This repository contains the source of the Gauge Structure Presentation Note Cosmochrony paper The Gauge Structure Sub-Programme — Presentation Note 3.

This work is a structured entry point to the gauge structure sub-programme of the Cosmochrony corpus, not a summary of results. It maps the constituent papers, records the status of every result as proved, conditional, structural, numerical, open, or withdrawn, and states the remaining open deliverables.

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?

Define $G_\Pi$ as the group of fibre symmetries of $\Pi$ that preserve the admissibility constraints. The sub-programme does not establish $G_\Pi$. The derivation of $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ is withdrawn at its source, and no replacement is available. This note records what the question has and has not established.

O31 is a withdrawal notice. It withdraws every derivation of a gauge factor and of the Standard Model gauge group as a composite, together with the determinant-one condition, the commutation of the putative factors and the direct-product structure. It also withdraws the uniqueness of $\mathrm{SU}(3)$ among connected compact subgroups acting irreducibly on $\mathbb{C}^3$, the irreducibility of the admissible transformation group, the co-admissibility of the individual profiles, the arithmetic criterion, the metaplectic identity as used, the $\Delta(27)$ spectral proposition, and a mistyped rank test.

Three counterexamples make this unavoidable rather than cautionary:

• $\mathrm{SO}(3) \subset \mathrm{SU}(3)$ is a proper connected compact subgroup acting irreducibly on $\mathbb{C}^3$ with determinant one, so those two conditions never single out $\mathrm{SU}(3)$;
• the diagonal torus $T^2 \subset \mathrm{SU}(3)$ satisfies every constraint that was imposed while acting reducibly;
• ${1, 2, q-3}$ is an admissible colour triplet for every prime $q \geq 7$, so triplets do not require $q \equiv 1 \pmod 3$.

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

1. $\mathrm{U}(1)$ from the Hopf phase fibre — the projection fibre $\Pi \simeq S^3$ carries a canonical Hopf fibration $S^1 \hookrightarrow S^3 \to S^2$; the circle fibre supports a $\mathrm{U}(1)$ phase symmetry whose winding number $w \in \pi_1(S^1) \cong \mathbb{Z}$ is the topological origin of electric charge (TopInv, Proposition 3.1, which does not invoke the canonical admissible model). Two premises there are structural inputs rather than results: the identification of the fibre with $S^3$, imported from the white paper, and the identification of the $S^1$ phase with electric charge. The sector is argued in the topological layer rather than in O31, so the O31 withdrawal notice does not reach it — which is a statement about independence from O31, not about its premises being derived.
2. $\mathrm{SU}(2)$, conditional on a supplied carrier — the admissible sector is $V_\rho \cong \mathbb{C}^2$, a supplied input; O23 (Theorem 3.1) proves that its traceless sector is $\mathfrak{su}(2) \cong \operatorname{Im}\mathbb{H}$, of real dimension three. O27 gives unique real-linear factorisation through the admissibility quotient and a one-dimensional complex intertwiner space; identifying that quotient with $\mathfrak{su}(2)$ is supplied by O23 and rests, in O27, on its admissible-saturation hypothesis, not derived. O27 carries a second hypothesis, parity transport, for the involution's descent. The vector lift into $V_\rho$ is open.
3. $\mathrm{SU}(3)$ — withdrawn. No route is available.

There is no composite. Even granting $\mathrm{U}(1)$ and a conditional $\mathrm{SU}(2)$, nothing establishes that they are factors of a single group, that they commute, or that the structure is a direct product.

The sub-programme sits at the intersection of Branch I (axiomatic primitive and fibre structure) and Branch II (spectral admissibility). It takes from Presentation Note 1 (spectral admissibility) the spin-$\tfrac12$ admissible sector $V_\rho \cong \mathbb{C}^2$ (a supplied input), the threshold $\Sigma_c(n_3) = 3$ (a supplied selection rule, O23), and $\operatorname{Im}\mathbb{H} \cong \mathfrak{su}(2)$. It takes no $\mathrm{SU}(3)$ input. It takes from Presentation Note 2 (emergent geometry) the effective projection space $H_{\mathrm{eff}}$, identified with $\mathrm{Sym}^2(V_\rho)$ by a supplied hypothesis, and the effective base manifold $\mathbb{R}\tau \times \mathrm{Heis}3(\mathbb{R})$, with no metric coefficient: Q5b gives, under the unestablished [H-L] and the open [H-lift], a rank-three co-metric with no coefficient values, and Q11 a rank-four form only for a supplied extension with independent temporal and spatial scales. The closure $g^{\mu\nu} = 2\eta^{\mu\nu}$ is not established. The admissible principal bundle $P{G\Pi}(M, G_\Pi)$ is defined in Q12 for a supplied compact $G_\Pi$; Q6a's two continuum hypotheses do not determine a connection, so the connection is supplied and read as the generator of parallel transport on those data.

Downstream, Q12, Q13 and Note 8 carry $G_\Pi$ as a supplied compact group, and their results hold for every such group; Q14 carries its colour module as an explicit hypothesis. None of them takes $G_\Pi$ as a derived input, so their results concern a supplied group, not an identified Standard Model group.

Status codes: P = proved, S = structural, N = numerical, C = conditional, O = open, W = withdrawn at the source.

1. A route to $\mathrm{SU}(3)$. The colour-triplet route is refuted by counterexample, not merely left incomplete. A replacement must single out $\mathrm{SU}(3)$ by an argument that excludes its subgroups, which irreducibility and determinant one demonstrably do not.
2. A route to a composite gauge group. Granting $\mathrm{U}(1)$ and a conditional $\mathrm{SU}(2)$, nothing establishes a single group, commuting factors, or a direct product.
3. $\pi_2(\mathcal{C}_{\mathrm{eff}}) = 0$ in general. The topological exclusion of magnetic monopoles is proved on the canonical admissible model; extension to the full effective configuration space is a well-posed open problem in homotopy theory.
4. $V - A$ maximality from admissibility. The structural mechanism for projective chirality is identified; its derivation of the specific $V - A$ form of the weak interaction current from admissibility data is an open analytical problem.

If you reference this work, please cite the Zenodo concept DOI, which always resolves to the latest version:

J. Beau: The Gauge Structure Sub-Programme — Presentation Note 3, Zenodo, 2026, 10.5281/zenodo.20435029.

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