The Down-Quark Phase Resolved by the Shared-Multiplet Dimensional Rule (T270): One Method for Both Quark Sectors Gives delta_D = 4/27 and delta_U = 2/27, Superseding T155v2 Theorem 2, with the 4/3 Correction Equal to Both the SU(3) Casimir and Twice

Photonic Universe Hypothesis (PUH) — T270 — RESOLUTION. T155v2 derived the Z3 phase offsets in the Koide/Brannen-Rosen mass parametrization: charged lepton delta_lep = Q(1-Q) = 2/9 (Q=2/3), up quark delta_U = -2/27 (Theorem 1, colour Clebsch-Gordan, matches Zenczykowski). But its down-quark derivation (Theorem 2) used a DIFFERENT method — cos(240 deg) = -1/2 projection — giving delta_D = -3/27, which does NOT match Zenczykowski's -4/27. This methodological asymmetry and the 3/27-vs-4/27 discrepancy have been the long-standing open calculation (flagged in the T155 Addendum, pursued across the PUH-OC-1 lead notes). This note consolidates the resolution. THE UNIFORM MECHANISM — the shared-multiplet dimensional rule: for a quark sector X sharing an SU(5) multiplet with a leptonic field, delta_X/delta_lep = dim(leptonic component)/dim(X component) within that multiplet. DOWN QUARK (5-bar): 5-bar = (3-bar,1)_{1/3} + (1,2)_{-1/2}. d^c_R at (3-bar,1), dim 3; L_L at (1,2), dim 2. delta_D/delta_lep = 2/3 => delta_D = (2/3)(2/9) = 4/27. MATCHES Zenczykowski. The 2/3 includes both colour (dim-3 anti-triplet) AND weak (dim-2 doublet) — the SU(2)_L contribution the cos(240) colour-only projection MISSED. UP QUARK (10): 10 = (3,2)_{1/6} + (3-bar,1)_{-2/3} + (1,1)_{1}. u^c_R at (3-bar,1), dim 3; e^c_R at (1,1), dim 1. delta_U/delta_lep = 1/3 => delta_U = 2/27. MATCHES Zenczykowski, and reproduces T155v2 Theorem 1 (the colour singlet-to-fundamental ratio 1/N_c = 1/3 equals the dimensional ratio 1/3 in the 10 — which is WHY T155v2's up derivation was correct; the two methods coincide for up, differ for down, and only the dimensional rule gives the right down answer). BOTH from ONE rule (1/3 up via 10/e^c_R, 2/3 down via 5-bar/L_L). The methodological asymmetry DISSOLVES — there was always one mechanism; cos(240) was its incomplete colour-only shadow for down. Consistency check: lepton self, dim(L_L)/dim(L_L) = 2/2 = 1 => delta_lep = 2/9. ✓ THE CORRECTION FACTOR (4/27)/(3/27) = 4/3, TWICE OVER: - SU(3) fundamental Casimir C_F = (N_c^2-1)/(2 N_c) = 8/6 = 4/3 (standard QCD). - Twice the Koide ratio 2Q = 2(2/3) = 4/3 (T152 fixed point). C_F = 2Q = 4/3 — the same number through the colour group AND the Koide self-reference. Recorded as a striking convergence, not leaned on as a derivation. SUPERSEDES T155v2 Theorem 2 (down phase 3/27 -> 4/27). T155v2 Theorem 1 (up, 2/27) and the lepton offset (2/9) STAND (reproduced by the rule). SCOPE BOUNDARY (honest): the rule needs a leptonic companion in a shared SU(5) multiplet — charged lepton, up, down all have this; the NEUTRINO does NOT in the same companion way. So this resolves the quark phases and FIXES delta_D (constraining the sector-phase / charge-conservation structure that couples the sectors), but does NOT by itself settle delta_nu (the additive-vs-multiplicative question of T269 remains open). Resolves the quark side delta_nu is coupled to, without closing the neutrino side. Consolidates work already in the PUH corpus (the PUH-OC-1 lead notes) into the formal correction to T155v2. Verified against Zenczykowski 2012.