The Gauge Structure Sub-Programme

The gauge structure sub-programme identifies the internal symmetry groups that survive as invariants of admissible non-injective projection. Its central question is: which internal symmetry groups are forced by the admissibility structure of $\Pi$, rather than postulated? The sub-programme establishes the following chain: \[ \Pi^{-1}(O_n) \;\Longrightarrow\; \text{phase fibre} \;\Longrightarrow\; \mathrm{U}(1) \;\Longrightarrow\; \operatorname{Im}\mathbb{H} \simeq \mathfrak{su}(2) \;\Longrightarrow\; \mathrm{SU}(2) \;\Longrightarrow\; \text{colour triplet} \;\Longrightarrow\; \mathrm{SU}(3). \] The status of each factor is distinct: $\mathrm{U}(1)$ is structural (Hopf fibre of $\Pi \simeq S^3$); $\mathrm{SU}(2)$ is closed analytically (O27–O30, Q6a); $\mathrm{SU}(3)$ is established unconditionally on the colour-adapted graph and supported on the standard graph at the rank, averaged, effective-exponent, and numerical levels. The remaining open component is the pointwise finite-$q$ profile equality on the standard graph. The sub-programme closes the gauge group identification. It does not derive Yang–Mills dynamics: that belongs to the gauge–gravity spectral stratification sub-programme.