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, identifies the internal phases, records the status of every result as proved, structural, numerical, or open, and states the remaining open deliverables.

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 the admissibility structure, not a postulate. This note concerns the identification of $G_\Pi$ only; the derivation of Yang–Mills dynamics belongs to the gauge–gravity spectral stratification sub-programme (Presentation Note 8).

Logical 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:

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.
2. $\mathrm{SU}(2)$ from quaternionic admissibility — the admissible sector is $V_\rho \cong \mathbb{C}^2$; the unique associative non-commutative minimal algebra compatible with admissibility is $\mathbb{H}$ (O23), and the quaternionic rigidity theorem (O27) forces every admissible morphism to factor through $\mathfrak{su}(2)$.
3. $\mathrm{SU}(3)$ from colour-triplet co-admissibility — metaplectic intertwining of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ forces unconditional triplet co-admissibility on the colour-adapted Cayley graph (O31); the standard graph is established at the rank, averaged, effective-exponent, and numerical levels (O31–O32).

Position in the Programme

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 $V_\rho \cong \mathbb{C}^2$, the threshold $\Sigma_c(n_3) = 3$, and $\operatorname{Im}\mathbb{H} \cong \mathfrak{su}(2)$. It takes from Presentation Note 2 (emergent geometry) the effective projection space $H_{\mathrm{eff}} = \mathrm{Sym}^2(V_\rho)$, the effective base manifold $\mathbb{R}\tau \times \mathrm{Heis}3(\mathbb{R})$, and the admissible principal bundle $P{G\Pi}(M, G_\Pi)$. It produces the gauge group $G_\Pi$ and the admissible principal bundle used downstream by Q12, Q13, Q14, and Note 8.

Constituent Papers

The note organises the constituent papers by sector:

Sector	Papers	Central output	Status
$\mathrm{U}(1)$ fibre	Foundation, TopInv, SMchargemass	Charge $= w \cdot e$; no monopoles	P (canonical)
$\mathrm{U}(1)$ bundle	Q6a, Q12	Admissible $\mathrm{U}(1)$ connection	S
$\mathrm{SU}(2)$ sector	O23, O27–O30, Q6a, Q7	$d_\rho = 2$; $\mathfrak{su}(2)$ closed	P
Charges	Q6a, TopInv, Q12	Bundle structure; holonomy charges	S
$\mathrm{SU}(3)$ adapted	O31	Triplet co-admissibility	P
$\mathrm{SU}(3)$ standard	O31–O32	$[\mathrm{H\text{-}color}]_{\mathrm{rank/avg/eff}}$	P/S
$[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$	O32	Finite-$q$ profile equality	O

Status codes: P = proved, S = structural, N = numerical, O = open.

Open Deliverables

1. $[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$. The $\mathrm{SU}(3)$ identification on the standard graph is established at the rank, averaged, effective-exponent, and numerical levels. Exact finite-$q$ profile equality remains analytically open; the spectral route is closed (O31), so the correct mechanism must act at the level of the BFS rank structure.
2. $\pi_2(\mathcal{C}_{\mathrm{eff}}) = 0$ 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.
3. $V - A$ 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.

Build

bash compile.sh

This runs pdflatex → bibtex → pdflatex → pdflatex on tex/GaugeStructureNote.tex and produces out/GaugeStructureNote.pdf.