The Gauge Structure Sub-Programme

Overview

The gauge structure sub-programme answers a single central question. The non-injective projection $\Pi: \chi \to O$ selects admissible configurations from the relational substrate; its fibre $\Pi^{-1}(O_n)$ carries internal structure: which internal symmetry groups are preserved as invariants of admissible projection, and why are they precisely $\mathrm{U}(1)$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$?

Define $G_\Pi$ as the group of fibre symmetries of $\Pi$ that preserve the admissibility constraints. The sub-programme establishes $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$: the gauge group of the Standard Model is a forced output of admissibility, not a separate postulate. The sub-programme sits at the intersection of the foundational branch (axiomatic primitive and fibre structure) and the spectral admissibility branch (Weil representation, co-admissibility). It covers gauge group identification only; the derivation of Yang–Mills dynamics belongs to the gauge–gravity spectral stratification sub-programme.

Three sectors, three mechanisms. $\mathrm{U}(1)$ arises from the Hopf phase fibre of $\Pi \simeq S^3$; charge quantisation follows from $\pi_1(S^1) \cong \mathbb{Z}$. $\mathrm{SU}(2)$ arises from the quaternionic minimality of the admissible algebra and the rigidity theorem of O27; it is closed analytically. $\mathrm{SU}(3)$ arises from the metaplectic co-admissibility of Weil-block colour triplets; it is unconditional on the colour-adapted graph and proved at all four analytical levels ($[\mathrm{H\text{-}color}]_{\mathrm{rank}}$, $[\mathrm{H\text{-}color}]_{\mathrm{avg}}$, $[\mathrm{H\text{-}color}]_{\mathrm{eff}}$, $[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$) on the standard graph, the last via Proposition 4.23 of O31 v1.5 (single-frequency BI fingerprint structure).

Quick facts

Author

Jérôme Beau

Work type

Synthesis (presentation note, working paper)

License

CC BY 4.0

Scope

12 papers (Foundation, TopologicalInvariants, SMchargemass, O23, O27–O32, Q6a, Q7, Q12); $\mathrm{U}(1)$, $\mathrm{SU}(2)$, $\mathrm{SU}(3)$ identification

Status

Gauge group identification closed at all three levels ($\mathrm{U}(1)$, $\mathrm{SU}(2)$, $\mathrm{SU}(3)$); $\mathrm{SU}(3)$ unconditional on the adapted graph and at all four analytical levels (rank, averaged, effective, pointwise) on the standard graph via O31 v1.5 single-frequency BI fingerprint; open: $\pi_2(\mathcal{C}_{\mathrm{eff}})=0$ in general, $V-A$ from admissibility

The structural chain

$\Pi \simeq S^3 \;\Longrightarrow\; S^1 \hookrightarrow S^3 \to S^2 \;\Longrightarrow\; \mathrm{U}(1) \;\Longrightarrow\; \mathfrak{su}(2) \text{ from } \operatorname{Im}\mathbb{H} \;\Longrightarrow\; \mathrm{SU}(2) \;\Longrightarrow\; \mathrm{SU}(3).$

Three factors arise from distinct structural mechanisms: the $\mathrm{U}(1)$ phase symmetry of the Hopf fibre, whose winding number $w \in \pi_1(S^1) \cong \mathbb{Z}$ is the topological origin of electric charge ($q_{\mathrm{eff}} = w \cdot e$); the $\mathrm{SU}(2)$ rigidity forced by $\mathbb{H}$ as the minimal admissible associative non-commutative algebra, with $\dim_\mathbb{R} \operatorname{Im}\mathbb{H} = 3$ giving three internal directions and the O27 rigidity theorem forcing every admissible morphism to factor through $\mathfrak{su}(2)$; and the $\mathrm{SU}(3)$ symmetry of co-admissible colour triplets $\{V_c, V_{\omega c}, V_{\omega^2 c}\}$ for $q \equiv 1 \pmod 3$, established unconditionally on the colour-adapted Cayley graph (O31) and at the rank, averaged, and effective-exponent levels on the standard graph (O31–O32).

Papers of the sub-programme

$\mathrm{U}(1)$ phase fibre and charge quantisation.

Foundation — identifies the projection fibre $\Pi^{-1}(O_n)$ with the Weil representation $V_\rho$ of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, carrying a canonical symplectic (hence phase) structure.
TopologicalInvariants — classifies the admissible homotopy invariants of the projection fibre. Electric charge $q_{\mathrm{eff}} = w \cdot e$ from $\pi_1(S^1) \cong \mathbb{Z}$; absence of magnetic monopoles from $\pi_2(\mathcal{C}_{\mathrm{adm}}) = 0$ on the canonical model; projective chirality as orientation dependence of admissibility.
SMchargemass — interprets mass, charge, and inertia as distinct symmetry classes of a single bounded-response mechanism; local $\mathrm{U}(1)$ gauge invariance as the residual phase freedom of the projected description.

$\mathrm{SU}(2)$ quaternionic admissibility.

O23 — $\mathbb{H}$ is the minimal admissible associative non-commutative algebra; $\dim_\mathbb{R} \operatorname{Im}\mathbb{H} = 3$ derives the threshold $\Sigma_c(n_3) = 3$ rather than postulating it.
O27 — quaternionic rigidity theorem: every admissible morphism $\Phi_{q,\rho}$ factors through $\mathfrak{su}(2)$, proved unconditionally.
O28 — asymptotic calibration: covariance $\mathcal{C}_c$ has rank 3 with invariant eigenvalue structure $[1 : \tfrac12 : \tfrac12]$.
O29 — spin-$\tfrac12$ sector identified via the symmetric rank formula; unique solution $d_\rho = 2$ confirmed at $q \in \{61, 101, 151\}$.
O30 — analytical derivation of the invariant eigenvalue ratio $\lambda_1/\lambda_2 = 2$ from the equatorial fibre constraint.
Q7 — $H_{\mathrm{eff}} = \mathrm{Sym}^2(V_\rho)$ is the unique irreducible 3D $\mathfrak{su}(2)$-module; spatial and internal $\mathfrak{su}(2)$ structures representation-theoretically locked.

Gauge charges and admissible bundle structure.

Q6a — full admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$ with $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$; gauge charges as holonomy invariants of the admissible connection.
Q12 — base–fibre decomposition and horizontal–vertical decoupling lemma: gauge sector $a_4$ and gravitational sector $a_2$ are independent at leading order; gauge group taken as input from Q6a and the O-series.

$\mathrm{SU}(3)$ colour-triplet co-admissibility.

O31 — metaplectic intertwining forces triplet co-admissibility unconditionally on the colour-adapted Cayley graph; the spectral route to $[\mathrm{H\text{-}color}]$ on the standard graph is closed; $\mathrm{SU}(3)$ identified as the symmetry of co-admissible triplets $\{V_c, V_{\omega c}, V_{\omega^2 c}\}$.
O32 — $[\mathrm{H\text{-}color}]$ closed at the rank, averaged, and effective-exponent levels; numerical passage at $q \in \{61, 151, 211, 307\}$; finite-$q$ pointwise profile equality proved analytically in O31 v1.5 (single-frequency BI fingerprint), residual $R_{\mathrm{var}}$ reinterpreted as finite-sample auxiliary-draw footprint.

Inputs and outputs

Upstream inputs. The Weil representation $V_\rho \simeq L^2(\mathbb{Z}/q\mathbb{Z})$ as a theorem of the foundational branch; the Born–Infeld bounded-flux constraint; non-injectivity as a structural necessity (ENI); and, from the spectral admissibility sub-programme (Presentation Note 1), the spin-$\tfrac12$ admissible sector, the threshold $\Sigma_c(n_3) = 3$, and $\operatorname{Im}\mathbb{H} \cong \mathfrak{su}(2)$. From the emergent geometry sub-programme (Presentation Note 2), the effective base manifold $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$ on which the principal bundle is defined.

Outputs. The gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$; the admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$; charge quantisation $q_{\mathrm{eff}} = w \cdot e$ with absence of magnetic monopoles on the canonical model; $d_\rho = 2$ and $\mathfrak{su}(2)$ closure; $\mathrm{SU}(3)$ unconditional on the standard graph via $[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$ (O31 v1.5, single-frequency BI fingerprint); projective chirality — consumed by Q12, Q13, Q14 and the gauge–gravity stratification sub-programme.

Status

The gauge group identification is closed at all three levels ($\mathrm{U}(1)$, $\mathrm{SU}(2)$, $\mathrm{SU}(3)$): $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ analytically via the Hopf phase fibre and the O27 rigidity theorem; $\mathrm{SU}(3)$ unconditionally on the colour-adapted graph (O31) and at all four analytical levels (rank, averaged, effective-exponent, and pointwise) on the standard graph (O31 v1.5–O32), the last via Proposition 4.23 of O31 (single-frequency BI fingerprint structure). Two items remain open: the extension of the topological exclusion of magnetic monopoles ($\pi_2(\mathcal{C}_{\mathrm{eff}}) = 0$) beyond the canonical model; and the derivation of the specific $V - A$ structure of the weak interaction current from admissibility data. This note does not derive Yang–Mills dynamics — that belongs to the gauge–gravity spectral stratification sub-programme (Presentation Note 8).