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 on the standard graph at all four analytical levels (rank, averaged, effective-exponent, and pointwise), the last via Proposition 4.23 of O31 v1.5 (single-frequency BI fingerprint structure). 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.