"""v488 -- FLAV.NUSCALE.04: the 3/5 CLEBSCH DOOR, ENUMERATED AND CLOSED. v482 named ONE escape for the integer seesaw rung (a third-generation Majorana factor r ~ 1.6696, 0.18% from g_car/N_fam = 5/3) and declined it for lack of a mechanism. This module executes the mechanism search at the group-theory level and finds a clean NEGATIVE: no minimal Pati-Salam / SO(10) Majorana channel can produce 5/3. The coincidence is decided dead; the honest endpoint stays the v481 one-parameter window candidate. [E] 1. NO RENORMALISABLE CHANNEL: sym^2(16) = 136 = 10 + 126; the only Higgs giving a renormalisable nu^c nu^c mass is the 126bar, and the 126 is NOT in the E8 hull {1, 10, 16, 45} (v247). M_R must come from the d = 5 operator (16_F 16bar_H)(16_F 16bar_H)/Lambda with the E8-allowed 16_H. [E] 2. CHANNEL LIST: 16 x 16bar = 1 + 45 + 210 (= 256); the E8 hull admits the singlet and 45 channels (210 absent). A 45 insertion acts on nu^c through its Cartan charges T_3R, B-L, Y (the SU(4) 15-direction is B-L up to normalisation). [E] 3. CHARGE TABLE: (B-L, T_3R, Y)(nu^c) = (+1, -1/2, 0), with Y = T_3R + (B-L)/2 verified on all 16 states and Tr_16 Y^2 = 10/3 (the v485 exact trace). [E] 4. THE {2,3}-SMOOTH THEOREM (the door closes): every available nu^c channel weight -- singlet 1, T_3R^2 = 1/4, (B-L)^2 = 1, T_3R(B-L) = -1/2, SU(4)-15 weight 3/8 -- is a {2,3}-smooth rational; the multiplicative closure of {2,3}-smooth rationals can never contain 5/3 (no factor 5). The UNIQUE natural 5 of the embedding is k_Y = Tr_16 Y^2 / Tr_16 T_3L^2 = 5/3 (Ginsparg, v470) -- a full-multiplet trace, and its carrier direction has Y(nu^c) = 0 EXACTLY: the only 5-carrying group number is structurally decoupled from the Majorana operator. [E] 5. GENERATION-BLINDNESS: the Clebsch multiplies the WHOLE family matrix c_ij of the d = 5 operator (the three 16's are identical reps); a per-generation ratio is a flavor texture, never a Clebsch -- the required diag(1, 1, 3/5) shape cannot arise from group theory by construction. [O] 6. COMPILER-FLAVOR INVENTORY (typed, declined): exact 3/5's in the frozen address tables sit only in WRONG slots (K lepton row (5,3,2): gen2/gen1; R (2,5,3) and L (8,5,3) lepton rows: gen3/gen2); the v482 escape needs gen3 vs a DEGENERATE gen1 = gen2 base, which no frozen table has; and no mechanism links M_R to the address tables. Recorded, declined per v354/v355. NET EFFECT: the named escape of FLAV.NUSCALE.03 is DECIDED NEGATIVE at the group-theory level -- the rung + 5/3 rescue is dead as a mechanism candidate; the v481 window candidate (M_R free inside the PS window; the Sigma m_nu = 0.059 eV floor as kill test) stands as the honest endpoint. No gate closes; the frozen record is untouched. Contract provenance: experiments/theory-contracts/ clebsch01_majorana_35_door.py (2026-07-15). Exact (Fraction/sympy); mirrorable, deferred to the next verified Wolfram engine run. """ from fractions import Fraction as F import sympy as sp from tfpt_constants import check, summary, reset, N_fam, g_car # the 16 of SO(10): (name, multiplicity, B-L, T_3R, Y) SIXTEEN = [ ("Q", 6, F(1, 3), F(0), F(1, 6)), ("u^c", 3, F(-1, 3), F(-1, 2), F(-2, 3)), ("d^c", 3, F(-1, 3), F(1, 2), F(1, 3)), ("L", 2, F(-1), F(0), F(-1, 2)), ("e^c", 1, F(1), F(1, 2), F(1)), ("nu^c", 1, F(1), F(-1, 2), F(0)), ] HULL = {1, 10, 16, 45} # E8-allowed SO(10) content (v247) # frozen flavor address tables (v12/v18/v20/v121); rows = up/down/lepton K_TAB = [[4, 2, 0], [4, 3, 2], [5, 3, 2]] Q_TAB = [[3, 1, 0], [3, 2, 0], [3, 2, 1]] R_TAB = [[1, 3, 0], [1, 5, 2], [2, 5, 3]] L_TAB = [[7, 3, 0], [7, 5, 2], [8, 5, 3]] def smooth23(x: F) -> bool: for n in (abs(x.numerator), x.denominator): for p in (2, 3): while n % p == 0: n //= p if n != 1: return False return True def run(): reset() print("v488 FLAV.NUSCALE.04: the 3/5 Clebsch door -- enumerated, closed") # 1. no renormalisable channel check("NO RENORMALISABLE CHANNEL [E]: sym^2(16) = 136 = 10 + 126; the " "only renormalisable nu^c nu^c mass needs the 126bar, which is " "NOT in the E8 hull {1, 10, 16, 45} (v247) -- M_R only via the " "d = 5 operator (16_F 16bar_H)^2/Lambda with the E8-allowed 16_H", 16 * 17 // 2 == 136 == 10 + 126 and 126 not in HULL) # 2. channel list check("CHANNEL LIST [E]: 16 x 16bar = 1 + 45 + 210 = 256; the E8 hull " "admits the singlet and 45 insertion channels (210 absent) -- a " "45 acts on nu^c through its Cartan charges T_3R, B-L, Y", 16 * 16 == 1 + 45 + 210 and 45 in HULL and 210 not in HULL) # 3. charge table nu = next(r for r in SIXTEEN if r[0] == "nu^c") tr_y2 = sum(m * y * y for _, m, _, _, y in SIXTEEN) check("CHARGE TABLE [E]: Y = T_3R + (B-L)/2 on all 16 states; " "Tr_16 Y^2 = 10/3 (v485); (B-L, T_3R, Y)(nu^c) = (+1, -1/2, 0)", all(y == t3r + bl / 2 for _, _, bl, t3r, y in SIXTEEN) and tr_y2 == F(10, 3) and nu[2:] == (F(1), F(-1, 2), F(0))) # 4. the {2,3}-smooth theorem bl, t3r, y = nu[2], nu[3], nu[4] weights = [F(1), t3r * t3r, bl * bl, t3r * bl, F(9, 24)] tr_t3l2 = 2 * (3 * F(1, 4) + F(1, 4)) k_y = tr_y2 / tr_t3l2 check("{2,3}-SMOOTH THEOREM [E]: every nonzero nu^c channel weight " "{1, 1/4, 1, -1/2, 3/8} is {2,3}-smooth, and the multiplicative " "closure of {2,3}-smooth rationals never contains 5/3 (no factor " "5); the UNIQUE natural 5 is k_Y = Tr Y^2/Tr T_3L^2 = 5/3 = " "g_car/N_fam (v470), a full-multiplet trace whose direction has " "Y(nu^c) = 0 EXACTLY -- structurally decoupled from M_R", all(smooth23(w) for w in weights) and not smooth23(F(5, 3)) and k_y == F(5, 3) and y == 0 and F(g_car, N_fam) == F(5, 3)) # 5. generation-blindness cij = sp.Matrix(sp.MatrixSymbol('c', 3, 3)) G = sp.Symbol('G') check("GENERATION-BLINDNESS [E]: M_R = G c_ij with the Clebsch G a " "family-space SCALAR (identical 16's); every generation ratio is " "a c-texture -- diag(1, 1, 3/5) cannot arise from group theory", sp.simplify((G * cij)[2, 2] / (G * cij)[0, 0] - cij[2, 2] / cij[0, 0]) == 0) # 6. compiler-flavor inventory hits = [] for name, tab in (("K", K_TAB), ("Q", Q_TAB), ("R", R_TAB), ("L", L_TAB)): lep = tab[2] for i, j, slot in ((2, 0, "gen3/gen1"), (2, 1, "gen3/gen2"), (1, 0, "gen2/gen1")): if lep[j] and F(lep[i], lep[j]) == F(3, 5): hits.append((name, slot)) degenerate_base_hit = any( tab[2][0] == tab[2][1] and tab[2][0] and F(tab[2][2], tab[2][0]) == F(3, 5) for tab in (K_TAB, Q_TAB, R_TAB, L_TAB)) check("COMPILER-FLAVOR INVENTORY [O] (typed, declined): exact 3/5's " "sit only in WRONG slots (K lepton (5,3,2): gen2/gen1; R (2,5,3) " "+ L (8,5,3): gen3/gen2); the v482 escape needs gen3 vs a " "DEGENERATE gen1 = gen2 base -- no frozen table has that shape, " "and no mechanism links M_R to the address tables (v354/v355)", hits == [("K", "gen2/gen1"), ("R", "gen3/gen2"), ("L", "gen3/gen2")] and not degenerate_base_hit) return summary("v488 FLAV.NUSCALE.04: the 3/5 Clebsch door CLOSES -- " "no renormalisable channel (126 E8-forbidden); all nu^c " "d=5 channel weights {2,3}-smooth, 5/3 unreachable; the " "unique 5 (k_Y = 5/3) decoupled by Y(nu^c) = 0; " "Clebsches generation-blind; compiler 3/5's in wrong " "slots -- the v482 rescue is DEAD as a mechanism " "candidate; the v481 window candidate stands") if __name__ == "__main__": raise SystemExit(1 if run() else 0)