From the Logic of Existence to the Gauge Symmetry Group: A Proof of Hypothesis H_SU2 from Axioms C1 and C2 (Projective Dynamic Logo Framework — Document D60)

Document D57 derived sin²θ_W(tree) = 1/4 as an unconditional theorem of the PDL axioms C1–C4, and formulated Hypothesis H_SU2 as a well-motivated conjecture: that axiom C1 forces the Klein four-group V4 to act trivially on the physical observable space of K4, making the effective symmetry group G_eff = S4/V4 ≅ S3. The present document proves this conjecture. The proof rests on two lemmas, each verified by independent exhaustive computation in Google Colab (scripts PDL_C1V4_script1.py and PDL_C1V4_script2.py) before any claim was incorporated into this document. A documented negative result is an integral part of the proof: the map s → −s (global sign flip) does not preserve the coherence condition C2 on K4; pulsation pairing is therefore not a naïve sign flip but the partition structure of the dynamic orbit O4 under V4. The proof proceeds as follows. The space Coh(K4) of coherent configurations decomposes into three S4-orbits of sizes 1+3+4; the dynamic orbit O4 is isomorphic to V4 as a regular V4-set. The three pulsation pairings of O4 coincide exactly with the three coset partitions of V4 under its three order-2 subgroups H1, H2, H3. An element g in S4 preserves all three pairings if and only if the induced permutation σ_g of V4 preserves all three coset structures — which holds if and only if g is in V4 (Lemma B). Furthermore, V4 fixes every element of O1 and O3 pointwise (Lemma C), so C1-admissibility is equivalent to V4-invariance on all of Coh(K4). The effective symmetry group is therefore G_eff = S4/V4 ≅ S3, an unconditional theorem of C1 and C2. Combined with D46, D57, D58, and D59, this completes the derivation of the Standard Model gauge group SU(3) × SU(2) × U(1) and its fundamental representation as unconditional theorems of C1–C4, with no remaining conjectural step in the gauge sector.