Document D57 derived sin²θW (tree) = 1/4 as an unconditional theorem of the PDL axioms C1–C4, and formulated Hypothesis HSU2 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 Gₑff = 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 PDLC1V4ₛcript1. py and PDLC1V4ₛcript2. 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 σg 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 Gₑff = 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.
Cédric Laubscher (Thu,) studied this question.