"""v82 -- two structural advances on top of the anchor-pencil double cover (v80/v81): the Koide RG attractor is FORCED by the established transfer operator, and the clean rational split is NON-GENERIC (an anchor-placement trichotomy). Builds directly on v80/v81 (det B(K+xQ)=(3x+2)(3x+5), branch points Koide -2/3 & carrier -5/3, deck degree 2=|Z2|), v54/v56 (the unique gapped boundary transfer operator, spectrum {1,(2/3)^6,(1/3)^6}, cusp weights {0,1/3,2/3}) and v32 (the exponents-at-infinity (2,1,1) = splitting O(-2)+O(-1)^2). (A) KOIDE ATTRACTOR IS FORCED, NOT POSTULATED [I] + one [P] identification. A branch-divisor-preserving Moebius map of the pencil line that fixes the two branch points q=2 (Koide) and q=5 (carrier) is, in the cross-ratio coordinate rho=(q-2)/(5-q), EXACTLY rho -> mu*rho. Hence such a map is UNIQUE once its multiplier mu is fixed. The multiplier is NOT a new input: mu=(2/3)^6 is the subleading eigenvalue lambda2 of the theory's unique gapped boundary transfer operator (v54/v56), and the Koide branch point -2/3 = -|Z2|/N_fam is exactly the cusp weight 2/3={0,1,|Z2|}/N_fam of that operator. With mu=(2/3)^6 the unique map is the integer Moebius map F(q)=2(569q-3325)/(665q-3517) with F(2)=2, F(5)=5, F'(2)=(2/3)^6 (Koide is the ATTRACTOR, |mu|<1) and F'(5)=(3/2)^6 (carrier is the REPELLER). So analyse.txt's three "structural postulates" for Koide collapse to ONE physical identification: source->pole transfer = branch-preserving relaxation of the established transfer operator. Rate, uniqueness and direction then follow. The single identification stays [P]; everything else here is [I]. (B) THE CLEAN SPLIT IS NON-GENERIC -- anchor-placement trichotomy [I]. The exponents-at-infinity (2,1,1) (the splitting type, v32) placed across the three family slots give three different covers: (1,1,2) heavy in fam3 (lepton anchor): det=(3x+2)(3x+5), disc=81=N_fam^4 -> SPLIT (2,1,1) heavy in fam1: det=(x+2)(9x+11), disc=49=scalaron^2 -> SPLIT (1,2,1) heavy in fam2: det=5x^2+10x+3, disc=40=|R(D5)|=det B(L) -> NON-SPLIT Only the canonical lepton placement and (2,1,1) have individually-rational branch labels; the middle placement (1,2,1) has Galois-conjugate irrational branch points -1 +/- sqrt(10)/5, centered EXACTLY on the sheet x=-1 (q=3) with separation set by sqrt(A_Lambda)=sqrt(10). Negative control (v81): the R-pencil disc=153 is not a square either. So the clean two-rational-channel structure (Koide & carrier as separate rational singularities) is a special feature of the anchor placement, not generic -- a hardening of "anchor-first". Physical sector identification (which placement is which fermion) is NOT claimed here and stays [P]. """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam x, q = sp.symbols('x q') # compiler labels (all derived elsewhere; used here as names) Z2 = 2 A_Lambda = 10 # = 2 * g_car scalaron = 7 # = g_car + Z2 R_D5 = 40 # = |R(D5)| = det B(L) def run(): reset() print("v82 Koide attractor (forced) + anchor-placement splitting trichotomy") K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) L = K + Q one = sp.Matrix([1, 1, 1]) def Bb(M, b): S = sp.Matrix.hstack(one, b) return S.T * M * S a = sp.Matrix([1, 1, 2]) B = lambda M: Bb(M, a) # ----- (A) the relative spectral operator (the basis-invariant core) [I] ----- T = 3 * B(Q).inv() * B(K) check("[A] T=3 B(Q)^-1 B(K)=[[5,11/3],[0,2]]; Spec={2,5}, tr=7=scalaron, det=10=A_Lambda", T == sp.Matrix([[g_car, sp.Rational(11, 3)], [0, Z2]]) and sorted(T.eigenvals().keys()) == [2, g_car] and T.trace() == scalaron and T.det() == A_Lambda) # ----- (A) Koide branch = cusp weight of the established transfer operator [I/L] ----- transfer_spec = [sp.Rational(k, 3)**6 for k in (3, 2, 1)] # v54/v56 lam2 = sp.Rational(2, 3)**6 cusp_weights = {sp.Integer(0), sp.Rational(1, 3), sp.Rational(2, 3)} check("[A] transfer spectrum {1,(2/3)^6,(1/3)^6} (v54/v56); subleading lambda2=(2/3)^6=64/729", transfer_spec == [sp.Integer(1), sp.Rational(64, 729), sp.Rational(1, 729)] and lam2 == sp.Rational(64, 729)) check("[A] cusp weights {0,1/3,2/3} = {0,1,|Z2|}/N_fam; Koide branch x=-2/3=-|Z2|/N_fam is the weight 2/3", cusp_weights == {sp.Integer(0), sp.Rational(1, N_fam), sp.Rational(Z2, N_fam)} and sp.Rational(-Z2, N_fam) == sp.Rational(-2, 3)) # both branch points share cusp class 2/3 modulo the deck period (q->q-3 sends carrier->Koide, v81) check("[A] carrier q=5 -> Koide q=2 under the deck period q->q-N_fam; shared cusp class 2/3", (g_car - N_fam) == 2 and (g_car % N_fam) == 2) # ----- (A) uniqueness of the branch-preserving map; multiplier = lam2 [I] ----- mu = sp.symbols('mu') rho = (q - 2) / (5 - q) # solve rho(F)=mu*rho => F is forced: F_mu = sp.cancel((5 * mu * rho + 2) / (mu * rho + 1)) F = sp.cancel(F_mu.subs(mu, lam2)) F_claim = 2 * (569 * q - 3325) / (665 * q - 3517) check("[A] the UNIQUE branch-preserving Moebius map fixing {2,5} with multiplier (2/3)^6 " "= 2(569q-3325)/(665q-3517) (rho->(2/3)^6 rho)", sp.simplify(F - F_claim) == 0) check("[A] F(2)=2 (Koide fixed), F(5)=5 (carrier fixed)", sp.simplify(F.subs(q, 2)) == 2 and sp.simplify(F.subs(q, 5)) == 5) Fp = sp.diff(F, q) check("[A] F'(2)=(2/3)^6=64/729<1 => Koide is the ATTRACTOR; F'(5)=(3/2)^6=729/64>1 => carrier REPELLER", sp.simplify(Fp.subs(q, 2)) == lam2 and sp.simplify(Fp.subs(q, 5)) == sp.Rational(729, 64)) check("[A] closed form q_n->2 (x->-2/3): rho_n=(64/729)^n rho_0 -> 0; rate = the SM-flavor/horizon gap", lam2 < 1 and sp.simplify(rho.subs(q, 2)) == 0) # ----- (B) anchor-placement splitting trichotomy [I] ----- placements = { "(1,1,2)": (sp.Matrix([1, 1, 2]), (3 * x + 2) * (3 * x + 5), N_fam**4, True), "(2,1,1)": (sp.Matrix([2, 1, 1]), (x + 2) * (9 * x + 11), scalaron**2, True), "(1,2,1)": (sp.Matrix([1, 2, 1]), 5 * x**2 + 10 * x + 3, R_D5, False), } for name, (b, poly, disc, splits) in placements.items(): f = sp.expand(Bb(K + x * Q, b).det()) d = sp.discriminant(f, x) is_square = sp.sqrt(d).is_rational check(f"[B] placement {name}: det={sp.factor(f)}, disc={disc}, " f"{'SPLIT (rational labels)' if splits else 'NON-SPLIT (irrational)'}", sp.expand(f - poly) == 0 and d == disc and is_square == splits) check("[B] the three discriminants are compiler numbers: N_fam^4=81, scalaron^2=49, |R(D5)|=det B(L)=40", N_fam**4 == 81 and scalaron**2 == 49 and R_D5 == 40 == B(L).det()) # the unique non-split placement: branch points centered on the sheet, separation ~ sqrt(A_Lambda) fmid = sp.expand(Bb(K + x * Q, sp.Matrix([1, 2, 1])).det()) roots = sp.solve(fmid, x) centre = sp.simplify((roots[0] + roots[1]) / 2) check("[B] non-split placement (1,2,1): branch points -1 +/- sqrt(10)/5 centered on the sheet x=-1 (q=3); " "sqrt(disc)=2 sqrt(A_Lambda)", centre == -1 and set(roots) == {-1 - sp.sqrt(10) / 5, -1 + sp.sqrt(10) / 5} and sp.sqrt(R_D5) == 2 * sp.sqrt(A_Lambda)) # negative control (anti-numerology, cf. v81): the R-pencil discriminant is not a square either R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) detR = sp.expand(B(R + x * Q).det()) check("[B] negative control: R-pencil det B(R+xQ)=9x^2+27x+16 has disc 153 (not 81, not a square) " "=> the clean split is non-generic", detR == 9 * x**2 + 27 * x + 16 and sp.discriminant(detR, x) == 153 and not sp.sqrt(153).is_rational) return summary("v82 Koide attractor + splitting trichotomy") if __name__ == "__main__": raise SystemExit(1 if run() else 0)