substrate.

The admissibility flow on the fibre structure has a finite set of fixed points corresponding to Lie groups that are compatible with the Weil representation constraint. Among these, $\mathrm{U}(1)$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$ (unconditional on the standard graph via O31 v1.5 single-frequency BI fingerprint) are identified as the three factors of the Standard Model gauge group.

The direct product structure $G_\mathrm{SM} = \mathrm{SU}(3)\times\mathrm{SU}(2)\times \mathrm{U}(1)$ is not imposed but follows from the independence of the three fixed points.

Relation to the Cosmochrony programme

Q6a is the gauge-theoretic counterpart to Q5b. Together they provide the two essential inputs for the complete effective field theory description:

• Q5b: provides the base Lorentzian manifold on which gauge fields propagate.
• Q6a: provides the gauge group $G_\Pi$ that acts on the fibre of the admissible bundle.
• Q6b: combines the geometric framework of Q5b with the gauge structure of Q6a.
• Q12: uses $G_\Pi$ from Q6a to derive Yang–Mills dynamics from the projective spectral entropy.

The O-series papers O23–O31 provide the technical scaffolding for the admissibility analysis that Q6a synthesises. O31 establishes the structural framework for [H-color]; O32 provides numerical evidence.

Open directions

• SU(3) uniqueness proof: prove that the SU(3) gauge structure emerges uniquely from the colour triplet co-admissibility structure without inputting the group as data (O31 §9.2; open).
• Hypercharge assignment: the specific U(1) hypercharge assignments of Standard Model fermions from the projection-fibre structure remain to be derived.
• Matter representations: deriving the specific matter representations (quarks, leptons) from the admissibility structure is deferred to future work.

References

Jérôme Beau. Gauge Structure from Admissible Non-Injective Projection, 2026. doi:10.5281/zenodo.19655294